TPTP Problem File: ITP236^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP236^2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_MinMax 00214_010433
% 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 : 0068_VEBT_MinMax_00214_010433 [Des22]
% Status : Theorem
% Rating : 0.67 v8.1.0
% Syntax : Number of formulae : 9644 (3028 unt; 639 typ; 0 def)
% Number of atoms : 28865 (9780 equ; 5 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 174781 (2571 ~; 357 |;2440 &;155889 @)
% ( 0 <=>;13524 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 8 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 4247 (4247 >; 0 *; 0 +; 0 <<)
% Number of symbols : 632 ( 628 usr; 23 con; 0-9 aty)
% Number of variables : 31643 (2756 ^;27301 !; 988 ?;31643 :)
% ( 598 !>; 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 22:12:25.402
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
thf(ty_t_VEBT__Definitions_OVEBT,type,
vEBT_VEBT: $tType ).
thf(ty_t_Code__Numeral_Ointeger,type,
code_integer: $tType ).
thf(ty_t_Code__Evaluation_Oterm,type,
code_term: $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 (621)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Oring__gcd,type,
ring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oabs__if,type,
abs_if:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_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_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_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_GCD_Osemiring__gcd__mult__normalize,type,
semiri6843258321239162965malize:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere166539214618696060dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere6658533253407199908up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__ab__group__add,type,
topolo1287966508704411220up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V7819770556892013058_space:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_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_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__Def_OGr,type,
bNF_Gr:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
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_Oand__not__num,type,
bit_and_not_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_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_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
bit_un2480387367778600638or_num: num > num > ( option @ num ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
thf(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
thf(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
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_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_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_Oadjust__div,type,
adjust_div: ( product_prod @ int @ int ) > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_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_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( filter @ B ) > ( filter @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
finite4664212375090638736ute_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
finite673082921795544331dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Finite__Set_Ofolding__idem__on,type,
finite1890593828518410140dem_on:
!>[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_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_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_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_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_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
thf(sy_c_List_Oextract,type,
extract:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
thf(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
thf(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
thf(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( B > A > B ) > B > ( list @ A ) > B ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
thf(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
sorted8670434370408473282of_set:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__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_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_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_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_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( 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_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_OBitM,type,
bitM: num > num ).
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_Ocase__num,type,
case_num:
!>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( option @ A ) > ( option @ Aa ) ) ).
thf(sy_c_Option_Ooption_Orec__option,type,
rec_option:
!>[C: $tType,A: $tType] : ( C > ( A > C ) > ( option @ A ) > C ) ).
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__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_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_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__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_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_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_OFrct,type,
frct: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_ORep__Rat,type,
rep_Rat: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
field_char_0_Rats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
thf(sy_c_Rat_Opositive,type,
positive: rat > $o ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
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_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_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OBex,type,
bex:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow2:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Ofilter,type,
filter3:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_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__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Member_Ovebt__member,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
vEBT_VEBT_add: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
vEBT_VEBT_greater: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
vEBT_VEBT_less: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
vEBT_VEBT_lesseq: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
vEBT_VEBT_max_in_set: ( set @ nat ) > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
vEBT_VEBT_min_in_set: ( set @ nat ) > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
vEBT_VEBT_mul: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift,type,
vEBT_V2048590022279873568_shift:
!>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
vEBT_VEBT_power: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
vEBT_vebt_maxt: vEBT_VEBT > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
vEBT_vebt_mint: vEBT_VEBT > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__extp,type,
max_extp:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_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 @ ( set @ A ) ) > ( set @ ( set @ ( set @ A ) ) ) ) ).
thf(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_deg,type,
deg: nat ).
thf(sy_v_m____,type,
m: nat ).
thf(sy_v_ma,type,
ma: nat ).
thf(sy_v_maxs____,type,
maxs: nat ).
thf(sy_v_mi,type,
mi: nat ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_summary,type,
summary: vEBT_VEBT ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_treeList,type,
treeList: list @ vEBT_VEBT ).
% Relevant facts (8186)
thf(fact_0__092_060open_062_092_060exists_062x_O_Aboth__member__options_A_ItreeList_A_B_Amaxs_J_Ax_092_060close_062,axiom,
? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ maxs ) @ X_1 ) ).
% \<open>\<exists>x. both_member_options (treeList ! maxs) x\<close>
thf(fact_1_not__min__Null__member,axiom,
! [T2: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ T2 )
=> ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_1 ) ) ).
% not_min_Null_member
thf(fact_2__092_060open_062Some_Amaxs_A_061_Avebt__maxt_Asummary_092_060close_062,axiom,
( ( some @ nat @ maxs )
= ( vEBT_vebt_maxt @ summary ) ) ).
% \<open>Some maxs = vebt_maxt summary\<close>
thf(fact_3_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T3: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ T3 @ X )
| ( vEBT_VEBT_membermima @ T3 @ X ) ) ) ) ).
% both_member_options_def
thf(fact_4__C5_C_I7_J,axiom,
( ( mi = ma )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ treeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ).
% "5"(7)
thf(fact_5__C5_C_I1_J,axiom,
! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ treeList ) )
=> ( vEBT_invar_vebt @ X2 @ n ) ) ).
% "5"(1)
thf(fact_6_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N: nat,TreeList: list @ vEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ N ) ) @ ( vEBT_VEBT_low @ X @ N ) ) ) ) ).
% in_children_def
thf(fact_7_both__member__options__equiv__member,axiom,
! [T2: vEBT_VEBT,N2: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X3 )
= ( vEBT_vebt_member @ T2 @ X3 ) ) ) ).
% both_member_options_equiv_member
thf(fact_8_valid__member__both__member__options,axiom,
! [T2: vEBT_VEBT,N2: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X3 )
=> ( vEBT_vebt_member @ T2 @ X3 ) ) ) ).
% valid_member_both_member_options
thf(fact_9_intind,axiom,
! [A: $tType,I: nat,N2: nat,P: A > $o,X3: A] :
( ( ord_less @ nat @ I @ N2 )
=> ( ( P @ X3 )
=> ( P @ ( nth @ A @ ( replicate @ A @ N2 @ X3 ) @ I ) ) ) ) ).
% intind
thf(fact_10_assms_I2_J,axiom,
mi != ma ).
% assms(2)
thf(fact_11_list__update__id,axiom,
! [A: $tType,Xs: list @ A,I: nat] :
( ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_12_nth__list__update__neq,axiom,
! [A: $tType,I: nat,J: nat,Xs: list @ A,X3: A] :
( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_13_min__Null__member,axiom,
! [T2: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_minNull @ T2 )
=> ~ ( vEBT_vebt_member @ T2 @ X3 ) ) ).
% min_Null_member
thf(fact_14__C5_C_I2_J,axiom,
vEBT_invar_vebt @ summary @ m ).
% "5"(2)
thf(fact_15_bit__split__inv,axiom,
! [X3: nat,D2: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X3 @ D2 ) @ ( vEBT_VEBT_low @ X3 @ D2 ) @ D2 )
= X3 ) ).
% bit_split_inv
thf(fact_16__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxs_O_ASome_Amaxs_A_061_Avebt__maxt_Asummary_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Maxs: nat] :
( ( some @ nat @ Maxs )
!= ( vEBT_vebt_maxt @ summary ) ) ).
% \<open>\<And>thesis. (\<And>maxs. Some maxs = vebt_maxt summary \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_17_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N2: nat,X3: nat] :
( ( vEBT_invar_vebt @ Tree @ N2 )
=> ( ( vEBT_vebt_member @ Tree @ X3 )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X3 )
| ( vEBT_VEBT_membermima @ Tree @ X3 ) ) ) ) ).
% member_valid_both_member_options
thf(fact_18_maxt__member,axiom,
! [T2: vEBT_VEBT,N2: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ Maxi ) )
=> ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ).
% maxt_member
thf(fact_19_list__update__overwrite,axiom,
! [A: $tType,Xs: list @ A,I: nat,X3: A,Y: A] :
( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ I @ Y )
= ( list_update @ A @ Xs @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_20_member__correct,axiom,
! [T2: vEBT_VEBT,N2: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( vEBT_vebt_member @ T2 @ X3 )
= ( member @ nat @ X3 @ ( vEBT_set_vebt @ T2 ) ) ) ) ).
% member_correct
thf(fact_21__092_060open_062both__member__options_Asummary_A_Ihigh_Ama_An_J_092_060close_062,axiom,
vEBT_V8194947554948674370ptions @ summary @ ( vEBT_VEBT_high @ ma @ n ) ).
% \<open>both_member_options summary (high ma n)\<close>
thf(fact_22_nth__replicate,axiom,
! [A: $tType,I: nat,N2: nat,X3: A] :
( ( ord_less @ nat @ I @ N2 )
=> ( ( nth @ A @ ( replicate @ A @ N2 @ X3 ) @ I )
= X3 ) ) ).
% nth_replicate
thf(fact_23__C5_C_I8_J,axiom,
ord_less_eq @ nat @ mi @ ma ).
% "5"(8)
thf(fact_24_greater__shift,axiom,
( ( ord_less @ nat )
= ( ^ [Y2: nat,X: nat] : ( vEBT_VEBT_greater @ ( some @ nat @ X ) @ ( some @ nat @ Y2 ) ) ) ) ).
% greater_shift
thf(fact_25_less__shift,axiom,
( ( ord_less @ nat )
= ( ^ [X: nat,Y2: nat] : ( vEBT_VEBT_less @ ( some @ nat @ X ) @ ( some @ nat @ Y2 ) ) ) ) ).
% less_shift
thf(fact_26__092_060open_062high_Ama_An_A_060_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062,axiom,
ord_less @ nat @ ( vEBT_VEBT_high @ ma @ n ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ summary ) ) ).
% \<open>high ma n < the (vebt_maxt summary)\<close>
thf(fact_27_mint__member,axiom,
! [T2: vEBT_VEBT,N2: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ Maxi ) )
=> ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ).
% mint_member
thf(fact_28_list__update__swap,axiom,
! [A: $tType,I: nat,I2: nat,Xs: list @ A,X3: A,X4: A] :
( ( I != I2 )
=> ( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ I2 @ X4 )
= ( list_update @ A @ ( list_update @ A @ Xs @ I2 @ X4 ) @ I @ X3 ) ) ) ).
% list_update_swap
thf(fact_29_maxt__corr__help,axiom,
! [T2: vEBT_VEBT,N2: nat,Maxi: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ Maxi ) )
=> ( ( vEBT_vebt_member @ T2 @ X3 )
=> ( ord_less_eq @ nat @ X3 @ Maxi ) ) ) ) ).
% maxt_corr_help
thf(fact_30_maxt__corr,axiom,
! [T2: vEBT_VEBT,N2: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X3 ) )
=> ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 ) ) ) ).
% maxt_corr
thf(fact_31_maxt__sound,axiom,
! [T2: vEBT_VEBT,N2: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 )
=> ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X3 ) ) ) ) ).
% maxt_sound
thf(fact_32_buildup__nothing__in__leaf,axiom,
! [N2: nat,X3: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X3 ) ).
% buildup_nothing_in_leaf
thf(fact_33_buildup__nothing__in__min__max,axiom,
! [N2: nat,X3: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X3 ) ).
% buildup_nothing_in_min_max
thf(fact_34_option_Oinject,axiom,
! [A: $tType,X22: A,Y22: A] :
( ( ( some @ A @ X22 )
= ( some @ A @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_35_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_36_valid__eq1,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_invar_vebt @ T2 @ D2 )
=> ( vEBT_VEBT_valid @ T2 @ D2 ) ) ).
% valid_eq1
thf(fact_37_valid__eq2,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_VEBT_valid @ T2 @ D2 )
=> ( vEBT_invar_vebt @ T2 @ D2 ) ) ).
% valid_eq2
thf(fact_38_inthall,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,N2: nat] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ N2 ) ) ) ) ).
% inthall
thf(fact_39_deg__not__0,axiom,
! [T2: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% deg_not_0
thf(fact_40__092_060open_062high_Ama_An_A_092_060le_062_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062,axiom,
ord_less_eq @ nat @ ( vEBT_VEBT_high @ ma @ n ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ summary ) ) ).
% \<open>high ma n \<le> the (vebt_maxt summary)\<close>
thf(fact_41_valid__tree__deg__neq__0,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_tree_deg_neq_0
thf(fact_42_valid__0__not,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_0_not
thf(fact_43_min__in__set__def,axiom,
( vEBT_VEBT_min_in_set
= ( ^ [Xs2: set @ nat,X: nat] :
( ( member @ nat @ X @ Xs2 )
& ! [Y2: nat] :
( ( member @ nat @ Y2 @ Xs2 )
=> ( ord_less_eq @ nat @ X @ Y2 ) ) ) ) ) ).
% min_in_set_def
thf(fact_44_max__in__set__def,axiom,
( vEBT_VEBT_max_in_set
= ( ^ [Xs2: set @ nat,X: nat] :
( ( member @ nat @ X @ Xs2 )
& ! [Y2: nat] :
( ( member @ nat @ Y2 @ Xs2 )
=> ( ord_less_eq @ nat @ Y2 @ X ) ) ) ) ) ).
% max_in_set_def
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X5: A] :
( ( F2 @ X5 )
= ( G @ X5 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_49_buildup__gives__valid,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N2 ) @ N2 ) ) ).
% buildup_gives_valid
thf(fact_50_set__vebt__set__vebt_H__valid,axiom,
! [T2: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( vEBT_set_vebt @ T2 )
= ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).
% set_vebt_set_vebt'_valid
thf(fact_51_mint__corr__help,axiom,
! [T2: vEBT_VEBT,N2: nat,Mini: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ Mini ) )
=> ( ( vEBT_vebt_member @ T2 @ X3 )
=> ( ord_less_eq @ nat @ Mini @ X3 ) ) ) ) ).
% mint_corr_help
thf(fact_52_replicate__eq__replicate,axiom,
! [A: $tType,M: nat,X3: A,N2: nat,Y: A] :
( ( ( replicate @ A @ M @ X3 )
= ( replicate @ A @ N2 @ Y ) )
= ( ( M = N2 )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( X3 = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_53_length__replicate,axiom,
! [A: $tType,N2: nat,X3: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N2 @ X3 ) )
= N2 ) ).
% length_replicate
thf(fact_54_length__list__update,axiom,
! [A: $tType,Xs: list @ A,I: nat,X3: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I @ X3 ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_list_update
thf(fact_55_mint__sound,axiom,
! [T2: vEBT_VEBT,N2: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 )
=> ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X3 ) ) ) ) ).
% mint_sound
thf(fact_56_mint__corr,axiom,
! [T2: vEBT_VEBT,N2: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X3 ) )
=> ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 ) ) ) ).
% mint_corr
thf(fact_57_Ball__set__replicate,axiom,
! [A: $tType,N2: nat,A2: A,P: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ( P @ A2 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_58_Bex__set__replicate,axiom,
! [A: $tType,N2: nat,A2: A,P: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ( P @ A2 )
& ( N2
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_59_in__set__replicate,axiom,
! [A: $tType,X3: A,N2: nat,Y: A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( replicate @ A @ N2 @ Y ) ) )
= ( ( X3 = Y )
& ( N2
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_60_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I: nat,X3: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( list_update @ A @ Xs @ I @ X3 )
= Xs ) ) ).
% list_update_beyond
thf(fact_61__C5_C_I4_J,axiom,
( m
= ( suc @ n ) ) ).
% "5"(4)
thf(fact_62_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs: list @ A,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ I )
= X3 ) ) ).
% nth_list_update_eq
thf(fact_63_lesseq__shift,axiom,
( ( ord_less_eq @ nat )
= ( ^ [X: nat,Y2: nat] : ( vEBT_VEBT_lesseq @ ( some @ nat @ X ) @ ( some @ nat @ Y2 ) ) ) ) ).
% lesseq_shift
thf(fact_64_set__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
= ( set2 @ A @ Xs ) ) ) ) ).
% set_swap
thf(fact_65_Ex__list__of__length,axiom,
! [A: $tType,N2: nat] :
? [Xs3: list @ A] :
( ( size_size @ ( list @ A ) @ Xs3 )
= N2 ) ).
% Ex_list_of_length
thf(fact_66_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_67_option_Osel,axiom,
! [A: $tType,X22: A] :
( ( the2 @ A @ ( some @ A @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_68_length__pos__if__in__set,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_69_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs3: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_70_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y3: list @ A,Z: list @ A] : Y3 = Z )
= ( ^ [Xs2: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I3 )
= ( nth @ A @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_71_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ? [X6: A] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ( P @ I3 @ ( nth @ A @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_72_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I4 )
= ( nth @ A @ Ys @ I4 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_73_replicate__length__same,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( X5 = X3 ) )
=> ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X3 )
= Xs ) ) ).
% replicate_length_same
thf(fact_74_replicate__eqI,axiom,
! [A: $tType,Xs: list @ A,N2: nat,X3: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N2 )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) )
=> ( Y4 = X3 ) )
=> ( Xs
= ( replicate @ A @ N2 @ X3 ) ) ) ) ).
% replicate_eqI
thf(fact_75_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) ) )
= ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_76_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,X3: A] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I4 ) ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_77_in__set__conv__nth,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_78_list__ball__nth,axiom,
! [A: $tType,N2: nat,Xs: list @ A,P: A > $o] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( P @ ( nth @ A @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_79_nth__mem,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ ( nth @ A @ Xs @ N2 ) @ ( set2 @ A @ Xs ) ) ) ).
% nth_mem
thf(fact_80_set__update__memI,axiom,
! [A: $tType,N2: nat,Xs: list @ A,X3: A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ X3 @ ( set2 @ A @ ( list_update @ A @ Xs @ N2 @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_81_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs: list @ A,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I @ X3 )
= Xs )
= ( ( nth @ A @ Xs @ I )
= X3 ) ) ) ).
% list_update_same_conv
thf(fact_82_nth__list__update,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ J )
= X3 ) )
& ( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_83_le0,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% le0
thf(fact_84_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_85_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_86_neq0__conv,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% neq0_conv
thf(fact_87_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_88_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_89_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_90__C5_C_I5_J,axiom,
( deg
= ( plus_plus @ nat @ n @ m ) ) ).
% "5"(5)
thf(fact_91_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N2 )
& ! [I5: nat] :
( ( ord_less @ nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_92_buildup__gives__empty,axiom,
! [N2: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% buildup_gives_empty
thf(fact_93_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_94_even__odd__cases,axiom,
! [X3: nat] :
( ! [N3: nat] :
( X3
!= ( plus_plus @ nat @ N3 @ N3 ) )
=> ~ ! [N3: nat] :
( X3
!= ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% even_odd_cases
thf(fact_95_add__shift,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ( plus_plus @ nat @ X3 @ Y )
= Z2 )
= ( ( vEBT_VEBT_add @ ( some @ nat @ X3 ) @ ( some @ nat @ Y ) )
= ( some @ nat @ Z2 ) ) ) ).
% add_shift
thf(fact_96_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_97_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_98_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_99_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_100_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_101_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_102_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_103_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_104_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_105_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_106_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_107_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_108_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_109_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_110_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A,Y: A] :
( ( ( plus_plus @ A @ X3 @ Y )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_111_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X3 @ Y ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_112_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add_0
thf(fact_113_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_114_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_115_lessI,axiom,
! [N2: nat] : ( ord_less @ nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_116_Suc__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_117_Suc__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_less_eq
thf(fact_118_Suc__le__mono,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( suc @ M ) )
= ( ord_less_eq @ nat @ N2 @ M ) ) ).
% Suc_le_mono
thf(fact_119_add__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( plus_plus @ nat @ M @ ( suc @ N2 ) )
= ( suc @ ( plus_plus @ nat @ M @ N2 ) ) ) ).
% add_Suc_right
thf(fact_120_add__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N2
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_121_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_122_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_123_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_124_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_125_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_126_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_127_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_128_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_129_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_130_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_131_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_132_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_133_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_134_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_135_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_136_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_137_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_138_add__gr__0,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% add_gr_0
thf(fact_139_one__is__add,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ M @ N2 ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% one_is_add
thf(fact_140_add__is__1,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus @ nat @ M @ N2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% add_is_1
thf(fact_141_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_142_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_143_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A2: nat] :
( ( A3
= ( plus_plus @ nat @ K @ A2 ) )
=> ( ( suc @ A3 )
= ( plus_plus @ nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_144_add__Suc,axiom,
! [M: nat,N2: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N2 )
= ( suc @ ( plus_plus @ nat @ M @ N2 ) ) ) ).
% add_Suc
thf(fact_145_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_146_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: A,K: A,B2: A,A2: A] :
( ( B3
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_147_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_148_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_149_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add.right_cancel
thf(fact_150_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B4: A] : ( plus_plus @ A @ B4 @ A4 ) ) ) ) ).
% add.commute
thf(fact_151_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.left_commute
thf(fact_152_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_153_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_154_Suc__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( suc @ X3 )
= ( suc @ Y ) )
=> ( X3 = Y ) ) ).
% Suc_inject
thf(fact_155_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_156_add__Suc__shift,axiom,
! [M: nat,N2: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N2 )
= ( plus_plus @ nat @ M @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_157_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_158_less__imp__Suc__add,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ? [K2: nat] :
( N2
= ( suc @ ( plus_plus @ nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_159_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N: nat] :
? [K3: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_160_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_161_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_162_less__natE,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ~ ! [Q2: nat] :
( N2
!= ( suc @ ( plus_plus @ nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_163_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_164_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_165_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_166_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_167_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_168_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_169_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_170_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_171_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_mono
thf(fact_172_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_173_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C3: A] :
( B2
!= ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% less_eqE
thf(fact_174_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_175_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
? [C4: A] :
( B4
= ( plus_plus @ A @ A4 @ C4 ) ) ) ) ) ).
% le_iff_add
thf(fact_176_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_177_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_178_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_179_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_180_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_181_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_182_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_183_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_184_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_strict_mono
thf(fact_185_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_186_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_187_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_188_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_189_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( ( zero_zero @ nat )
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_190_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_191_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_192_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_193_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat4: nat] :
( Y
!= ( suc @ Nat4 ) ) ) ).
% old.nat.exhaust
thf(fact_194_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_195_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [X5: nat] : ( P @ X5 @ ( zero_zero @ nat ) )
=> ( ! [Y4: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y4 ) )
=> ( ! [X5: nat,Y4: nat] :
( ( P @ X5 @ Y4 )
=> ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N2 ) ) ) ) ).
% diff_induct
thf(fact_196_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_197_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_198_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_199_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_200_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_201_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_202_add__eq__self__zero,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus @ nat @ M @ N2 )
= M )
=> ( N2
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_203_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_204_Suc__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N2 )
=> ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_lessD
thf(fact_205_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_206_Suc__lessI,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( ( suc @ M )
!= N2 )
=> ( ord_less @ nat @ ( suc @ M ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_207_less__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less @ nat @ M @ N2 )
=> ( M = N2 ) ) ) ).
% less_SucE
thf(fact_208_less__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ nat @ M @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_209_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ N2 )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_210_less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( suc @ N2 ) )
= ( ( ord_less @ nat @ M @ N2 )
| ( M = N2 ) ) ) ).
% less_Suc_eq
thf(fact_211_not__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less @ nat @ M @ N2 ) )
= ( ord_less @ nat @ N2 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_212_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ N2 )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_213_Suc__less__eq2,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N2 ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less @ nat @ N2 @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_214_less__antisym,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M ) )
=> ( M = N2 ) ) ) ).
% less_antisym
thf(fact_215_Suc__less__SucD,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
=> ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_less_SucD
thf(fact_216_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_217_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I4 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ( P @ I4 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I4 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_218_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_219_not__less__less__Suc__eq,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_220_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
=> ( ord_less @ nat @ I @ K ) ) ).
% add_lessD1
thf(fact_221_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_222_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_223_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_224_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_225_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_226_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_227_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K @ N2 ) )
=> ( ord_less @ nat @ M @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_228_Suc__leD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% Suc_leD
thf(fact_229_le__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq @ nat @ M @ N2 )
=> ( M
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_230_le__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_231_Suc__le__D,axiom,
! [N2: nat,M5: nat] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ M5 )
=> ? [M3: nat] :
( M5
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_232_le__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_eq @ nat @ M @ N2 )
| ( M
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_233_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_234_not__less__eq__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_eq @ nat @ M @ N2 ) )
= ( ord_less_eq @ nat @ ( suc @ N2 ) @ M ) ) ).
% not_less_eq_eq
thf(fact_235_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M6: nat] :
( ( ord_less_eq @ nat @ ( suc @ M6 ) @ N3 )
=> ( P @ M6 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_236_nat__induct__at__least,axiom,
! [M: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_237_transitive__stepwise__le,axiom,
! [M: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ! [X5: nat] : ( R @ X5 @ X5 )
=> ( ! [X5: nat,Y4: nat,Z3: nat] :
( ( R @ X5 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X5 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_238_add__leE,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N2 )
=> ~ ( ( ord_less_eq @ nat @ M @ N2 )
=> ~ ( ord_less_eq @ nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_239_le__add1,axiom,
! [N2: nat,M: nat] : ( ord_less_eq @ nat @ N2 @ ( plus_plus @ nat @ N2 @ M ) ) ).
% le_add1
thf(fact_240_le__add2,axiom,
! [N2: nat,M: nat] : ( ord_less_eq @ nat @ N2 @ ( plus_plus @ nat @ M @ N2 ) ) ).
% le_add2
thf(fact_241_add__leD1,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N2 )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% add_leD1
thf(fact_242_add__leD2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N2 )
=> ( ord_less_eq @ nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_243_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus @ nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_244_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_245_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_246_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_247_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_248_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N: nat] :
? [K3: nat] :
( N
= ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_249_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B3 )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_250_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_251_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_252_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_253_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_254_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_255_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_256_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X3 @ Y )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_257_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X3 @ Y )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_258_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_259_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_260_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_261_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_262_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_263_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ! [C3: A] :
( ( B2
= ( plus_plus @ A @ A2 @ C3 ) )
=> ( C3
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_264_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_265_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_266_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less @ nat @ N2 @ N4 )
=> ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ N4 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_267_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N2: nat,M: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ M ) )
= ( ord_less @ nat @ N2 @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_268_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ nat @ N2 @ N4 )
=> ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( F2 @ N4 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_269_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
=> ( ( ord_less_eq @ nat @ N2 @ N4 )
=> ( ord_less_eq @ A @ ( F2 @ N4 ) @ ( F2 @ N2 ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_270_less__Suc__eq__0__disj,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( suc @ N2 ) )
= ( ( M
= ( zero_zero @ nat ) )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less @ nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_271_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_272_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_273_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
= ( ? [M2: nat] :
( N2
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_274_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_275_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ( plus_plus @ nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_276_le__imp__less__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less @ nat @ M @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_277_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N: nat] : ( ord_less_eq @ nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_278_less__Suc__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( suc @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_279_le__less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% le_less_Suc_eq
thf(fact_280_Suc__le__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
=> ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_le_lessD
thf(fact_281_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I @ N3 )
=> ( ( ord_less @ nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_282_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I @ N3 )
=> ( ( ord_less @ nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_283_Suc__le__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
= ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_le_eq
thf(fact_284_Suc__leI,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_leI
thf(fact_285_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( ord_less @ nat @ ( F2 @ M3 ) @ ( F2 @ N3 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_286_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_287_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_288_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_289_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_290_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_291_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_292_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N2 )
& ! [I5: nat] :
( ( ord_less_eq @ nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_293_set__update__subsetI,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A,X3: A,I: nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X3 ) ) @ A3 ) ) ) ).
% set_update_subsetI
thf(fact_294_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_295_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_296_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
& ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_297_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_298_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_299_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X3 )
=> ( X3 = Y ) ) ) ) ).
% order_antisym
thf(fact_300_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_301_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_302_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_303_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
& ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_304_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_305_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_306_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_307_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funD
thf(fact_308_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funE
thf(fact_309_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B] :
( ! [X5: A] : ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% le_funI
thf(fact_310_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F3 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_311_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
& ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_312_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less_eq @ B @ X5 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_313_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ C @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_314_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( X3 = Y )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% order_eq_refl
thf(fact_315_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
| ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% linorder_linear
thf(fact_316_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less_eq @ B @ X5 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_317_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F2: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_318_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% linorder_le_cases
thf(fact_319_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
= ( X3 = Y ) ) ) ) ).
% order_antisym_conv
thf(fact_320_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X3: A] :
( ( ( zero_zero @ A )
= X3 )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_321_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X3: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% lt_ex
thf(fact_322_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X3: A] :
? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% gt_ex
thf(fact_323_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X3 @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_324_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( X3 != Y ) ) ) ).
% less_imp_neq
thf(fact_325_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_326_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_327_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_328_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A] :
( ! [X5: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X5 )
=> ( P @ Y5 ) )
=> ( P @ X5 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_329_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X3: A] :
( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( X3 = Y ) ) ) ) ).
% antisym_conv3
thf(fact_330_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% linorder_cases
thf(fact_331_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_332_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_333_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X7: A] : ( P2 @ X7 ) )
= ( ^ [P3: A > $o] :
? [N: A] :
( ( P3 @ N )
& ! [M2: A] :
( ( ord_less @ A @ M2 @ N )
=> ~ ( P3 @ M2 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_334_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A5: A,B5: A] :
( ( ord_less @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A] : ( P @ A5 @ A5 )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_335_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_336_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( ( ord_less @ A @ Y @ X3 )
| ( X3 = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_337_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_338_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_339_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_340_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A2: A] :
( ! [X5: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F2 @ Y5 ) @ ( F2 @ X5 ) )
=> ( P @ Y5 ) )
=> ( P @ X5 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct
thf(fact_341_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A2: A] :
( ! [X5: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F2 @ Y5 ) @ ( F2 @ X5 ) )
=> ( P @ Y5 ) )
=> ( P @ X5 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_342_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
=> ( ~ ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% linorder_neqE
thf(fact_343_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% order_less_asym
thf(fact_344_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
= ( ( ord_less @ A @ X3 @ Y )
| ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% linorder_neq_iff
thf(fact_345_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_346_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% order_less_trans
thf(fact_347_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less @ B @ X5 @ Y4 )
=> ( ord_less @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_348_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F2: A > B,C2: B] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less @ A @ X5 @ Y4 )
=> ( ord_less @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_349_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% order_less_irrefl
thf(fact_350_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less @ B @ X5 @ Y4 )
=> ( ord_less @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_351_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less @ A @ X5 @ Y4 )
=> ( ord_less @ C @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_352_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% order_less_not_sym
thf(fact_353_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,P: $o] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ X3 )
=> P ) ) ) ).
% order_less_imp_triv
thf(fact_354_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
| ( X3 = Y )
| ( ord_less @ A @ Y @ X3 ) ) ) ).
% linorder_less_linear
thf(fact_355_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( X3 != Y ) ) ) ).
% order_less_imp_not_eq
thf(fact_356_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( Y != X3 ) ) ) ).
% order_less_imp_not_eq2
thf(fact_357_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% order_less_imp_not_less
thf(fact_358_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less @ nat @ M @ N2 )
| ( ord_less @ nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_359_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_360_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_361_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_362_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_363_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M6: nat] :
( ( ord_less @ nat @ M6 @ N3 )
=> ( P @ M6 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_364_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M6: nat] :
( ( ord_less @ nat @ M6 @ N3 )
& ~ ( P @ M6 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_365_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less @ nat @ X3 @ Y )
=> ( ord_less @ nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_366_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X3: A] :
( ! [X5: A] :
( ~ ( P @ X5 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X5 ) )
& ~ ( P @ Y5 ) ) )
=> ( P @ X3 ) ) ).
% infinite_descent_measure
thf(fact_367_le__refl,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% le_refl
thf(fact_368_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_369_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_370_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_371_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
| ( ord_less_eq @ nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_372_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ Y5 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_373_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X3: A,Y: A] :
( ( ( size_size @ A @ X3 )
!= ( size_size @ A @ Y ) )
=> ( X3 != Y ) ) ) ).
% size_neq_size_imp_neq
thf(fact_374_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ).
% zero_le
thf(fact_375_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ~ ( ord_less @ A @ X3 @ Y ) ) ) ).
% leD
thf(fact_376_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% leI
thf(fact_377_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_378_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ X3 @ Y )
= ( X3 = Y ) ) ) ) ).
% antisym_conv1
thf(fact_379_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( X3 = Y ) ) ) ) ).
% antisym_conv2
thf(fact_380_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,Y: A] :
( ! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ord_less_eq @ A @ Y @ X5 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_ge
thf(fact_381_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z2: A] :
( ! [X5: A] :
( ( ord_less @ A @ X5 @ Y )
=> ( ord_less_eq @ A @ X5 @ Z2 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_le
thf(fact_382_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
& ~ ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ).
% less_le_not_le
thf(fact_383_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X3: A] :
( ~ ( ord_less_eq @ A @ Y @ X3 )
=> ( ord_less @ A @ X3 @ Y ) ) ) ).
% not_le_imp_less
thf(fact_384_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_385_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_386_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_387_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_388_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
& ~ ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_389_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X3 )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_390_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X3 @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_391_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_392_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_393_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_394_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_395_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
& ~ ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_396_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_397_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_398_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
| ( X = Y2 ) ) ) ) ) ).
% order_le_less
thf(fact_399_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
& ( X != Y2 ) ) ) ) ) ).
% order_less_le
thf(fact_400_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X3 @ Y ) )
= ( ord_less @ A @ Y @ X3 ) ) ) ).
% linorder_not_le
thf(fact_401_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% linorder_not_less
thf(fact_402_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% order_less_imp_le
thf(fact_403_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_404_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_405_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% order_le_less_trans
thf(fact_406_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% order_less_le_trans
thf(fact_407_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less @ B @ X5 @ Y4 )
=> ( ord_less @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_408_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ C @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_409_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less_eq @ B @ X5 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_410_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less @ A @ X5 @ Y4 )
=> ( ord_less @ C @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_411_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
| ( ord_less @ A @ Y @ X3 ) ) ) ).
% linorder_le_less_linear
thf(fact_412_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less @ A @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_413_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( N2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ).
% gr_zeroI
thf(fact_414_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_415_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M: A,N2: A] :
( ( ord_less @ A @ M @ N2 )
=> ( N2
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_416_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N2 )
= ( N2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_417_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_418_gr0I,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% gr0I
thf(fact_419_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_420_not__less0,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_421_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_422_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( N2
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_423_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M6: nat] :
( ( ord_less @ nat @ M6 @ N3 )
& ~ ( P @ M6 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_424_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X3: A] :
( ! [X5: A] :
( ( ( V @ X5 )
= ( zero_zero @ nat ) )
=> ( P @ X5 ) )
=> ( ! [X5: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X5 ) )
=> ( ~ ( P @ X5 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X5 ) )
& ~ ( P @ Y5 ) ) ) )
=> ( P @ X3 ) ) ) ).
% infinite_descent0_measure
thf(fact_425_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_426_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_427_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_428_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_429_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
& ( M2 != N ) ) ) ) ).
% nat_less_le
thf(fact_430_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_431_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
| ( M2 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_432_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less @ nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_433_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less @ nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_434_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less @ nat @ I4 @ J2 )
=> ( ord_less @ nat @ ( F2 @ I4 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_435_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_436_assms_I1_J,axiom,
vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).
% assms(1)
thf(fact_437_mint__corr__help__empty,axiom,
! [T2: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) )
=> ( ( vEBT_VEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mint_corr_help_empty
thf(fact_438_maxt__corr__help__empty,axiom,
! [T2: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( none @ nat ) )
=> ( ( vEBT_VEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% maxt_corr_help_empty
thf(fact_439_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ Y @ E ) ) )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% field_le_epsilon
thf(fact_440_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,N2: nat] :
( ( P @ K )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ? [Y5: A] :
( ( P @ Y5 )
& ~ ( ord_less_eq @ nat @ ( F2 @ Y5 ) @ ( F2 @ X5 ) ) ) )
=> ? [Y4: A] :
( ( P @ Y4 )
& ~ ( ord_less @ nat @ ( F2 @ Y4 ) @ ( plus_plus @ nat @ ( F2 @ K ) @ N2 ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_441_option_Osize__gen_I2_J,axiom,
! [A: $tType,X3: A > nat,X22: A] :
( ( size_option @ A @ X3 @ ( some @ A @ X22 ) )
= ( plus_plus @ nat @ ( X3 @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_442_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X3 @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_443_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_444_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_445_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_446_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X: A] :
~ ( member @ A @ X @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_447_deg__deg__n,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
=> ( Deg = N2 ) ) ).
% deg_deg_n
thf(fact_448_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) )
=> ? [Info2: option @ ( product_prod @ nat @ nat ),TreeList3: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N2 ) ) @ TreeList3 @ S2 ) ) ) ).
% deg_SUcn_Node
thf(fact_449_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X: A] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_450_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_451_psubsetI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% psubsetI
thf(fact_452_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_453_subsetI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( member @ A @ X5 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% subsetI
thf(fact_454_mi__eq__ma__no__ch,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
=> ( ( Mi = Ma )
=> ( ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) )
& ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_455_not__Some__eq,axiom,
! [A: $tType,X3: option @ A] :
( ( ! [Y2: A] :
( X3
!= ( some @ A @ Y2 ) ) )
= ( X3
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_456_not__None__eq,axiom,
! [A: $tType,X3: option @ A] :
( ( X3
!= ( none @ A ) )
= ( ? [Y2: A] :
( X3
= ( some @ A @ Y2 ) ) ) ) ).
% not_None_eq
thf(fact_457_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the2 @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_458_add__def,axiom,
( vEBT_VEBT_add
= ( vEBT_V2048590022279873568_shift @ nat @ ( plus_plus @ nat ) ) ) ).
% add_def
thf(fact_459_psubsetE,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% psubsetE
thf(fact_460_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_461_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% less_fun_def
thf(fact_462_psubset__imp__subset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_463_psubset__subset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 )
=> ( ord_less @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% psubset_subset_trans
thf(fact_464_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_465_subset__psubset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less @ ( set @ A ) @ B3 @ C5 )
=> ( ord_less @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% subset_psubset_trans
thf(fact_466_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_467_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_468_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z: set @ A] : Y3 = Z )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_469_subset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% subset_trans
thf(fact_470_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_471_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_472_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A6 )
=> ( member @ A @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_473_equalityD2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_474_equalityD1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_475_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X: A] :
( ( member @ A @ X @ A6 )
=> ( member @ A @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_476_equalityE,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_477_subsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_478_in__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B3 ) ) ) ).
% in_mono
thf(fact_479_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_480_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_481_combine__options__cases,axiom,
! [A: $tType,B: $tType,X3: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
( ( ( X3
= ( none @ A ) )
=> ( P @ X3 @ Y ) )
=> ( ( ( Y
= ( none @ B ) )
=> ( P @ X3 @ Y ) )
=> ( ! [A5: A,B5: B] :
( ( X3
= ( some @ A @ A5 ) )
=> ( ( Y
= ( some @ B @ B5 ) )
=> ( P @ X3 @ Y ) ) )
=> ( P @ X3 @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_482_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
! [X7: option @ A] : ( P2 @ X7 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
& ! [X: A] : ( P3 @ ( some @ A @ X ) ) ) ) ) ).
% split_option_all
thf(fact_483_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
? [X7: option @ A] : ( P2 @ X7 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
| ? [X: A] : ( P3 @ ( some @ A @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_484_option_Oexhaust,axiom,
! [A: $tType,Y: option @ A] :
( ( Y
!= ( none @ A ) )
=> ~ ! [X23: A] :
( Y
!= ( some @ A @ X23 ) ) ) ).
% option.exhaust
thf(fact_485_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X22: A] :
( ( Option
= ( some @ A @ X22 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_486_option_Odistinct_I1_J,axiom,
! [A: $tType,X22: A] :
( ( none @ A )
!= ( some @ A @ X22 ) ) ).
% option.distinct(1)
thf(fact_487_vebt__maxt_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( some @ nat @ Ma ) ) ).
% vebt_maxt.simps(3)
thf(fact_488_vebt__mint_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( some @ nat @ Mi ) ) ).
% vebt_mint.simps(3)
thf(fact_489_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_490_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_491_option_Osize__gen_I1_J,axiom,
! [A: $tType,X3: A > nat] :
( ( size_option @ A @ X3 @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size_gen(1)
thf(fact_492_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_493_option_Osize__neq,axiom,
! [A: $tType,X3: option @ A] :
( ( size_size @ ( option @ A ) @ X3 )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_494_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
=> ( ~ ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_495_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_496_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ).
% linordered_field_no_lb
thf(fact_497_not__psubset__empty,axiom,
! [A: $tType,A3: set @ A] :
~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_498_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X: A] : ( member @ A @ X @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_499_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_500_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_501_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_502_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ? [X5: A] :
( ( P @ X5 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M @ X5 ) @ ( M @ Y5 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_503_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less @ nat @ ( F2 @ Y4 ) @ B2 ) )
=> ? [X5: A] :
( ( P @ X5 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( F2 @ Y5 ) @ ( F2 @ X5 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_504_option_Osize_I4_J,axiom,
! [A: $tType,X22: A] :
( ( size_size @ ( option @ A ) @ ( some @ A @ X22 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(4)
thf(fact_505_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va: list @ vEBT_VEBT,Vb: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ Va @ Vb ) @ X3 )
= ( ( X3 = Mi )
| ( X3 = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_506_vebt__member_Osimps_I4_J,axiom,
! [V2: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X3 ) ).
% vebt_member.simps(4)
thf(fact_507_vebt__member_Osimps_I3_J,axiom,
! [V2: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X3 ) ).
% vebt_member.simps(3)
thf(fact_508_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_509_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_510_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_511_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B7 ) )
= ( ( A2 = A7 )
& ( B2 = B7 ) ) ) ).
% old.prod.inject
thf(fact_512_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_513_nth__enumerate__eq,axiom,
! [A: $tType,M: nat,Xs: list @ A,N2: nat] :
( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) @ M )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N2 @ M ) @ ( nth @ A @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_514_set__vebt__finite,axiom,
! [T2: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( finite_finite2 @ nat @ ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).
% set_vebt_finite
thf(fact_515_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
= ( P @ B5 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ ( zero_zero @ nat ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
=> ( P @ A5 @ ( plus_plus @ nat @ A5 @ B5 ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_516_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_517_length__enumerate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( size_size @ ( list @ ( product_prod @ nat @ A ) ) @ ( enumerate @ A @ N2 @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_enumerate
thf(fact_518_psubset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less @ ( set @ A ) @ B3 @ C5 )
=> ( ord_less @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% psubset_trans
thf(fact_519_psubsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% psubsetD
thf(fact_520_VEBT__internal_Ooption__shift_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu2: A > A > A,Uv2: option @ A] :
( X3
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) )
=> ( ! [Uw2: A > A > A,V3: A] :
( X3
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) )
=> ~ ! [F4: A > A > A,A5: A,B5: A] :
( X3
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F4 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ A5 ) @ ( some @ A @ B5 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_521_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu2: A > A > $o,Uv2: option @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) )
=> ( ! [Uw2: A > A > $o,V3: A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) )
=> ~ ! [F4: A > A > $o,X5: A,Y4: A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F4 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X5 ) @ ( some @ A @ Y4 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_522_finite__list,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [Xs3: list @ A] :
( ( set2 @ A @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_523_vebt__member_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X3 ) ).
% vebt_member.simps(2)
thf(fact_524_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_525_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ S3 )
& ~ ? [Xa: A] :
( ( member @ A @ Xa @ S3 )
& ( ord_less @ A @ Xa @ X5 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_526_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X8: set @ A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ X8 )
& ( ord_less @ A @ X5 @ Xa ) ) )
=> ~ ( finite_finite2 @ A @ X8 ) ) ) ) ).
% infinite_growing
thf(fact_527_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_528_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [A: $tType,F2: A > A > A,A2: A,B2: A] :
( ( vEBT_V2048590022279873568_shift @ A @ F2 @ ( some @ A @ A2 ) @ ( some @ A @ B2 ) )
= ( some @ A @ ( F2 @ A2 @ B2 ) ) ) ).
% VEBT_internal.option_shift.simps(3)
thf(fact_529_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
! [A: $tType,Uu: A > A > A,Uv: option @ A] :
( ( vEBT_V2048590022279873568_shift @ A @ Uu @ ( none @ A ) @ Uv )
= ( none @ A ) ) ).
% VEBT_internal.option_shift.simps(1)
thf(fact_530_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_531_surj__pair,axiom,
! [A: $tType,B: $tType,P4: product_prod @ A @ B] :
? [X5: A,Y4: B] :
( P4
= ( product_Pair @ A @ B @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_532_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_533_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B7 ) )
=> ~ ( ( A2 = A7 )
=> ( B2 != B7 ) ) ) ).
% Pair_inject
thf(fact_534_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B5: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% prod_cases3
thf(fact_535_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B5: B,C3: C,D3: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_536_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
~ ! [A5: A,B5: B,C3: C,D3: D,E: E2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D3 @ E ) ) ) ) ) ).
% prod_cases5
thf(fact_537_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) )] :
~ ! [A5: A,B5: B,C3: C,D3: D,E: E2,F4: F] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F ) @ D3 @ ( product_Pair @ E2 @ F @ E @ F4 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_538_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) )] :
~ ! [A5: A,B5: B,C3: C,D3: D,E: E2,F4: F,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F @ G3 ) @ E @ ( product_Pair @ F @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_539_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
=> ( P @ X3 ) ) ).
% prod_induct3
thf(fact_540_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B5: B,C3: C,D3: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D3 ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct4
thf(fact_541_prod__induct5,axiom,
! [E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
( ! [A5: A,B5: B,C3: C,D3: D,E: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D3 @ E ) ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct5
thf(fact_542_prod__induct6,axiom,
! [F: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) )] :
( ! [A5: A,B5: B,C3: C,D3: D,E: E2,F4: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F ) @ D3 @ ( product_Pair @ E2 @ F @ E @ F4 ) ) ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct6
thf(fact_543_prod__induct7,axiom,
! [G3: $tType,F: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) )] :
( ! [A5: A,B5: B,C3: C,D3: D,E: E2,F4: F,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F @ G3 ) @ E @ ( product_Pair @ F @ G3 @ F4 @ G4 ) ) ) ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct7
thf(fact_544_vebt__maxt_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
= ( none @ nat ) ) ).
% vebt_maxt.simps(2)
thf(fact_545_vebt__mint_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
= ( none @ nat ) ) ).
% vebt_mint.simps(2)
thf(fact_546_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [A: $tType,Uw: A > A > A,V2: A] :
( ( vEBT_V2048590022279873568_shift @ A @ Uw @ ( some @ A @ V2 ) @ ( none @ A ) )
= ( none @ A ) ) ).
% VEBT_internal.option_shift.simps(2)
thf(fact_547_VEBT__internal_Ooption__shift_Oelims,axiom,
! [A: $tType,X3: A > A > A,Xa2: option @ A,Xb: option @ A,Y: option @ A] :
( ( ( vEBT_V2048590022279873568_shift @ A @ X3 @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2
= ( none @ A ) )
=> ( Y
!= ( none @ A ) ) )
=> ( ( ? [V3: A] :
( Xa2
= ( some @ A @ V3 ) )
=> ( ( Xb
= ( none @ A ) )
=> ( Y
!= ( none @ A ) ) ) )
=> ~ ! [A5: A] :
( ( Xa2
= ( some @ A @ A5 ) )
=> ! [B5: A] :
( ( Xb
= ( some @ A @ B5 ) )
=> ( Y
!= ( some @ A @ ( X3 @ A5 @ B5 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_548_vebt__buildup_Ocases,axiom,
! [X3: nat] :
( ( X3
!= ( zero_zero @ nat ) )
=> ( ( X3
!= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ! [Va2: nat] :
( X3
!= ( suc @ ( suc @ Va2 ) ) ) ) ) ).
% vebt_buildup.cases
thf(fact_549_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_550_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_551_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_552_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_553_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_554_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_555_infinite__nat__iff__unbounded__le,axiom,
! [S3: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S3 ) )
= ( ! [M2: nat] :
? [N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
& ( member @ nat @ N @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_556_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N5: set @ nat] :
? [M2: nat] :
! [X: nat] :
( ( member @ nat @ X @ N5 )
=> ( ord_less_eq @ nat @ X @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_557_unbounded__k__infinite,axiom,
! [K: nat,S3: set @ nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ K @ M3 )
=> ? [N6: nat] :
( ( ord_less @ nat @ M3 @ N6 )
& ( member @ nat @ N6 @ S3 ) ) )
=> ~ ( finite_finite2 @ nat @ S3 ) ) ).
% unbounded_k_infinite
thf(fact_558_bounded__nat__set__is__finite,axiom,
! [N7: set @ nat,N2: nat] :
( ! [X5: nat] :
( ( member @ nat @ X5 @ N7 )
=> ( ord_less @ nat @ X5 @ N2 ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% bounded_nat_set_is_finite
thf(fact_559_finite__maxlen,axiom,
! [A: $tType,M7: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M7 )
=> ? [N3: nat] :
! [X2: list @ A] :
( ( member @ ( list @ A ) @ X2 @ M7 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X2 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_560_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X5: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ).
% subrelI
thf(fact_561_finite__set__choice,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ? [X_12: B] : ( P @ X5 @ X_12 ) )
=> ? [F4: A > B] :
! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( P @ X2 @ ( F4 @ X2 ) ) ) ) ) ).
% finite_set_choice
thf(fact_562_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% finite
thf(fact_563_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M7: nat] :
( ( P @ X3 )
=> ( ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq @ nat @ X5 @ M7 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq @ nat @ X2 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_564_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
~ ! [F4: nat > A > A,A5: nat,B5: nat,Acc: A] :
( X3
!= ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F4 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A5 @ ( product_Pair @ nat @ A @ B5 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_565_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A3 )
& ( ord_less_eq @ A @ X5 @ A2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_566_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A3 )
& ( ord_less_eq @ A @ A2 @ X5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_567_finite__transitivity__chain,axiom,
! [A: $tType,A3: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X5: A] :
~ ( R @ X5 @ X5 )
=> ( ! [X5: A,Y4: A,Z3: A] :
( ( R @ X5 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X5 @ Z3 ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A3 )
& ( R @ X5 @ Y5 ) ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_568_infinite__imp__nonempty,axiom,
! [A: $tType,S3: set @ A] :
( ~ ( finite_finite2 @ A @ S3 )
=> ( S3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_569_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_570_rev__finite__subset,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_571_infinite__super,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ T4 )
=> ( ~ ( finite_finite2 @ A @ S3 )
=> ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% infinite_super
thf(fact_572_finite__subset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_subset
thf(fact_573_finite__psubset__induct,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ! [B8: set @ A] :
( ( ord_less @ ( set @ A ) @ B8 @ A8 )
=> ( P @ B8 ) )
=> ( P @ A8 ) ) )
=> ( P @ A3 ) ) ) ).
% finite_psubset_induct
thf(fact_574_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N5: set @ nat] :
? [M2: nat] :
! [X: nat] :
( ( member @ nat @ X @ N5 )
=> ( ord_less @ nat @ X @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_575_infinite__nat__iff__unbounded,axiom,
! [S3: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S3 ) )
= ( ! [M2: nat] :
? [N: nat] :
( ( ord_less @ nat @ M2 @ N )
& ( member @ nat @ N @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_576_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ord_less @ B @ ( F2 @ X2 ) @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S3 ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_577_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S3: set @ A,Y: A,F2: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y @ S3 )
=> ( ord_less_eq @ B @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S3 ) ) @ ( F2 @ Y ) ) ) ) ) ) ).
% arg_min_least
thf(fact_578_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less @ nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N2 )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ K2 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_579_subset__emptyI,axiom,
! [A: $tType,A3: set @ A] :
( ! [X5: A] :
~ ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_580_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_581_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_12: A] : ( P @ ( zero_zero @ nat ) @ X_12 )
=> ( ! [X5: A,N3: nat] :
( ( P @ N3 @ X5 )
=> ? [Y5: A] :
( ( P @ ( suc @ N3 ) @ Y5 )
& ( Q @ N3 @ X5 @ Y5 ) ) )
=> ? [F4: nat > A] :
! [N6: nat] :
( ( P @ N6 @ ( F4 @ N6 ) )
& ( Q @ N6 @ ( F4 @ N6 ) @ ( F4 @ ( suc @ N6 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_582_list__decode_Ocases,axiom,
! [X3: nat] :
( ( X3
!= ( zero_zero @ nat ) )
=> ~ ! [N3: nat] :
( X3
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_583_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% verit_sum_simplify
thf(fact_584_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_585_field__lbound__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D1: A,D22: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 )
=> ? [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
& ( ord_less @ A @ E @ D1 )
& ( ord_less @ A @ E @ D22 ) ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_586_less__by__empty,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] :
( ( A3
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B3 ) ) ).
% less_by_empty
thf(fact_587_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(2)
thf(fact_588_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_589_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(1)
thf(fact_590_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_591_prod__decode__aux_Ocases,axiom,
! [X3: product_prod @ nat @ nat] :
~ ! [K2: nat,M3: nat] :
( X3
!= ( product_Pair @ nat @ nat @ K2 @ M3 ) ) ).
% prod_decode_aux.cases
thf(fact_592_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S3 ) @ S3 ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_593_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B7: B,A7: B] :
( ( ~ ( ord_less_eq @ B @ B7 @ A7 ) )
= ( ord_less @ B @ A7 @ B7 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_594_triangle__Suc,axiom,
! [N2: nat] :
( ( nat_triangle @ ( suc @ N2 ) )
= ( plus_plus @ nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).
% triangle_Suc
thf(fact_595_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_596_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z3 )
=> ~ ( ord_less_eq @ A @ T2 @ X2 ) ) ) ).
% minf(8)
thf(fact_597_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z3 )
=> ( ord_less_eq @ A @ X2 @ T2 ) ) ) ).
% minf(6)
thf(fact_598_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ Z3 @ X2 )
=> ( ord_less_eq @ A @ T2 @ X2 ) ) ) ).
% pinf(8)
thf(fact_599_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ Z3 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ T2 ) ) ) ).
% pinf(6)
thf(fact_600_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ A @ C3 @ B2 )
& ! [X2: A] :
( ( ( ord_less_eq @ A @ A2 @ X2 )
& ( ord_less @ A @ X2 @ C3 ) )
=> ( P @ X2 ) )
& ! [D4: A] :
( ! [X5: A] :
( ( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less @ A @ X5 @ D4 ) )
=> ( P @ X5 ) )
=> ( ord_less_eq @ A @ D4 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_601_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_602_find__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X3: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X3 ) )
= ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I3 ) )
& ( X3
= ( nth @ A @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I3 )
=> ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_603_find__Some__iff2,axiom,
! [A: $tType,X3: A,P: A > $o,Xs: list @ A] :
( ( ( some @ A @ X3 )
= ( find @ A @ P @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I3 ) )
& ( X3
= ( nth @ A @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I3 )
=> ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_604_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_605_find__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( find @ A @ P @ Xs )
= ( find @ A @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_606_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit5016429287641298734tinuum @ A )
=> ! [A2: A] :
? [B5: A] :
( ( ord_less @ A @ A2 @ B5 )
| ( ord_less @ A @ B5 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_607_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q3 @ X5 ) ) )
=> ? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ Z3 @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P5 @ X2 )
& ( Q3 @ X2 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_608_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q3 @ X5 ) ) )
=> ? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ Z3 @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P5 @ X2 )
| ( Q3 @ X2 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_609_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ Z3 @ X2 )
=> ( X2 != T2 ) ) ) ).
% pinf(3)
thf(fact_610_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ Z3 @ X2 )
=> ( X2 != T2 ) ) ) ).
% pinf(4)
thf(fact_611_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ Z3 @ X2 )
=> ~ ( ord_less @ A @ X2 @ T2 ) ) ) ).
% pinf(5)
thf(fact_612_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ Z3 @ X2 )
=> ( ord_less @ A @ T2 @ X2 ) ) ) ).
% pinf(7)
thf(fact_613_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F5: D] :
? [Z3: C] :
! [X2: C] :
( ( ord_less @ C @ Z3 @ X2 )
=> ( F5 = F5 ) ) ) ).
% pinf(11)
thf(fact_614_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q3 @ X5 ) ) )
=> ? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z3 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P5 @ X2 )
& ( Q3 @ X2 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_615_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q3 @ X5 ) ) )
=> ? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z3 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P5 @ X2 )
| ( Q3 @ X2 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_616_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z3 )
=> ( X2 != T2 ) ) ) ).
% minf(3)
thf(fact_617_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z3 )
=> ( X2 != T2 ) ) ) ).
% minf(4)
thf(fact_618_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z3 )
=> ( ord_less @ A @ X2 @ T2 ) ) ) ).
% minf(5)
thf(fact_619_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z3 )
=> ~ ( ord_less @ A @ T2 @ X2 ) ) ) ).
% minf(7)
thf(fact_620_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F5: D] :
? [Z3: C] :
! [X2: C] :
( ( ord_less @ C @ X2 @ Z3 )
=> ( F5 = F5 ) ) ) ).
% minf(11)
thf(fact_621_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_622_find__None__iff2,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( none @ A )
= ( find @ A @ P @ Xs ) )
= ( ~ ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( P @ X ) ) ) ) ).
% find_None_iff2
thf(fact_623_find__None__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( find @ A @ P @ Xs )
= ( none @ A ) )
= ( ~ ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( P @ X ) ) ) ) ).
% find_None_iff
thf(fact_624_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_625__C5_C_I10_J,axiom,
( ( mi != ma )
=> ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( ( ( vEBT_VEBT_high @ ma @ n )
= I5 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I5 ) @ ( vEBT_VEBT_low @ ma @ n ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ n )
= I5 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I5 ) @ ( vEBT_VEBT_low @ X2 @ n ) ) )
=> ( ( ord_less @ nat @ mi @ X2 )
& ( ord_less_eq @ nat @ X2 @ ma ) ) ) ) ) ) ).
% "5"(10)
thf(fact_626_vebt__member_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B5: $o,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X5 ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X5 ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ X5 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ X5 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ).
% vebt_member.cases
thf(fact_627_VEBT__internal_Omembermima_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
( X3
!= ( 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,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ X5 ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ X5 ) )
=> ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ X5 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_628_count__notin,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( count_list @ A @ Xs @ X3 )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_629_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 ) )
= ( ? [Y2: A,N: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N ) @ Y2 ) @ R2 )
& ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys
= ( list_update @ A @ Xs @ N @ Y2 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_630_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_631_pair__less__iff1,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X3 @ Y ) @ ( product_Pair @ nat @ nat @ X3 @ Z2 ) ) @ fun_pair_less )
= ( ord_less @ nat @ Y @ Z2 ) ) ).
% pair_less_iff1
thf(fact_632_gen__length__def,axiom,
! [A: $tType] :
( ( gen_length @ A )
= ( ^ [N: nat,Xs2: list @ A] : ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% gen_length_def
thf(fact_633_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_634_Leaf__0__not,axiom,
! [A2: $o,B2: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) ) ).
% Leaf_0_not
thf(fact_635_power__shift,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ( power_power @ nat @ X3 @ Y )
= Z2 )
= ( ( vEBT_VEBT_power @ ( some @ nat @ X3 ) @ ( some @ nat @ Y ) )
= ( some @ nat @ Z2 ) ) ) ).
% power_shift
thf(fact_636__C5_C_I3_J,axiom,
( ( size_size @ ( list @ vEBT_VEBT ) @ treeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) ) ).
% "5"(3)
thf(fact_637__C5_C_I9_J,axiom,
ord_less @ nat @ ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ) ).
% "5"(9)
thf(fact_638_VEBT_Oinject_I2_J,axiom,
! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
( ( ( vEBT_Leaf @ X21 @ X222 )
= ( vEBT_Leaf @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% VEBT.inject(2)
thf(fact_639_local_Opower__def,axiom,
( vEBT_VEBT_power
= ( vEBT_V2048590022279873568_shift @ nat @ ( power_power @ nat ) ) ) ).
% local.power_def
thf(fact_640_high__bound__aux,axiom,
! [Ma: nat,N2: nat,M: nat] :
( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% high_bound_aux
thf(fact_641_member__bound,axiom,
! [Tree: vEBT_VEBT,X3: nat,N2: nat] :
( ( vEBT_vebt_member @ Tree @ X3 )
=> ( ( vEBT_invar_vebt @ Tree @ N2 )
=> ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% member_bound
thf(fact_642__C5_C_I6_J,axiom,
! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I5 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ summary @ I5 ) ) ) ).
% "5"(6)
thf(fact_643_numeral__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N2: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less_eq @ num @ M @ N2 ) ) ) ).
% numeral_le_iff
thf(fact_644_numeral__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N2: num] :
( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less @ num @ M @ N2 ) ) ) ).
% numeral_less_iff
thf(fact_645_numeral__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [M: num,N2: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N2 ) ) ) ) ).
% numeral_plus_numeral
thf(fact_646_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V2: num,W2: num,Z2: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Z2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W2 ) ) @ Z2 ) ) ) ).
% add_numeral_left
thf(fact_647_mi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
=> ( ( 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_648_Suc__numeral,axiom,
! [N2: num] :
( ( suc @ ( numeral_numeral @ nat @ N2 ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N2 @ one2 ) ) ) ).
% Suc_numeral
thf(fact_649_add__2__eq__Suc_H,axiom,
! [N2: nat] :
( ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ N2 ) ) ) ).
% add_2_eq_Suc'
thf(fact_650_add__2__eq__Suc,axiom,
! [N2: nat] :
( ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
= ( suc @ ( suc @ N2 ) ) ) ).
% add_2_eq_Suc
thf(fact_651_numeral__Bit0,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( numeral_numeral @ A @ ( bit0 @ N2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% numeral_Bit0
thf(fact_652_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_653_less__2__cases,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( N2
= ( zero_zero @ nat ) )
| ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases
thf(fact_654_less__2__cases__iff,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ( N2
= ( zero_zero @ nat ) )
| ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases_iff
thf(fact_655_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X3: nat,N2: nat,M: nat] :
( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_656_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X3: nat,N2: nat,M: nat] :
( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_low @ X3 @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_657_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_658_Suc__nat__number__of__add,axiom,
! [V2: num,N2: nat] :
( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N2 ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V2 @ one2 ) ) @ N2 ) ) ).
% Suc_nat_number_of_add
thf(fact_659_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_660_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,D3: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D3 ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).
% VEBT_internal.valid'.cases
thf(fact_661_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_662_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_663_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N2 ) ) ) ).
% zero_neq_numeral
thf(fact_664_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_665_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_666_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_667_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_668_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% VEBT_internal.minNull.simps(3)
thf(fact_669_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% VEBT_internal.minNull.simps(2)
thf(fact_670_VEBT__internal_OminNull_Osimps_I1_J,axiom,
vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% VEBT_internal.minNull.simps(1)
thf(fact_671_invar__vebt_Ointros_I2_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N2 )
=> ( ( Deg
= ( plus_plus @ nat @ N2 @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_672_invar__vebt_Ointros_I3_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N2 ) )
=> ( ( Deg
= ( plus_plus @ nat @ N2 @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_673_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_674_in__set__zipE,axiom,
! [A: $tType,B: $tType,X3: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ~ ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ~ ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_675_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X3: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_676_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X3: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_677_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,X3: A,Ys: list @ B,Y: B] :
( ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ X3 ) @ ( list_update @ B @ Ys @ I @ Y ) )
= ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I @ ( product_Pair @ A @ B @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_678_zero__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% zero_le_numeral
thf(fact_679_not__numeral__le__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_le_zero
thf(fact_680_not__numeral__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_less_zero
thf(fact_681_zero__less__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% zero_less_numeral
thf(fact_682_set__encode__eq,axiom,
! [A3: set @ nat,B3: set @ nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ( finite_finite2 @ nat @ B3 )
=> ( ( ( nat_set_encode @ A3 )
= ( nat_set_encode @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% set_encode_eq
thf(fact_683_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_684_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 ) )
=> ~ ! [X5: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_685_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X3: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ~ ! [Y4: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_686_VEBT__internal_Onaive__member_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B5: $o,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X5 ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Ux2 ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) @ X5 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_687_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_688_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_689_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A1: vEBT_VEBT,A22: nat] :
( ( ? [A4: $o,B4: $o] :
( A1
= ( vEBT_Leaf @ A4 @ B4 ) )
& ( A22
= ( suc @ ( zero_zero @ nat ) ) ) )
| ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A22 @ TreeList @ Summary3 ) )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X @ N ) )
& ( vEBT_invar_vebt @ Summary3 @ N )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
& ( A22
= ( plus_plus @ nat @ N @ N ) )
& ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
| ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A22 @ TreeList @ Summary3 ) )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X @ N ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) )
& ( A22
= ( plus_plus @ nat @ N @ ( suc @ N ) ) )
& ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
| ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X @ N ) )
& ( vEBT_invar_vebt @ Summary3 @ N )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
& ( A22
= ( plus_plus @ nat @ N @ N ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A22 ) )
& ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
& ! [X: nat] :
( ( ( ( vEBT_VEBT_high @ X @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) )
| ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X @ N ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) )
& ( A22
= ( plus_plus @ nat @ N @ ( suc @ N ) ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A22 ) )
& ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
& ! [X: nat] :
( ( ( ( vEBT_VEBT_high @ X @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_690_invar__vebt_Ocases,axiom,
! [A12: vEBT_VEBT,A23: nat] :
( ( vEBT_invar_vebt @ A12 @ A23 )
=> ( ( ? [A5: $o,B5: $o] :
( A12
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( A23
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3 = N3 )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M3 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M3 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3 = N3 )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M3 ) )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I5 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
= I5 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M3 ) )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I5 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
= I5 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_691_VEBT__internal_OminNull_Ocases,axiom,
! [X3: vEBT_VEBT] :
( ( X3
!= ( vEBT_Leaf @ $false @ $false ) )
=> ( ! [Uv2: $o] :
( X3
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X3
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.cases
thf(fact_692_num_Osize_I5_J,axiom,
! [X22: num] :
( ( size_size @ num @ ( bit0 @ X22 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(5)
thf(fact_693_invar__vebt_Ointros_I4_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N2 )
=> ( ( Deg
= ( plus_plus @ nat @ N2 @ M ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_694_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X3 )
=> ( ! [Uv2: $o] :
( X3
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X3
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_695_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X3 )
=> ( ( X3
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_696_invar__vebt_Ointros_I5_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N2 ) )
=> ( ( Deg
= ( plus_plus @ nat @ N2 @ M ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_697_set__encode__inf,axiom,
! [A3: set @ nat] :
( ~ ( finite_finite2 @ nat @ A3 )
=> ( ( nat_set_encode @ A3 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_698_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_699_count__le__length,axiom,
! [A: $tType,Xs: list @ A,X3: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X3 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% count_le_length
thf(fact_700_vebt__mint_Ocases,axiom,
! [X3: vEBT_VEBT] :
( ! [A5: $o,B5: $o] :
( X3
!= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% vebt_mint.cases
thf(fact_701_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X3 )
= Y )
=> ( ( ( X3
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y )
=> ( ( ? [Uv2: $o] :
( X3
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> Y )
=> ( ( ? [Uu2: $o] :
( X3
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> Y )
=> ( ( ? [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
= ( 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] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> Y ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_702_length__code,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% length_code
thf(fact_703_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_704_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_705_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_706_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( X3 = Y ) ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_707_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_708_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N2: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_709_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A,N2: nat] :
( ( ( power_power @ A @ A2 @ N2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% power_eq_0_iff
thf(fact_710_member__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
& ( ( X3 = Mi )
| ( X3 = Ma )
| ( ( ord_less @ nat @ X3 @ Ma )
& ( ord_less @ nat @ Mi @ X3 )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_711_both__member__options__ding,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X3: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
=> ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ).
% both_member_options_ding
thf(fact_712_nat__zero__less__power__iff,axiom,
! [X3: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X3 @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_713_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_714_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X: nat,N: nat] : ( divide_divide @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% high_def
thf(fact_715_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_716_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_right
thf(fact_717_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( divide_divide @ A @ C2 @ A2 )
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_left
thf(fact_718_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_719_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_720_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_721_power__0__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat] :
( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N2 ) )
= ( zero_zero @ A ) ) ) ).
% power_0_Suc
thf(fact_722_power__zero__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [K: num] :
( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ K ) )
= ( zero_zero @ A ) ) ) ).
% power_zero_numeral
thf(fact_723_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_724_power__Suc__0,axiom,
! [N2: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_725_nat__power__eq__Suc__0__iff,axiom,
! [X3: nat,M: nat] :
( ( ( power_power @ nat @ X3 @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( X3
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_726_add__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% add_divide_distrib
thf(fact_727_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_728_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_729_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_730_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_731_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_732_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_733_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_right_mono
thf(fact_734_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_735_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% divide_neg_neg
thf(fact_736_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_737_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_738_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% divide_pos_pos
thf(fact_739_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_740_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) )
& ( C2
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_741_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_742_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_743_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_744_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X3: A,W2: A,Z2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_le
thf(fact_745_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A,W2: A,Z2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z2 )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less
thf(fact_746_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A,W2: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less @ A @ W2 @ Z2 )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_747_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_748_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_749_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_750_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_751_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_752_field__sum__of__halves,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= X3 ) ) ).
% field_sum_of_halves
thf(fact_753_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_754_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_755_field__less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ A @ X3 @ ( divide_divide @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% field_less_half_sum
thf(fact_756_power__not__zero,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A,N2: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A2 @ N2 )
!= ( zero_zero @ A ) ) ) ) ).
% power_not_zero
thf(fact_757_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).
% zero_le_power
thf(fact_758_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N2: nat] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ).
% power_mono
thf(fact_759_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).
% zero_less_power
thf(fact_760_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N2 ) )
=> ( ord_less @ nat @ M @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_761_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: nat,B2: A] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_762_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: nat,B2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N2 ) ) @ ( power_power @ A @ B2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% power_le_imp_le_base
thf(fact_763_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: nat,B2: A] :
( ( ( power_power @ A @ A2 @ ( suc @ N2 ) )
= ( power_power @ A @ B2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( A2 = B2 ) ) ) ) ) ).
% power_inject_base
thf(fact_764_zero__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N2 )
= ( zero_zero @ A ) ) ) ) ).
% zero_power
thf(fact_765_power__gt__expt,axiom,
! [N2: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
=> ( ord_less @ nat @ K @ ( power_power @ nat @ N2 @ K ) ) ) ).
% power_gt_expt
thf(fact_766_nat__one__le__power,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N2 ) ) ) ).
% nat_one_le_power
thf(fact_767_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ( power_power @ A @ A2 @ N2 )
= ( power_power @ A @ B2 @ N2 ) )
= ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_768_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: nat,B2: A] :
( ( ( power_power @ A @ A2 @ N2 )
= ( power_power @ A @ B2 @ N2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_769_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_770_less__exp,axiom,
! [N2: nat] : ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% less_exp
thf(fact_771_power2__nat__le__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N2 )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% power2_nat_le_imp_le
thf(fact_772_power2__nat__le__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% power2_nat_le_eq_le
thf(fact_773_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ord_less_eq @ nat @ M @ ( power_power @ nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_774_power2__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( 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 @ X3 @ Y ) ) ) ) ).
% power2_le_imp_le
thf(fact_775_power2__eq__imp__eq,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,Y: A] :
( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( X3 = Y ) ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_776_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_777_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N2: nat] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_778_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_779_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( power_power @ A @ X3 @ ( 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 @ X3 @ Y ) ) ) ) ).
% power2_less_imp_less
thf(fact_780_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_781_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_782_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_783_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X3
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_784_both__member__options__from__chilf__to__complete__tree,axiom,
! [X3: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( 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 @ TreeList2 @ Summary ) @ X3 ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_785_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= M ) ).
% add_self_div_2
thf(fact_786_div2__Suc__Suc,axiom,
! [M: nat] :
( ( divide_divide @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% div2_Suc_Suc
thf(fact_787_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: 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 @ TreeList2 @ Summary ) @ X3 )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
| ( X3 = Mi )
| ( X3 = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_788_semiring__norm_I76_J,axiom,
! [N2: num] : ( ord_less @ num @ one2 @ ( bit0 @ N2 ) ) ).
% semiring_norm(76)
thf(fact_789_div__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( divide_divide @ nat @ M @ N2 )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_790_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_791_set__n__deg__not__0,axiom,
! [TreeList2: list @ vEBT_VEBT,N2: nat,M: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 ) ) ) ).
% set_n_deg_not_0
thf(fact_792_Suc__n__div__2__gt__zero,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_793_div__2__gt__zero,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_794_deg1Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
= ( ? [A4: $o,B4: $o] :
( T2
= ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).
% deg1Leaf
thf(fact_795_deg__1__Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
=> ? [A5: $o,B5: $o] :
( T2
= ( vEBT_Leaf @ A5 @ B5 ) ) ) ).
% deg_1_Leaf
thf(fact_796_deg__1__Leafy,axiom,
! [T2: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( N2
= ( one_one @ nat ) )
=> ? [A5: $o,B5: $o] :
( T2
= ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ).
% deg_1_Leafy
thf(fact_797_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% div_by_1
thf(fact_798_semiring__norm_I78_J,axiom,
! [M: num,N2: num] :
( ( ord_less @ num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less @ num @ M @ N2 ) ) ).
% semiring_norm(78)
thf(fact_799_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less @ num @ M @ one2 ) ).
% semiring_norm(75)
thf(fact_800_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_801_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_802_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_803_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_804_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_805_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_806_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_807_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_808_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_809_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ( power_power @ A @ A2 @ M )
= ( power_power @ A @ A2 @ N2 ) )
= ( M = N2 ) ) ) ) ).
% power_inject_exp
thf(fact_810_less__one,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( one_one @ nat ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_811_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_812_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_813_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_814_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_815_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_816_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_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__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_819_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X3: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ X3 ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less @ nat @ X3 @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_820_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_821_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_822_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_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_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_825_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M: nat,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N2 ) )
= ( ord_less @ nat @ N2 @ M ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_826_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X3: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X3 ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less_eq @ nat @ X3 @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_827_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_828_one__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N2 ) ) ) ) ).
% one_plus_numeral
thf(fact_829_numeral__plus__one,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ N2 @ one2 ) ) ) ) ).
% numeral_plus_one
thf(fact_830_numeral__le__one__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ num @ N2 @ one2 ) ) ) ).
% numeral_le_one_iff
thf(fact_831_one__less__numeral__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less @ num @ one2 @ N2 ) ) ) ).
% one_less_numeral_iff
thf(fact_832_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_833_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M: nat,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N2 ) )
= ( ord_less_eq @ nat @ N2 @ M ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_834_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X3: A] :
( ( ( one_one @ A )
= X3 )
= ( X3
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_835_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_836_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_837_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_838_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_839_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_840_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_841_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_842_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_843_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_844_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_845_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_846_one__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% one_le_numeral
thf(fact_847_not__numeral__less__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) ) ) ).
% not_numeral_less_one
thf(fact_848_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_849_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_850_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_851_one__plus__numeral__commute,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X3: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X3 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X3 ) @ ( one_one @ A ) ) ) ) ).
% one_plus_numeral_commute
thf(fact_852_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).
% one_le_power
thf(fact_853_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_854_div__eq__dividend__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ( divide_divide @ nat @ M @ N2 )
= M )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% div_eq_dividend_iff
thf(fact_855_div__less__dividend,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N2 ) @ M ) ) ) ).
% div_less_dividend
thf(fact_856_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_857_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_858_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_859_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_860_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,D2: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D2 )
= ( D2
= ( one_one @ nat ) ) ) ).
% VEBT_internal.valid'.simps(1)
thf(fact_861_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_862_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: 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 @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_863_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_864_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_865_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_866_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_867_power__0__left,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N2 )
= ( one_one @ A ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N2 )
= ( zero_zero @ A ) ) ) ) ) ).
% power_0_left
thf(fact_868_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( suc @ N2 ) ) ) ) ) ).
% power_gt1
thf(fact_869_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) )
=> ( ord_less @ nat @ M @ N2 ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_870_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N7: nat,A2: A] :
( ( ord_less @ nat @ N2 @ N7 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ A2 @ N7 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_871_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N7: nat,A2: A] :
( ( ord_less_eq @ nat @ N2 @ N7 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ A2 @ N7 ) ) ) ) ) ).
% power_increasing
thf(fact_872_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_873_vebt__member_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X3: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A2 @ B2 ) @ X3 )
= ( ( ( X3
= ( zero_zero @ nat ) )
=> A2 )
& ( ( X3
!= ( zero_zero @ nat ) )
=> ( ( ( X3
= ( one_one @ nat ) )
=> B2 )
& ( X3
= ( one_one @ nat ) ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_874_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X3: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A2 @ B2 ) @ X3 )
= ( ( ( X3
= ( zero_zero @ nat ) )
=> A2 )
& ( ( X3
!= ( zero_zero @ nat ) )
=> ( ( ( X3
= ( one_one @ nat ) )
=> B2 )
& ( X3
= ( one_one @ nat ) ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_875_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_876_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_877_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: 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 @ N2 ) ) @ A2 ) ) ) ) ).
% power_Suc_le_self
thf(fact_878_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ N2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_879_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N7: nat,A2: A] :
( ( ord_less @ nat @ N2 @ N7 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N7 ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_880_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N7: nat,A2: A] :
( ( ord_less_eq @ nat @ N2 @ N7 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N7 ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ) ).
% power_decreasing
thf(fact_881_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_882_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ).
% self_le_power
thf(fact_883_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ).
% one_less_power
thf(fact_884_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_885_vebt__maxt_Osimps_I1_J,axiom,
! [B2: $o,A2: $o] :
( ( B2
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( A2
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_maxt.simps(1)
thf(fact_886_vebt__mint_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( A2
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( B2
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_mint.simps(1)
thf(fact_887_div__le__mono,axiom,
! [M: nat,N2: nat,K: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N2 @ K ) ) ) ).
% div_le_mono
thf(fact_888_div__le__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N2 ) @ M ) ).
% div_le_dividend
thf(fact_889_ex__power__ivl2,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_890_ex__power__ivl1,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ? [N3: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_891_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N2: nat] :
( ( ( divide_divide @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
= ( ( ord_less @ nat @ M @ N2 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_892_Suc__div__le__mono,axiom,
! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N2 ) @ ( divide_divide @ nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_div_le_mono
thf(fact_893_vebt__maxt_Oelims,axiom,
! [X3: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_maxt @ X3 )
= Y )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ~ ( ( B5
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B5
=> ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat] :
( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( some @ nat @ Ma2 ) ) ) ) ) ) ).
% vebt_maxt.elims
thf(fact_894_vebt__mint_Oelims,axiom,
! [X3: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_mint @ X3 )
= Y )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ~ ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( ( B5
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B5
=> ( Y
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat] :
( ? [Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( some @ nat @ Mi2 ) ) ) ) ) ) ).
% vebt_mint.elims
thf(fact_895_div__greater__zero__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N2 ) )
= ( ( ord_less_eq @ nat @ N2 @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_896_div__le__mono2,axiom,
! [M: nat,N2: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N2 ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_897_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_898_nat__induct2,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct2
thf(fact_899_div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M: nat,N2: nat] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ).
% div_exp_eq
thf(fact_900_exp__add__not__zero__imp__right,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N2: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_right
thf(fact_901_exp__add__not__zero__imp__left,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N2: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_left
thf(fact_902_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_903_low__inv,axiom,
! [X3: nat,N2: nat,Y: nat] :
( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ X3 ) @ N2 )
= X3 ) ) ).
% low_inv
thf(fact_904_high__inv,axiom,
! [X3: nat,N2: nat,Y: nat] :
( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ X3 ) @ N2 )
= Y ) ) ).
% high_inv
thf(fact_905_enat__ord__number_I1_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N2 ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) ) ).
% enat_ord_number(1)
thf(fact_906_enat__ord__number_I2_J,axiom,
! [M: num,N2: num] :
( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N2 ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) ) ).
% enat_ord_number(2)
thf(fact_907_mul__shift,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ( times_times @ nat @ X3 @ Y )
= Z2 )
= ( ( vEBT_VEBT_mul @ ( some @ nat @ X3 ) @ ( some @ nat @ Y ) )
= ( some @ nat @ Z2 ) ) ) ).
% mul_shift
thf(fact_908_mul__def,axiom,
( vEBT_VEBT_mul
= ( vEBT_V2048590022279873568_shift @ nat @ ( times_times @ nat ) ) ) ).
% mul_def
thf(fact_909_i0__less,axiom,
! [N2: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 )
= ( N2
!= ( zero_zero @ extended_enat ) ) ) ).
% i0_less
thf(fact_910_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_911_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_912_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_913_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_914_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_915_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_916_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_917_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_918_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult_1
thf(fact_919_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% half_negative_int_iff
thf(fact_920_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ).
% times_divide_eq_left
thf(fact_921_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_922_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% divide_divide_eq_right
thf(fact_923_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% times_divide_eq_right
thf(fact_924_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( times_times @ nat @ M @ K )
= ( times_times @ nat @ N2 @ K ) )
= ( ( M = N2 )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_925_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N2 ) )
= ( ( M = N2 )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_926_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_927_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( times_times @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N2
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_928_nat__1__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M @ N2 ) )
= ( ( M
= ( one_one @ nat ) )
& ( N2
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_929_nat__mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times @ nat @ M @ N2 )
= ( one_one @ nat ) )
= ( ( M
= ( one_one @ nat ) )
& ( N2
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_930_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A2: A,C2: A] :
( ( ( times_times @ A @ A2 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_931_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_932_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,A2: A] :
( ( ( times_times @ A @ C2 @ A2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_933_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_934_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X3: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_935_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_936_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_937_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_938_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_939_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_940_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_941_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_942_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_943_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult2
thf(fact_944_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult1
thf(fact_945_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A2: A,B2: A,V2: num] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_946_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V2: num,B2: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C2 ) ) ) ) ).
% distrib_left_numeral
thf(fact_947_mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times @ nat @ M @ N2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% mult_eq_1_iff
thf(fact_948_one__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( times_times @ nat @ M @ N2 ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% one_eq_mult_iff
thf(fact_949_mult__less__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_950_nat__0__less__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_951_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_952_mult__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( times_times @ nat @ M @ ( suc @ N2 ) )
= ( plus_plus @ nat @ M @ ( times_times @ nat @ M @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_953_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( K
= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( zero_zero @ nat ) ) )
& ( ( K
!= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( divide_divide @ nat @ M @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
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_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_959_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_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__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self1
thf(fact_963_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self2
thf(fact_964_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self3
thf(fact_965_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self4
thf(fact_966_one__le__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N2 ) )
= ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
& ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_967_mult__le__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_968_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_969_div__mult__self__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ M @ N2 ) @ N2 )
= M ) ) ).
% div_mult_self_is_m
thf(fact_970_div__mult__self1__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ N2 @ M ) @ N2 )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_971_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_972_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_973_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A4: A,B4: A] : ( times_times @ A @ B4 @ A4 ) ) ) ) ).
% mult.commute
thf(fact_974_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.left_commute
thf(fact_975_not__iless0,axiom,
! [N2: extended_enat] :
~ ( ord_less @ extended_enat @ N2 @ ( zero_zero @ extended_enat ) ) ).
% not_iless0
thf(fact_976_enat__less__induct,axiom,
! [P: extended_enat > $o,N2: extended_enat] :
( ! [N3: extended_enat] :
( ! [M6: extended_enat] :
( ( ord_less @ extended_enat @ M6 @ N3 )
=> ( P @ M6 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% enat_less_induct
thf(fact_977_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_978_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_979_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_980_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_981_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_982_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_983_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_984_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,E3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ E3 ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_985_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% distrib_right
thf(fact_986_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% distrib_left
thf(fact_987_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_988_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_989_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_990_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( A2 != B2 )
& ( C2 != D2 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A2 @ D2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_991_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W2: A,Y: A,X3: A,Z2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W2 @ Y ) @ ( times_times @ A @ X3 @ Z2 ) )
= ( plus_plus @ A @ ( times_times @ A @ W2 @ Z2 ) @ ( times_times @ A @ X3 @ Y ) ) )
= ( ( W2 = X3 )
| ( Y = Z2 ) ) ) ) ).
% crossproduct_eq
thf(fact_992_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,Y: A,Z2: A,W2: A] :
( ( times_times @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ Z2 @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ Y @ W2 ) ) ) ) ).
% times_divide_times_eq
thf(fact_993_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,Y: A,Z2: A,W2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ Z2 @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X3 @ W2 ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ).
% divide_divide_times_eq
thf(fact_994_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( divide_divide @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_995_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times @ nat @ ( suc @ K ) @ M )
= ( times_times @ nat @ ( suc @ K ) @ N2 ) )
= ( M = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_996_mult__0,axiom,
! [N2: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_997_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N2 ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_998_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_999_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1000_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1001_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_1002_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1003_add__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N2 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1004_add__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M @ N2 ) @ K )
= ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1005_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1006_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1007_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times @ nat @ N2 @ ( one_one @ nat ) )
= N2 ) ).
% nat_mult_1_right
thf(fact_1008_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1009_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_1010_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_1011_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_1012_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_1013_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_1014_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_1015_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_1016_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_1017_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1018_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_1019_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_1020_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1021_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_1022_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_1023_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1024_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_1025_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_1026_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_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1036_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1037_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1038_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1039_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1040_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1041_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1042_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1043_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1044_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R2: A,A2: A,B2: A,C2: A,D2: A] :
( ( R2
!= ( zero_zero @ A ) )
=> ( ( ( A2 = B2 )
& ( C2 != D2 ) )
=> ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R2 @ C2 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R2 @ D2 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1045_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: A,N2: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N2 ) ) ) ) ) ).
% less_1_mult
thf(fact_1046_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X3: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X3 @ Y )
= ( divide_divide @ A @ W2 @ Z2 ) )
= ( ( times_times @ A @ X3 @ Z2 )
= ( times_times @ A @ W2 @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1047_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ( divide_divide @ A @ B2 @ C2 )
= A2 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_1048_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_1049_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( B2
= ( times_times @ A @ A2 @ C2 ) )
=> ( ( divide_divide @ A @ B2 @ C2 )
= A2 ) ) ) ) ).
% divide_eq_imp
thf(fact_1050_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C2 )
= B2 )
=> ( A2
= ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_1051_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ C2 )
= A2 )
= ( B2
= ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1052_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( A2
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( times_times @ A @ A2 @ C2 )
= B2 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1053_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A,N2: nat] :
( ( power_power @ A @ A2 @ ( suc @ N2 ) )
= ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).
% power_Suc
thf(fact_1054_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N2: nat] :
( ( power_power @ A @ A2 @ ( suc @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ N2 ) @ A2 ) ) ) ).
% power_Suc2
thf(fact_1055_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1056_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: nat,N2: nat] :
( ( power_power @ A @ A2 @ ( plus_plus @ nat @ M @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).
% power_add
thf(fact_1057_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1058_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1059_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N2 ) )
= ( M = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1060_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1061_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1062_mult__Suc,axiom,
! [M: nat,N2: nat] :
( ( times_times @ nat @ ( suc @ M ) @ N2 )
= ( plus_plus @ nat @ N2 @ ( times_times @ nat @ M @ N2 ) ) ) ).
% mult_Suc
thf(fact_1063_mult__eq__self__implies__10,axiom,
! [M: nat,N2: nat] :
( ( M
= ( times_times @ nat @ M @ N2 ) )
=> ( ( N2
= ( one_one @ nat ) )
| ( M
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1064_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( times_times @ nat @ I @ N2 ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N2 ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1065_times__div__less__eq__dividend,axiom,
! [N2: nat,M: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ ( divide_divide @ nat @ M @ N2 ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1066_div__times__less__eq__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N2 ) @ N2 ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1067_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N2: nat] :
( ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( times_times @ A @ A2 @ ( power_power @ A @ ( power_power @ A @ A2 @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_1068_Suc__double__not__eq__double,axiom,
! [M: nat,N2: nat] :
( ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
!= ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% Suc_double_not_eq_double
thf(fact_1069_double__not__eq__Suc__double,axiom,
! [M: nat,N2: nat] :
( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
!= ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).
% double_not_eq_Suc_double
thf(fact_1070_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1071_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1072_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_1073_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1074_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1075_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_1076_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1077_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1078_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1079_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1080_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_1081_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_1082_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1083_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1084_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ A2 ) ) ) ) ).
% mult_left_le
thf(fact_1085_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_1086_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X3 @ Y ) @ X3 ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1087_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X3 ) @ X3 ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1088_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1089_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_1090_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) )
= ( ( X3
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1091_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X3: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_1092_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1093_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_1094_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_1095_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_1096_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_1097_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_1098_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X3: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ Z2 ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1099_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ ( times_times @ A @ Z2 @ Y ) @ X3 )
=> ( ord_less @ A @ Z2 @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1100_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_1101_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_1102_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( numeral_numeral @ A @ W2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1103_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ( numeral_numeral @ A @ W2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1104_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A2: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1105_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A2: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) )
= A2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1106_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X3: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_1107_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,X3: A,Z2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Y ) @ Z2 )
= ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% add_frac_num
thf(fact_1108_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X3: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ X3 @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% add_num_frac
thf(fact_1109_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X3 @ ( divide_divide @ A @ Y @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ) ).
% add_divide_eq_iff
thf(fact_1110_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% divide_add_eq_iff
thf(fact_1111_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_1112_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_1113_one__less__mult,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_1114_n__less__m__mult__n,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N2 @ ( times_times @ nat @ M @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1115_n__less__n__mult__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N2 @ ( times_times @ nat @ N2 @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1116_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1117_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( divide_divide @ nat @ M @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1118_div__less__iff__less__mult,axiom,
! [Q4: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q4 ) @ N2 )
= ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ Q4 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1119_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1120_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1121_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1122_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1123_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1124_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1125_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1126_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1127_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: 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 @ X3 ) @ Y ) ) )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_1128_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_1129_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_1130_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_1131_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_1132_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_1133_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_1134_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_1135_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X3: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ Z2 ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1136_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ Y ) @ X3 )
=> ( ord_less_eq @ A @ Z2 @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1137_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_1138_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X3: A,A2: A,Y: A,U: A,V2: A] :
( ( ord_less_eq @ A @ X3 @ A2 )
=> ( ( ord_less_eq @ A @ Y @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X3 ) @ ( times_times @ A @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1139_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_1140_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_1141_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N2: 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 @ N2 ) ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ).
% power_Suc_less
thf(fact_1142_mult__2,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 )
= ( plus_plus @ A @ Z2 @ Z2 ) ) ) ).
% mult_2
thf(fact_1143_mult__2__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z2: A] :
( ( times_times @ A @ Z2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ Z2 @ Z2 ) ) ) ).
% mult_2_right
thf(fact_1144_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_1145_div__nat__eqI,axiom,
! [N2: nat,Q4: nat,M: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ Q4 ) @ M )
=> ( ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ ( suc @ Q4 ) ) )
=> ( ( divide_divide @ nat @ M @ N2 )
= Q4 ) ) ) ).
% div_nat_eqI
thf(fact_1146_less__eq__div__iff__mult__less__eq,axiom,
! [Q4: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 )
=> ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N2 @ Q4 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q4 ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1147_split__div,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( divide_divide @ nat @ M @ N2 ) )
= ( ( ( N2
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ! [I3: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N2 )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N2 @ I3 ) @ J3 ) )
=> ( P @ I3 ) ) ) ) ) ) ).
% split_div
thf(fact_1148_dividend__less__div__times,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N2 ) @ N2 ) ) ) ) ).
% dividend_less_div_times
thf(fact_1149_dividend__less__times__div,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ N2 @ ( divide_divide @ nat @ M @ N2 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1150_nat__bit__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_bit_induct
thf(fact_1151_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X3: A,A2: A,Y: A,U: A,V2: A] :
( ( ord_less @ A @ X3 @ A2 )
=> ( ( ord_less @ A @ Y @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X3 ) @ ( times_times @ A @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1152_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_1153_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_1154_split__div_H,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( divide_divide @ nat @ M @ N2 ) )
= ( ( ( N2
= ( zero_zero @ nat ) )
& ( P @ ( zero_zero @ nat ) ) )
| ? [Q5: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ Q5 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ ( suc @ Q5 ) ) )
& ( P @ Q5 ) ) ) ) ).
% split_div'
thf(fact_1155_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X3: A,Y: A] :
( ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( 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 ) ) @ X3 ) @ Y ) ) ) ) ).
% power2_sum
thf(fact_1156_zero__le__even__power_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% zero_le_even_power'
thf(fact_1157_triangle__def,axiom,
( nat_triangle
= ( ^ [N: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% triangle_def
thf(fact_1158_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_1159_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_1160_arith__geo__mean,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,X3: A,Y: A] :
( ( ( power_power @ A @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ X3 @ Y ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ U @ ( divide_divide @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_1161_sum__squares__bound,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) @ Y ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_squares_bound
thf(fact_1162_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_1163_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_1164_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_1165_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_1166_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_1167_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_1168_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_1169_zle__add1__eq__le,axiom,
! [W2: int,Z2: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) )
= ( ord_less_eq @ int @ W2 @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1170_set__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less @ int @ ( bit_se5668285175392031749et_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% set_bit_negative_int_iff
thf(fact_1171_unset__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less @ int @ ( bit_se2638667681897837118et_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% unset_bit_negative_int_iff
thf(fact_1172_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L )
=> ( ( divide_divide @ int @ K @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_pos_pos_trivial
thf(fact_1173_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_neg_neg_trivial
thf(fact_1174_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_1175_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_1176_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_1177_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1178_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I @ K ) )
= ( ord_less_eq @ int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1179_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_1180_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_1181_div__positive__int,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ L @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) ) ) ) ).
% div_positive_int
thf(fact_1182_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) )
= ( ( K
= ( zero_zero @ int ) )
| ( L
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) )
| ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ).
% div_int_pos_iff
thf(fact_1183_zless__imp__add1__zle,axiom,
! [W2: int,Z2: int] :
( ( ord_less @ int @ W2 @ Z2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1184_zdiv__mono2__neg,axiom,
! [A2: int,B7: int,B2: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ( ( ord_less_eq @ int @ B7 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B7 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1185_zdiv__mono1__neg,axiom,
! [A2: int,A7: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A7 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A7 @ B2 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1186_int__div__pos__eq,axiom,
! [A2: int,B2: int,Q4: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q4 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1187_int__div__neg__eq,axiom,
! [A2: int,B2: int,Q4: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q4 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1188_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide @ int @ I @ K )
= ( zero_zero @ int ) )
= ( ( K
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
& ( ord_less @ int @ I @ K ) )
| ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
& ( ord_less @ int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1189_zdiv__mono2,axiom,
! [A2: int,B7: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ( ( ord_less_eq @ int @ B7 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B7 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1190_zdiv__mono1,axiom,
! [A2: int,A7: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A7 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A7 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_1191_split__zdiv,axiom,
! [P: int > $o,N2: int,K: int] :
( ( P @ ( divide_divide @ int @ N2 @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I3: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_zdiv
thf(fact_1192_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1193_add1__zle__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z2 )
= ( ord_less @ int @ W2 @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1194_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ Z2 @ ( zero_zero @ int ) ) ) ).
% odd_less_0_iff
thf(fact_1195_pos__zmult__eq__1__iff,axiom,
! [M: int,N2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M )
=> ( ( ( times_times @ int @ M @ N2 )
= ( one_one @ int ) )
= ( ( M
= ( one_one @ int ) )
& ( N2
= ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1196_zless__add1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) )
= ( ( ord_less @ int @ W2 @ Z2 )
| ( W2 = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1197_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less @ int @ K @ I )
=> ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I4: int] :
( ( ord_less @ int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1198_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_1199_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_1200_int__div__less__self,axiom,
! [X3: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K )
=> ( ord_less @ int @ ( divide_divide @ int @ X3 @ K ) @ X3 ) ) ) ).
% int_div_less_self
thf(fact_1201_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_1202_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_1203_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_1204_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times @ int @ K @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_1205_bot__enat__def,axiom,
( ( bot_bot @ extended_enat )
= ( zero_zero @ extended_enat ) ) ).
% bot_enat_def
thf(fact_1206_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( plus_plus @ int @ X5 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus @ int @ X2 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1207_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less @ int @ I @ J )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ord_less @ int @ ( times_times @ int @ K @ I ) @ ( times_times @ int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1208_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_1209_q__pos__lemma,axiom,
! [B7: int,Q6: int,R3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B7 @ Q6 ) @ R3 ) )
=> ( ( ord_less @ int @ R3 @ B7 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q6 ) ) ) ) ).
% q_pos_lemma
thf(fact_1210_zdiv__mono2__lemma,axiom,
! [B2: int,Q4: int,R2: int,B7: int,Q6: int,R3: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B7 @ Q6 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B7 @ Q6 ) @ R3 ) )
=> ( ( ord_less @ int @ R3 @ B7 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ( ( ord_less_eq @ int @ B7 @ B2 )
=> ( ord_less_eq @ int @ Q4 @ Q6 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_1211_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q4: int,R2: int,B7: int,Q6: int,R3: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B7 @ Q6 ) @ R3 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B7 @ Q6 ) @ R3 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ( ( ord_less_eq @ int @ B7 @ B2 )
=> ( ord_less_eq @ int @ Q6 @ Q4 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_1212_unique__quotient__lemma,axiom,
! [B2: int,Q6: int,R3: int,Q4: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R3 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
=> ( ( ord_less @ int @ R3 @ B2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ord_less_eq @ int @ Q6 @ Q4 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_1213_unique__quotient__lemma__neg,axiom,
! [B2: int,Q6: int,R3: int,Q4: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R3 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( ord_less @ int @ B2 @ R3 )
=> ( ord_less_eq @ int @ Q4 @ Q6 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_1214_enat__0__less__mult__iff,axiom,
! [M: extended_enat,N2: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ ( times_times @ extended_enat @ M @ N2 ) )
= ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ M )
& ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 ) ) ) ).
% enat_0_less_mult_iff
thf(fact_1215_realpow__pos__nth2,axiom,
! [A2: real,N2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
& ( ( power_power @ real @ R4 @ ( suc @ N2 ) )
= A2 ) ) ) ).
% realpow_pos_nth2
thf(fact_1216_realpow__pos__nth__unique,axiom,
! [N2: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [X5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X5 )
& ( ( power_power @ real @ X5 @ N2 )
= A2 )
& ! [Y5: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y5 )
& ( ( power_power @ real @ Y5 @ N2 )
= A2 ) )
=> ( Y5 = X5 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1217_realpow__pos__nth,axiom,
! [N2: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
& ( ( power_power @ real @ R4 @ N2 )
= A2 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1218_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ X3 ) @ ( times_times @ A @ Z2 @ Y ) )
= ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1219_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ Y @ Z2 ) )
= ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1220_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q4: A,R2: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q4 @ R2 ) )
= ( R2
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_1221_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X: nat,N: nat] : ( modulo_modulo @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% low_def
thf(fact_1222_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1223_set__decode__Suc,axiom,
! [N2: nat,X3: nat] :
( ( member @ nat @ ( suc @ N2 ) @ ( nat_set_decode @ X3 ) )
= ( member @ nat @ N2 @ ( nat_set_decode @ ( divide_divide @ nat @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_1224_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_1225_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ Y @ Z2 ) )
= ( ord_less @ A @ X3 @ Y ) ) ) ) ).
% mult_less_iff1
thf(fact_1226_num_Osize__gen_I2_J,axiom,
! [X22: num] :
( ( size_num @ ( bit0 @ X22 ) )
= ( plus_plus @ nat @ ( size_num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(2)
thf(fact_1227_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd @ nat @ M @ ( one_one @ nat ) )
= ( M
= ( one_one @ nat ) ) ) ).
% nat_dvd_1_iff_1
thf(fact_1228_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_1229_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ A2 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_1230_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_1231_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_1232_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_1233_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_1234_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% mod_by_0
thf(fact_1235_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_1236_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_1237_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( M
= ( suc @ ( zero_zero @ nat ) ) ) ) ).
% dvd_1_iff_1
thf(fact_1238_div__dvd__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( divide_divide @ A @ C2 @ A2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ) ).
% div_dvd_div
thf(fact_1239_not__real__square__gt__zero,axiom,
! [X3: real] :
( ( ~ ( ord_less @ real @ ( zero_zero @ real ) @ ( times_times @ real @ X3 @ X3 ) ) )
= ( X3
= ( zero_zero @ real ) ) ) ).
% not_real_square_gt_zero
thf(fact_1240_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_1241_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_1242_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( dvd_dvd @ nat @ M @ N2 ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1243_mod__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( modulo_modulo @ nat @ M @ N2 )
= M ) ) ).
% mod_less
thf(fact_1244_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1245_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_1246_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_1247_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_1248_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_1249_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1250_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A2 ) @ B2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1251_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_1252_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_1253_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_1254_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_1255_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_1256_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_1257_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_1258_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_1259_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_1260_div__add,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% div_add
thf(fact_1261_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_1262_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_1263_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self4
thf(fact_1264_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self3
thf(fact_1265_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self2
thf(fact_1266_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self1
thf(fact_1267_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_1268_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_1269_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_1270_set__encode__inverse,axiom,
! [A3: set @ nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ( nat_set_decode @ ( nat_set_encode @ A3 ) )
= A3 ) ) ).
% set_encode_inverse
thf(fact_1271_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_1272_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_1273_even__Suc,axiom,
! [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).
% even_Suc
thf(fact_1274_even__Suc__Suc__iff,axiom,
! [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ N2 ) ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% even_Suc_Suc_iff
thf(fact_1275_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N2: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_1276_Suc__mod__mult__self4,axiom,
! [N2: nat,K: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ K ) @ M ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_mod_mult_self4
thf(fact_1277_Suc__mod__mult__self3,axiom,
! [K: nat,N2: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N2 ) @ M ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_mod_mult_self3
thf(fact_1278_Suc__mod__mult__self2,axiom,
! [M: nat,N2: nat,K: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N2 @ K ) ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_mod_mult_self2
thf(fact_1279_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ K @ N2 ) ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_mod_mult_self1
thf(fact_1280_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_1281_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_1282_odd__Suc__div__two,axiom,
! [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% odd_Suc_div_two
thf(fact_1283_even__Suc__div__two,axiom,
! [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% even_Suc_div_two
thf(fact_1284_mod2__Suc__Suc,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% mod2_Suc_Suc
thf(fact_1285_Suc__times__numeral__mod__eq,axiom,
! [K: num,N2: nat] :
( ( ( numeral_numeral @ nat @ K )
!= ( one_one @ nat ) )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N2 ) ) @ ( numeral_numeral @ nat @ K ) )
= ( one_one @ nat ) ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_1286_set__decode__0,axiom,
! [X3: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X3 ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X3 ) ) ) ).
% set_decode_0
thf(fact_1287_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_1288_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_1289_power__less__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq
thf(fact_1290_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_1291_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_1292_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_1293_not__mod2__eq__Suc__0__eq__0,axiom,
! [N2: nat] :
( ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
!= ( suc @ ( zero_zero @ nat ) ) )
= ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ nat ) ) ) ).
% not_mod2_eq_Suc_0_eq_0
thf(fact_1294_add__self__mod__2,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ nat ) ) ).
% add_self_mod_2
thf(fact_1295_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_1296_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_1297_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_1298_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_1299_even__power,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A2 @ N2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% even_power
thf(fact_1300_mod2__gr__0,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ nat ) ) ) ).
% mod2_gr_0
thf(fact_1301_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_1302_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_1303_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_1304_real__arch__pow,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ? [N3: nat] : ( ord_less @ real @ Y @ ( power_power @ real @ X3 @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1305_real__arch__pow__inv,axiom,
! [Y: real,X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ? [N3: nat] : ( ord_less @ real @ ( power_power @ real @ X3 @ N3 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1306_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_1307_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_1308_zdvd__period,axiom,
! [A2: int,D2: int,X3: int,T2: int,C2: int] :
( ( dvd_dvd @ int @ A2 @ D2 )
=> ( ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ X3 @ T2 ) )
= ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ ( plus_plus @ int @ X3 @ ( times_times @ int @ C2 @ D2 ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_1309_zdvd__reduce,axiom,
! [K: int,N2: int,M: int] :
( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N2 @ ( times_times @ int @ K @ M ) ) )
= ( dvd_dvd @ int @ K @ N2 ) ) ).
% zdvd_reduce
thf(fact_1310_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X: real,Y2: real] :
( ( ord_less @ real @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% less_eq_real_def
thf(fact_1311_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_1312_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A4: A,B4: A] :
( ( modulo_modulo @ A @ B4 @ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_1313_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_1314_dvd__refl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ A2 @ A2 ) ) ).
% dvd_refl
thf(fact_1315_dvd__trans,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ C2 )
=> ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_trans
thf(fact_1316_dvd__mod__iff,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( dvd_dvd @ A @ C2 @ A2 ) ) ) ) ).
% dvd_mod_iff
thf(fact_1317_dvd__mod__imp__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ A2 ) ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_1318_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq @ int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1319_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_1320_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ N2 ) )
= ( ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_1321_mod__add__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_add_right_eq
thf(fact_1322_mod__add__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_add_left_eq
thf(fact_1323_mod__add__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,A7: A,B2: A,B7: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A7 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B7 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A7 @ B7 ) @ C2 ) ) ) ) ) ).
% mod_add_cong
thf(fact_1324_mod__add__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_add_eq
thf(fact_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__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A4: A,B4: A] :
( ( A4
= ( zero_zero @ A ) )
=> ( B4
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_1327_mod__Suc__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ N2 ) ) ).
% mod_Suc_Suc_eq
thf(fact_1328_mod__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ).
% mod_Suc_eq
thf(fact_1329_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P4: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ P4 @ ( times_times @ A @ A2 @ B2 ) )
=> ~ ! [X5: A,Y4: A] :
( ( P4
= ( times_times @ A @ X5 @ Y4 ) )
=> ( ( dvd_dvd @ A @ X5 @ A2 )
=> ~ ( dvd_dvd @ A @ Y4 @ B2 ) ) ) ) ) ).
% dvd_productE
thf(fact_1330_division__decomp,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
=> ? [B9: A,C6: A] :
( ( A2
= ( times_times @ A @ B9 @ C6 ) )
& ( dvd_dvd @ A @ B9 @ B2 )
& ( dvd_dvd @ A @ C6 @ C2 ) ) ) ) ).
% division_decomp
thf(fact_1331_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ~ ! [K2: A] :
( A2
!= ( times_times @ A @ B2 @ K2 ) ) ) ) ).
% dvdE
thf(fact_1332_dvdI,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [A2: A,B2: A,K: A] :
( ( A2
= ( times_times @ A @ B2 @ K ) )
=> ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% dvdI
thf(fact_1333_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B4: A,A4: A] :
? [K3: A] :
( A4
= ( times_times @ A @ B4 @ K3 ) ) ) ) ) ).
% dvd_def
thf(fact_1334_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult
thf(fact_1335_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult2
thf(fact_1336_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ A2 @ C2 ) ) ) ).
% dvd_mult_left
thf(fact_1337_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_1338_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_1339_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ B2 @ C2 ) ) ) ).
% dvd_mult_right
thf(fact_1340_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_1341_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A2 ) ) ).
% one_dvd
thf(fact_1342_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_1343_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_1344_dvd__add__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_add_right_iff
thf(fact_1345_dvd__add__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_add_left_iff
thf(fact_1346_dvd__add,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ C2 )
=> ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ) ).
% dvd_add
thf(fact_1347_dvd__div__eq__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( ( divide_divide @ A @ A2 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_1348_dvd__div__eq__cancel,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
=> ( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( A2 = B2 ) ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_1349_div__div__div__same,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [D2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ D2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_div_div_same
thf(fact_1350_mod__less__eq__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N2 ) @ M ) ).
% mod_less_eq_dividend
thf(fact_1351_gcd__nat_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1352_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_1353_gcd__nat_Oextremum__unique,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_unique
thf(fact_1354_gcd__nat_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
& ( ( zero_zero @ nat )
!= A2 ) ) ).
% gcd_nat.extremum_strict
thf(fact_1355_gcd__nat_Oextremum,axiom,
! [A2: nat] : ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% gcd_nat.extremum
thf(fact_1356_zdvd__antisym__nonneg,axiom,
! [M: int,N2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( dvd_dvd @ int @ M @ N2 )
=> ( ( dvd_dvd @ int @ N2 @ M )
=> ( M = N2 ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_1357_zdvd__mult__cancel,axiom,
! [K: int,M: int,N2: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N2 ) )
=> ( ( K
!= ( zero_zero @ int ) )
=> ( dvd_dvd @ int @ M @ N2 ) ) ) ).
% zdvd_mult_cancel
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,
! [M: nat,N2: nat] :
( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% 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,C2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ B2 @ C2 ) )
=> ~ ! [D3: A] :
( B2
!= ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ D3 ) ) ) ) ) ).
% mod_eqE
thf(fact_1364_div__add1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_add1_eq
thf(fact_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 ) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X2: B] :
( ( ord_less @ B @ X2 @ Z3 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ) ).
% minf(10)
thf(fact_1367_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X2: B] :
( ( ord_less @ B @ X2 @ Z3 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ).
% minf(9)
thf(fact_1368_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X2: B] :
( ( ord_less @ B @ Z3 @ X2 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ) ).
% pinf(10)
thf(fact_1369_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X2: B] :
( ( ord_less @ B @ Z3 @ X2 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ 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_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B2 @ A2 )
= ( times_times @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_1372_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A2 @ B2 )
= ( times_times @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_1373_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_1374_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_1375_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_1376_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_1377_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_1378_mod__Suc,axiom,
! [M: nat,N2: nat] :
( ( ( ( suc @ ( modulo_modulo @ nat @ M @ N2 ) )
= N2 )
=> ( ( modulo_modulo @ nat @ ( suc @ M ) @ N2 )
= ( zero_zero @ nat ) ) )
& ( ( ( suc @ ( modulo_modulo @ nat @ M @ N2 ) )
!= N2 )
=> ( ( modulo_modulo @ nat @ ( suc @ M ) @ N2 )
= ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) ) ).
% mod_Suc
thf(fact_1379_finite__set__decode,axiom,
! [N2: nat] : ( finite_finite2 @ nat @ ( nat_set_decode @ N2 ) ) ).
% finite_set_decode
thf(fact_1380_mod__induct,axiom,
! [P: nat > $o,N2: nat,P4: nat,M: nat] :
( ( P @ N2 )
=> ( ( ord_less @ nat @ N2 @ P4 )
=> ( ( ord_less @ nat @ M @ P4 )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ P4 )
=> ( ( P @ N3 )
=> ( P @ ( modulo_modulo @ nat @ ( suc @ N3 ) @ P4 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_1381_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ) ).
% dvd_div_mult
thf(fact_1382_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% div_mult_swap
thf(fact_1383_div__div__eq__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_1384_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_1385_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 )
=> ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_1386_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,D2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( dvd_dvd @ A @ D2 @ C2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( divide_divide @ A @ C2 @ D2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_1387_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [M3: nat] : ( P @ M3 @ ( zero_zero @ nat ) )
=> ( ! [M3: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo @ nat @ M3 @ N3 ) )
=> ( P @ M3 @ N3 ) ) )
=> ( P @ M @ N2 ) ) ) ).
% gcd_nat_induct
thf(fact_1388_mod__less__divisor,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ ( modulo_modulo @ nat @ M @ N2 ) @ N2 ) ) ).
% mod_less_divisor
thf(fact_1389_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_1390_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_1391_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= ( divide_divide @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_div_cancel
thf(fact_1392_div__plus__div__distrib__dvd__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_1393_div__plus__div__distrib__dvd__left,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_1394_mod__Suc__le__divisor,axiom,
! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ ( suc @ N2 ) ) @ N2 ) ).
% mod_Suc_le_divisor
thf(fact_1395_mod__eq__0D,axiom,
! [M: nat,D2: nat] :
( ( ( modulo_modulo @ nat @ M @ D2 )
= ( zero_zero @ nat ) )
=> ? [Q2: nat] :
( M
= ( times_times @ nat @ D2 @ Q2 ) ) ) ).
% mod_eq_0D
thf(fact_1396_le__imp__power__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N2: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).
% le_imp_power_dvd
thf(fact_1397_power__le__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N2: nat,B2: A,M: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N2 ) @ B2 )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ B2 ) ) ) ) ).
% power_le_dvd
thf(fact_1398_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X3: A,Y: A,N2: nat,M: nat] :
( ( dvd_dvd @ A @ X3 @ Y )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( dvd_dvd @ A @ ( power_power @ A @ X3 @ N2 ) @ ( power_power @ A @ Y @ M ) ) ) ) ) ).
% dvd_power_le
thf(fact_1399_nat__mod__eq__iff,axiom,
! [X3: nat,N2: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X3 @ N2 )
= ( modulo_modulo @ nat @ Y @ N2 ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus @ nat @ X3 @ ( times_times @ nat @ N2 @ Q1 ) )
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N2 @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_1400_nat__dvd__not__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N2 )
=> ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_1401_dvd__pos__nat,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( dvd_dvd @ nat @ M @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) ) ) ).
% dvd_pos_nat
thf(fact_1402_zdvd__imp__le,axiom,
! [Z2: int,N2: int] :
( ( dvd_dvd @ int @ Z2 @ N2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less_eq @ int @ Z2 @ N2 ) ) ) ).
% zdvd_imp_le
thf(fact_1403_zdvd__not__zless,axiom,
! [M: int,N2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M )
=> ( ( ord_less @ int @ M @ N2 )
=> ~ ( dvd_dvd @ int @ N2 @ M ) ) ) ).
% zdvd_not_zless
thf(fact_1404_bezout__add__nat,axiom,
! [A2: nat,B2: nat] :
? [D3: nat,X5: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D3 @ A2 )
& ( dvd_dvd @ nat @ D3 @ B2 )
& ( ( ( times_times @ nat @ A2 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D3 ) )
| ( ( times_times @ nat @ B2 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y4 ) @ D3 ) ) ) ) ).
% bezout_add_nat
thf(fact_1405_bezout__lemma__nat,axiom,
! [D2: nat,A2: nat,B2: nat,X3: nat,Y: nat] :
( ( dvd_dvd @ nat @ D2 @ A2 )
=> ( ( dvd_dvd @ nat @ D2 @ B2 )
=> ( ( ( ( times_times @ nat @ A2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y ) @ D2 ) )
| ( ( times_times @ nat @ B2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y ) @ D2 ) ) )
=> ? [X5: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D2 @ A2 )
& ( dvd_dvd @ nat @ D2 @ ( plus_plus @ nat @ A2 @ B2 ) )
& ( ( ( times_times @ nat @ A2 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ Y4 ) @ D2 ) )
| ( ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y4 ) @ D2 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1406_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_1407_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_1408_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_1409_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_1410_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num,Q4: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ Q4 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(2)
thf(fact_1411_cong__exp__iff__simps_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ one2 ) )
= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(1)
thf(fact_1412_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_1413_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_1414_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_1415_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_1416_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_1417_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A2 @ C2 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_1418_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A2 @ C2 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_1419_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_mult1_eq
thf(fact_1420_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [C3: A] :
( B2
!= ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% unit_dvdE
thf(fact_1421_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X: A] : ( P @ ( times_times @ A @ L @ X ) ) )
= ( ? [X: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X @ ( zero_zero @ A ) ) )
& ( P @ X ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_1422_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= C2 )
= ( B2
= ( times_times @ A @ C2 @ A2 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_1423_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_1424_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_1425_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D2 )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= ( divide_divide @ A @ D2 @ C2 ) )
= ( ( times_times @ A @ B2 @ C2 )
= ( times_times @ A @ A2 @ D2 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_1426_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_1427_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_1428_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_1429_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% unit_div_commute
thf(fact_1430_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_1431_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( A2
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( times_times @ A @ A2 @ B2 )
= C2 ) ) ) ) ).
% unit_eq_div2
thf(fact_1432_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= C2 )
= ( A2
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_1433_mod__le__divisor,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N2 ) @ N2 ) ) ).
% mod_le_divisor
thf(fact_1434_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,N2: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N2 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_1435_div__less__mono,axiom,
! [A3: nat,B3: nat,N2: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( modulo_modulo @ nat @ A3 @ N2 )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B3 @ N2 )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A3 @ N2 ) @ ( divide_divide @ nat @ B3 @ N2 ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1436_mod__eq__nat1E,axiom,
! [M: nat,Q4: nat,N2: nat] :
( ( ( modulo_modulo @ nat @ M @ Q4 )
= ( modulo_modulo @ nat @ N2 @ Q4 ) )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ~ ! [S2: nat] :
( M
!= ( plus_plus @ nat @ N2 @ ( times_times @ nat @ Q4 @ S2 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_1437_mod__eq__nat2E,axiom,
! [M: nat,Q4: nat,N2: nat] :
( ( ( modulo_modulo @ nat @ M @ Q4 )
= ( modulo_modulo @ nat @ N2 @ Q4 ) )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ~ ! [S2: nat] :
( N2
!= ( plus_plus @ nat @ M @ ( times_times @ nat @ Q4 @ S2 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_1438_nat__mod__eq__lemma,axiom,
! [X3: nat,N2: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X3 @ N2 )
= ( modulo_modulo @ nat @ Y @ N2 ) )
=> ( ( ord_less_eq @ nat @ Y @ X3 )
=> ? [Q2: nat] :
( X3
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N2 @ Q2 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_1439_dvd__imp__le,axiom,
! [K: nat,N2: nat] :
( ( dvd_dvd @ nat @ K @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ nat @ K @ N2 ) ) ) ).
% dvd_imp_le
thf(fact_1440_div__mod__decomp,axiom,
! [A3: nat,N2: nat] :
( A3
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A3 @ N2 ) @ N2 ) @ ( modulo_modulo @ nat @ A3 @ N2 ) ) ) ).
% div_mod_decomp
thf(fact_1441_mod__mult2__eq,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( modulo_modulo @ nat @ M @ ( times_times @ nat @ N2 @ Q4 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M @ N2 ) @ Q4 ) ) @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ).
% mod_mult2_eq
thf(fact_1442_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( dvd_dvd @ nat @ M @ N2 ) ) ) ).
% dvd_mult_cancel
thf(fact_1443_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( dvd_dvd @ nat @ M @ N2 ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_1444_bezout__add__strong__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ? [D3: nat,X5: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D3 @ A2 )
& ( dvd_dvd @ nat @ D3 @ B2 )
& ( ( times_times @ nat @ A2 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D3 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1445_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_1446_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [B5: A] :
( ( B5
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B5 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= B5 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B5 )
= A2 )
=> ( ( ( times_times @ A @ A2 @ B5 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A2 )
!= ( times_times @ A @ C2 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_1447_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_1448_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_1449_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_1450_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X3: A,M: nat,N2: nat] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ X3 @ N2 ) )
= ( ( dvd_dvd @ A @ X3 @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ) ).
% dvd_power_iff
thf(fact_1451_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N2: nat,X3: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
| ( X3
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X3 @ ( power_power @ A @ X3 @ N2 ) ) ) ) ).
% dvd_power
thf(fact_1452_split__mod,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( modulo_modulo @ nat @ M @ N2 ) )
= ( ( ( N2
= ( zero_zero @ nat ) )
=> ( P @ M ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ! [I3: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N2 )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N2 @ I3 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_1453_dvd__mult__cancel1,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ M @ N2 ) @ M )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel1
thf(fact_1454_dvd__mult__cancel2,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ N2 @ M ) @ M )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel2
thf(fact_1455_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N2: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N2 ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% power_dvd_imp_le
thf(fact_1456_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_1457_product__nth,axiom,
! [A: $tType,B: $tType,N2: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N2 @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) @ N2 )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1458_power__mono__odd,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,A2: A,B2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ).
% power_mono_odd
thf(fact_1459_Suc__times__mod__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M @ N2 ) ) @ M )
= ( one_one @ nat ) ) ) ).
% Suc_times_mod_eq
thf(fact_1460_odd__pos,axiom,
! [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% odd_pos
thf(fact_1461_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ( dvd_dvd @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% dvd_power_iff_le
thf(fact_1462_even__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
| ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_unset_bit_iff
thf(fact_1463_even__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( M
!= ( zero_zero @ nat ) ) ) ) ) ).
% even_set_bit_iff
thf(fact_1464_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_1465_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_1466_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_1467_div__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N2: nat,M: nat] :
( ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_1468_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ~ ! [B5: A] :
( A2
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_1469_zero__le__even__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).
% zero_le_even_power
thf(fact_1470_zero__le__odd__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,A2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% zero_le_odd_power
thf(fact_1471_zero__le__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_le_power_eq
thf(fact_1472_verit__le__mono__div,axiom,
! [A3: nat,B3: nat,N2: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A3 @ N2 )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B3 @ N2 )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B3 @ N2 ) ) ) ) ).
% verit_le_mono_div
thf(fact_1473_even__set__encode__iff,axiom,
! [A3: set @ nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A3 ) )
= ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 ) ) ) ) ).
% even_set_encode_iff
thf(fact_1474_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_1475_zero__less__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) )
= ( ( N2
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( A2
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_less_power_eq
thf(fact_1476_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: A,X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ( modulo_modulo @ A @ X3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ A @ X3 @ M ) )
| ( ( modulo_modulo @ A @ X3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X3 @ M ) @ M ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_1477_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A2: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N2 ) @ 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 @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_1478_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A2: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N2 ) @ 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 @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_1479_power__le__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( A2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_1480_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_1481_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A2: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N2 ) @ 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 @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_1482_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q4: int,R2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q4 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_1483_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N2 ) @ 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 @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_1484_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M: nat,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( ( ord_less @ nat @ N2 @ M )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M @ N2 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_1485_even__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
!= ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_flip_bit_iff
thf(fact_1486_one__mod__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% one_mod_2_pow_eq
thf(fact_1487_set__encode__insert,axiom,
! [A3: set @ nat,N2: nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ~ ( member @ nat @ N2 @ A3 )
=> ( ( nat_set_encode @ ( insert @ nat @ N2 @ A3 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).
% set_encode_insert
thf(fact_1488_insert__absorb2,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A3 ) )
= ( insert @ A @ X3 @ A3 ) ) ).
% insert_absorb2
thf(fact_1489_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_1490_insertCI,axiom,
! [A: $tType,A2: A,B3: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_1491_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_1492_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_1493_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_1494_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_1495_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_1496_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_1497_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_1498_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_1499_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_1500_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_1501_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_1502_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_1503_finite__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) )
= ( finite_finite2 @ A @ A3 ) ) ).
% finite_insert
thf(fact_1504_insert__subset,axiom,
! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 )
= ( ( member @ A @ X3 @ B3 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_1505_diff__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( minus_minus @ nat @ M @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1506_Suc__diff__diff,axiom,
! [M: nat,N2: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1507_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_1508_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_1509_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1510_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1511_flip__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less @ int @ ( bit_se8732182000553998342ip_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% flip_bit_negative_int_iff
thf(fact_1512_of__bool__less__eq__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P: $o,Q: $o] :
( ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
= ( P
=> Q ) ) ) ).
% of_bool_less_eq_iff
thf(fact_1513_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_1514_of__bool__eq__0__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: $o] :
( ( ( zero_neq_one_of_bool @ A @ P )
= ( zero_zero @ A ) )
= ~ P ) ) ).
% of_bool_eq_0_iff
thf(fact_1515_of__bool__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P: $o,Q: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
= ( ~ P
& Q ) ) ) ).
% of_bool_less_iff
thf(fact_1516_of__bool__eq__1__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: $o] :
( ( ( zero_neq_one_of_bool @ A @ P )
= ( one_one @ A ) )
= P ) ) ).
% of_bool_eq_1_iff
thf(fact_1517_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_1518_signed__take__bit__of__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% signed_take_bit_of_0
thf(fact_1519_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_1520_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_1521_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_1522_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_1523_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_1524_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_1525_div__diff,axiom,
! [A: $tType] :
( ( idom_modulo @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% div_diff
thf(fact_1526_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1527_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A3: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A3 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1528_zero__less__of__bool__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) )
= P ) ) ).
% zero_less_of_bool_iff
thf(fact_1529_zero__less__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% zero_less_diff
thf(fact_1530_diff__is__0__eq,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1531_diff__is__0__eq_H,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( minus_minus @ nat @ M @ N2 )
= ( zero_zero @ nat ) ) ) ).
% diff_is_0_eq'
thf(fact_1532_of__bool__less__one__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) )
= ~ P ) ) ).
% of_bool_less_one_iff
thf(fact_1533_of__bool__not__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P: $o] :
( ( zero_neq_one_of_bool @ A @ ~ P )
= ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ) ).
% of_bool_not_iff
thf(fact_1534_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1535_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
= ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1536_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1537_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( suc @ N2 ) @ ( one_one @ nat ) )
= N2 ) ).
% diff_Suc_1
thf(fact_1538_Suc__0__mod__eq,axiom,
! [N2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( zero_neq_one_of_bool @ nat
@ ( N2
!= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% Suc_0_mod_eq
thf(fact_1539_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L )
=> ( ( modulo_modulo @ int @ K @ L )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_1540_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K )
=> ( ( modulo_modulo @ int @ K @ L )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_1541_signed__take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N2 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_Suc_1
thf(fact_1542_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1543_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J @ K ) ) @ I )
= ( minus_minus @ nat @ ( suc @ J ) @ ( plus_plus @ nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1544_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ I @ ( suc @ ( minus_minus @ nat @ J @ K ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1545_signed__take__bit__Suc__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_1546_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1547_set__replicate,axiom,
! [A: $tType,N2: nat,X3: A] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N2 @ X3 ) )
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_1548_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_1549_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_1550_odd__Suc__minus__one,axiom,
! [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
= N2 ) ) ).
% odd_Suc_minus_one
thf(fact_1551_even__diff__nat,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N2 ) )
= ( ( ord_less @ nat @ M @ N2 )
| ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ).
% even_diff_nat
thf(fact_1552_semiring__parity__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) )
= ( N2
= ( zero_zero @ nat ) ) ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_1553_one__div__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( zero_neq_one_of_bool @ A
@ ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% one_div_2_pow_eq
thf(fact_1554_bits__1__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N2: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( zero_neq_one_of_bool @ A
@ ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% bits_1_div_exp
thf(fact_1555_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_1556_mod__int__unique,axiom,
! [K: int,L: int,Q4: int,R2: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
=> ( ( modulo_modulo @ int @ K @ L )
= R2 ) ) ).
% mod_int_unique
thf(fact_1557_gcd__nat_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ~ ( ( dvd_dvd @ nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% gcd_nat.asym
thf(fact_1558_gcd__nat_Orefl,axiom,
! [A2: nat] : ( dvd_dvd @ nat @ A2 @ A2 ) ).
% gcd_nat.refl
thf(fact_1559_gcd__nat_Otrans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ B2 @ C2 )
=> ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ).
% gcd_nat.trans
thf(fact_1560_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : Y3 = Z )
= ( ^ [A4: nat,B4: nat] :
( ( dvd_dvd @ nat @ A4 @ B4 )
& ( dvd_dvd @ nat @ B4 @ A4 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_1561_gcd__nat_Oirrefl,axiom,
! [A2: nat] :
~ ( ( dvd_dvd @ nat @ A2 @ A2 )
& ( A2 != A2 ) ) ).
% gcd_nat.irrefl
thf(fact_1562_gcd__nat_Oantisym,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% gcd_nat.antisym
thf(fact_1563_gcd__nat_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( ( ( dvd_dvd @ nat @ B2 @ C2 )
& ( B2 != C2 ) )
=> ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_1564_gcd__nat_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( ( dvd_dvd @ nat @ B2 @ C2 )
& ( B2 != C2 ) )
=> ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_1565_gcd__nat_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( ( dvd_dvd @ nat @ B2 @ C2 )
=> ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_1566_gcd__nat_Ostrict__iff__not,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
= ( ( dvd_dvd @ nat @ A2 @ B2 )
& ~ ( dvd_dvd @ nat @ B2 @ A2 ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_1567_gcd__nat_Oorder__iff__strict,axiom,
( ( dvd_dvd @ nat )
= ( ^ [A4: nat,B4: nat] :
( ( ( dvd_dvd @ nat @ A4 @ B4 )
& ( A4 != B4 ) )
| ( A4 = B4 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_1568_gcd__nat_Ostrict__iff__order,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
= ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_1569_gcd__nat_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( dvd_dvd @ nat @ A2 @ B2 ) ) ).
% gcd_nat.strict_implies_order
thf(fact_1570_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( A2 != B2 ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_1571_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_1572_dvd__antisym,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd @ nat @ M @ N2 )
=> ( ( dvd_dvd @ nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% dvd_antisym
thf(fact_1573_dvd__diff__nat,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd @ nat @ K @ M )
=> ( ( dvd_dvd @ nat @ K @ N2 )
=> ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ).
% dvd_diff_nat
thf(fact_1574_eucl__rel__int,axiom,
! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K @ L ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% eucl_rel_int
thf(fact_1575_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ? [B10: set @ A] :
( ( A3
= ( insert @ A @ A2 @ B10 ) )
& ~ ( member @ A @ A2 @ B10 ) ) ) ).
% mk_disjoint_insert
thf(fact_1576_insert__commute,axiom,
! [A: $tType,X3: A,Y: A,A3: set @ A] :
( ( insert @ A @ X3 @ ( insert @ A @ Y @ A3 ) )
= ( insert @ A @ Y @ ( insert @ A @ X3 @ A3 ) ) ) ).
% insert_commute
thf(fact_1577_insert__eq__iff,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A,B3: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ B2 @ B3 )
=> ( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ B3 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B3 ) )
& ( ( A2 != B2 )
=> ? [C7: set @ A] :
( ( A3
= ( insert @ A @ B2 @ C7 ) )
& ~ ( member @ A @ B2 @ C7 )
& ( B3
= ( insert @ A @ A2 @ C7 ) )
& ~ ( member @ A @ A2 @ C7 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1578_insert__absorb,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_1579_insert__ident,axiom,
! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ~ ( member @ A @ X3 @ B3 )
=> ( ( ( insert @ A @ X3 @ A3 )
= ( insert @ A @ X3 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_1580_Set_Oset__insert,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( member @ A @ X3 @ A3 )
=> ~ ! [B10: set @ A] :
( ( A3
= ( insert @ A @ X3 @ B10 ) )
=> ( member @ A @ X3 @ B10 ) ) ) ).
% Set.set_insert
thf(fact_1581_insertI2,axiom,
! [A: $tType,A2: A,B3: set @ A,B2: A] :
( ( member @ A @ A2 @ B3 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% insertI2
thf(fact_1582_insertI1,axiom,
! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ).
% insertI1
thf(fact_1583_insertE,axiom,
! [A: $tType,A2: A,B2: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_1584_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P4: $o,Q4: $o] :
( ( ( zero_neq_one_of_bool @ A @ P4 )
= ( zero_neq_one_of_bool @ A @ Q4 ) )
= ( P4 = Q4 ) ) ) ).
% of_bool_eq_iff
thf(fact_1585_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1586_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% diff_right_commute
thf(fact_1587_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( A2 = B2 )
= ( C2 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_1588_full__exhaustive__int_H_Ocases,axiom,
! [X3: product_prod @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
~ ! [F4: ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: int,I4: int] :
( X3
!= ( product_Pair @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D3 @ I4 ) ) ) ).
% full_exhaustive_int'.cases
thf(fact_1589_exhaustive__int_H_Ocases,axiom,
! [X3: product_prod @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
~ ! [F4: int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: int,I4: int] :
( X3
!= ( product_Pair @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D3 @ I4 ) ) ) ).
% exhaustive_int'.cases
thf(fact_1590_small__lazy_H_Ocases,axiom,
! [X3: product_prod @ int @ int] :
~ ! [D3: int,I4: int] :
( X3
!= ( product_Pair @ int @ int @ D3 @ I4 ) ) ).
% small_lazy'.cases
thf(fact_1591_unique__quotient,axiom,
! [A2: int,B2: int,Q4: int,R2: int,Q6: int,R3: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q6 @ R3 ) )
=> ( Q4 = Q6 ) ) ) ).
% unique_quotient
thf(fact_1592_unique__remainder,axiom,
! [A2: int,B2: int,Q4: int,R2: int,Q6: int,R3: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q6 @ R3 ) )
=> ( R2 = R3 ) ) ) ).
% unique_remainder
thf(fact_1593_of__bool__conj,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P
& Q ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_conj
thf(fact_1594_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D2 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_1595_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_1596_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_1597_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1598_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [A4: A,B4: A] :
( ( minus_minus @ A @ A4 @ B4 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1599_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D2 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_1600_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
= ( ord_less @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_1601_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_1602_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_1603_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% right_diff_distrib'
thf(fact_1604_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A2 )
= ( minus_minus @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) ) ) ) ).
% left_diff_distrib'
thf(fact_1605_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% right_diff_distrib
thf(fact_1606_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% left_diff_distrib
thf(fact_1607_add__diff__add,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) )
= ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( minus_minus @ A @ C2 @ D2 ) ) ) ) ).
% add_diff_add
thf(fact_1608_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( minus_minus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_1609_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= C2 )
= ( A2
= ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_1610_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( A2
= ( minus_minus @ A @ C2 @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_1611_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_1612_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_1613_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_1614_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1615_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ( plus_plus @ A @ C2 @ B2 )
= A2 )
=> ( C2
= ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_1616_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% diff_diff_eq
thf(fact_1617_diff__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% diff_divide_distrib
thf(fact_1618_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_1619_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_1620_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C2: A,D2: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C2 )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1621_insert__not__empty,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ A3 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_1622_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_1623_finite_OinsertI,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) ) ) ).
% finite.insertI
thf(fact_1624_insert__subsetI,axiom,
! [A: $tType,X3: A,A3: set @ A,X8: set @ A] :
( ( member @ A @ X3 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X8 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ X8 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_1625_insert__mono,axiom,
! [A: $tType,C5: set @ A,D6: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ D6 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C5 ) @ ( insert @ A @ A2 @ D6 ) ) ) ).
% insert_mono
thf(fact_1626_subset__insert,axiom,
! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_1627_subset__insertI,axiom,
! [A: $tType,B3: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( insert @ A @ A2 @ B3 ) ) ).
% subset_insertI
thf(fact_1628_subset__insertI2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% subset_insertI2
thf(fact_1629_dvd__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( dvd_dvd @ A @ X3 @ Y )
=> ( ( dvd_dvd @ A @ X3 @ Z2 )
=> ( dvd_dvd @ A @ X3 @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ) ).
% dvd_diff
thf(fact_1630_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1631_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_1632_diffs0__imp__equal,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N2 @ M )
= ( zero_zero @ nat ) )
=> ( M = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1633_diff__less__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( ord_less @ nat @ M @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1634_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1635_dvd__minus__self,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ M ) )
= ( ( ord_less @ nat @ N2 @ M )
| ( dvd_dvd @ nat @ M @ N2 ) ) ) ).
% dvd_minus_self
thf(fact_1636_eq__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ( minus_minus @ nat @ M @ K )
= ( minus_minus @ nat @ N2 @ K ) )
= ( M = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1637_le__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1638_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( minus_minus @ nat @ M @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1639_diff__le__mono,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1640_diff__le__self,axiom,
! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ).
% diff_le_self
thf(fact_1641_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq @ nat @ A2 @ C2 )
=> ( ( ord_less_eq @ nat @ B2 @ C2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A2 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
= ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1642_diff__le__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1643_dvd__diffD,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) )
=> ( ( dvd_dvd @ nat @ K @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( dvd_dvd @ nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_1644_dvd__diffD1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) )
=> ( ( dvd_dvd @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( dvd_dvd @ nat @ K @ N2 ) ) ) ) ).
% dvd_diffD1
thf(fact_1645_less__eq__dvd__minus,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( dvd_dvd @ nat @ M @ N2 )
= ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1646_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) )
= ( minus_minus @ nat @ M @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1647_diff__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) )
= ( minus_minus @ nat @ M @ N2 ) ) ).
% diff_cancel2
thf(fact_1648_diff__add__inverse,axiom,
! [N2: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ N2 )
= M ) ).
% diff_add_inverse
thf(fact_1649_diff__add__inverse2,axiom,
! [M: nat,N2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ N2 )
= M ) ).
% diff_add_inverse2
thf(fact_1650_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) )
= ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_1651_diff__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K )
= ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1652_bezout1__nat,axiom,
! [A2: nat,B2: nat] :
? [D3: nat,X5: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D3 @ A2 )
& ( dvd_dvd @ nat @ D3 @ B2 )
& ( ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X5 ) @ ( times_times @ nat @ B2 @ Y4 ) )
= D3 )
| ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X5 ) @ ( times_times @ nat @ A2 @ Y4 ) )
= D3 ) ) ) ).
% bezout1_nat
thf(fact_1653_eucl__rel__int__by0,axiom,
! [K: int] : ( eucl_rel_int @ K @ ( zero_zero @ int ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K ) ) ).
% eucl_rel_int_by0
thf(fact_1654_div__int__unique,axiom,
! [K: int,L: int,Q4: int,R2: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
=> ( ( divide_divide @ int @ K @ L )
= Q4 ) ) ).
% div_int_unique
thf(fact_1655_zero__less__eq__of__bool,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% zero_less_eq_of_bool
thf(fact_1656_of__bool__less__eq__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) ) ) ).
% of_bool_less_eq_one
thf(fact_1657_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ~ ( ( P4
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P4
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_1658_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ( P4
=> ( P @ ( one_one @ A ) ) )
& ( ~ P4
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_1659_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_1660_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_1661_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B4: A] : ( ord_less @ A @ ( minus_minus @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_1662_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A2 )
= C2 )
= ( B2
= ( plus_plus @ A @ C2 @ A2 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1663_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_1664_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1665_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1666_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1667_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 )
= ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1668_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1669_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1670_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% le_add_diff
thf(fact_1671_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_1672_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% le_diff_eq
thf(fact_1673_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_1674_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N2: A,J: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N2 )
=> ( ( ord_less_eq @ A @ N2 @ ( plus_plus @ A @ J @ K ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N2 )
=> ( ( ord_less_eq @ A @ N2 @ ( plus_plus @ A @ J @ K ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N2 @ K ) @ J ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1675_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N2 )
=> ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N2 @ K ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_1676_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% less_diff_eq
thf(fact_1677_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( ord_less @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_1678_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_1679_mult__diff__mult,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [X3: A,Y: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X3 @ Y ) @ ( times_times @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ X3 @ ( minus_minus @ A @ Y @ B2 ) ) @ ( times_times @ A @ ( minus_minus @ A @ X3 @ A2 ) @ B2 ) ) ) ) ).
% mult_diff_mult
thf(fact_1680_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X3: A,Y: A] :
( ( minus_minus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) )
= ( times_times @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( minus_minus @ A @ X3 @ Y ) ) ) ) ).
% square_diff_square_factored
thf(fact_1681_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ).
% eq_add_iff2
thf(fact_1682_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C2 )
= D2 ) ) ) ).
% eq_add_iff1
thf(fact_1683_finite_Ocases,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A8: set @ A] :
( ? [A5: A] :
( A2
= ( insert @ A @ A5 @ A8 ) )
=> ~ ( finite_finite2 @ A @ A8 ) ) ) ) ).
% finite.cases
thf(fact_1684_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A4: set @ A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ? [A6: set @ A,B4: A] :
( ( A4
= ( insert @ A @ B4 @ A6 ) )
& ( finite_finite2 @ A @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_1685_finite__induct,axiom,
! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X5 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X5 @ F6 ) ) ) ) )
=> ( P @ F5 ) ) ) ) ).
% finite_induct
thf(fact_1686_finite__ne__induct,axiom,
! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( F5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] : ( P @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X5: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( F6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X5 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X5 @ F6 ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1687_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] :
( ! [A8: set @ A] :
( ~ ( finite_finite2 @ A @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X5 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X5 @ F6 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1688_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( modulo_modulo @ int @ K @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_1689_neg__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less @ int @ L @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% neg_mod_bound
thf(fact_1690_subset__singleton__iff,axiom,
! [A: $tType,X8: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X8 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X8
= ( bot_bot @ ( set @ A ) ) )
| ( X8
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_1691_subset__singletonD,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( A3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
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,
! [M: nat,N2: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N2 ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1694_Suc__diff__Suc,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ N2 @ M )
=> ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N2 ) ) )
= ( minus_minus @ nat @ M @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_1695_diff__less,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ) ) ).
% diff_less
thf(fact_1696_Suc__diff__le,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N2 )
= ( suc @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1697_diff__less__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_less_eq @ nat @ C2 @ A2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1698_less__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( ord_less @ nat @ M @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1699_diff__add__0,axiom,
! [N2: nat,M: nat] :
( ( minus_minus @ nat @ N2 @ ( plus_plus @ nat @ N2 @ M ) )
= ( zero_zero @ nat ) ) ).
% diff_add_0
thf(fact_1700_add__diff__inverse__nat,axiom,
! [M: nat,N2: nat] :
( ~ ( ord_less @ nat @ M @ N2 )
=> ( ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ M @ N2 ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1701_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1702_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( minus_minus @ nat @ J @ I )
= K )
= ( J
= ( plus_plus @ nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1703_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K )
= ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1704_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
= ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1705_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1706_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
= ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1707_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N2: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N2 ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1708_mod__geq,axiom,
! [M: nat,N2: nat] :
( ~ ( ord_less @ nat @ M @ N2 )
=> ( ( modulo_modulo @ nat @ M @ N2 )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ).
% mod_geq
thf(fact_1709_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M2: nat,N: nat] : ( if @ nat @ ( ord_less @ nat @ M2 @ N ) @ M2 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ) ).
% mod_if
thf(fact_1710_le__mod__geq,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( modulo_modulo @ nat @ M @ N2 )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ).
% le_mod_geq
thf(fact_1711_mod__eq__dvd__iff__nat,axiom,
! [N2: nat,M: nat,Q4: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( ( modulo_modulo @ nat @ M @ Q4 )
= ( modulo_modulo @ nat @ N2 @ Q4 ) )
= ( dvd_dvd @ nat @ Q4 @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_1712_ordered__ring__class_Ole__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1713_ordered__ring__class_Ole__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C2 ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1714_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
= ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ).
% less_add_iff2
thf(fact_1715_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C2 ) @ D2 ) ) ) ).
% less_add_iff1
thf(fact_1716_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ X3 @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_1717_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X3 @ ( divide_divide @ A @ Y @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_1718_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X3: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1719_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A2: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) )
= A2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A2 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1720_square__diff__one__factored,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: A] :
( ( minus_minus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( one_one @ A ) )
= ( times_times @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) @ ( minus_minus @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% square_diff_one_factored
thf(fact_1721_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S3: set @ B,P: ( set @ B ) > $o,F2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B,S5: set @ B] :
( ( finite_finite2 @ B @ S5 )
=> ( ! [Y5: B] :
( ( member @ B @ Y5 @ S5 )
=> ( ord_less_eq @ A @ ( F2 @ Y5 ) @ ( F2 @ X5 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert @ B @ X5 @ S5 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_1722_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B5: A,A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( ord_less @ A @ B5 @ X2 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert @ A @ B5 @ A8 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1723_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B5: A,A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( ord_less @ A @ X2 @ B5 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert @ A @ B5 @ A8 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1724_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D6: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D6 )
=> ! [X2: A,K4: A] :
( ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X2 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K4 @ D6 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_1725_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D6: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D6 )
=> ! [X2: A,K4: A] :
( ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X2 @ T2 ) )
= ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K4 @ D6 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_1726_finite__subset__induct,axiom,
! [A: $tType,F5: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A5 @ A3 )
=> ( ~ ( member @ A @ A5 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ A5 @ F6 ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1727_finite__subset__induct_H,axiom,
! [A: $tType,F5: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A5 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ F6 @ A3 )
=> ( ~ ( member @ A @ A5 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ A5 @ F6 ) ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1728_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_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__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_1731_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_1732_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q4: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( K
= ( times_times @ int @ Q4 @ L ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_1733_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_1734_neg__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ K @ L ) @ ( zero_zero @ int ) ) ) ).
% neg_mod_sign
thf(fact_1735_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_1736_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_1737_zmod__trivial__iff,axiom,
! [I: int,K: int] :
( ( ( modulo_modulo @ int @ I @ K )
= I )
= ( ( K
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
& ( ord_less @ int @ I @ K ) )
| ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
& ( ord_less @ int @ K @ I ) ) ) ) ).
% zmod_trivial_iff
thf(fact_1738_mod__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( ( L
= ( zero_zero @ int ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_1739_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_1740_zdiv__mono__strict,axiom,
! [A3: int,B3: int,N2: int] :
( ( ord_less @ int @ A3 @ B3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( ( modulo_modulo @ int @ A3 @ N2 )
= ( zero_zero @ int ) )
=> ( ( ( modulo_modulo @ int @ B3 @ N2 )
= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( divide_divide @ int @ A3 @ N2 ) @ ( divide_divide @ int @ B3 @ N2 ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_1741_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less @ nat @ A2 @ B2 )
& ~ ( P @ ( zero_zero @ nat ) ) )
| ? [D5: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1742_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( ( ( ord_less @ nat @ A2 @ B2 )
=> ( P @ ( zero_zero @ nat ) ) )
& ! [D5: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1743_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
= ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1744_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1745_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1746_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1747_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1748_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( M
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1749_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1750_dvd__minus__add,axiom,
! [Q4: nat,N2: nat,R2: nat,M: nat] :
( ( ord_less_eq @ nat @ Q4 @ N2 )
=> ( ( ord_less_eq @ nat @ Q4 @ ( times_times @ nat @ R2 @ M ) )
=> ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ Q4 ) )
= ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ ( times_times @ nat @ R2 @ M ) @ Q4 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_1751_set__update__subset__insert,axiom,
! [A: $tType,Xs: list @ A,I: nat,X3: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X3 ) ) @ ( insert @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_1752_mod__nat__eqI,axiom,
! [R2: nat,N2: nat,M: nat] :
( ( ord_less @ nat @ R2 @ N2 )
=> ( ( ord_less_eq @ nat @ R2 @ M )
=> ( ( dvd_dvd @ nat @ N2 @ ( minus_minus @ nat @ M @ R2 ) )
=> ( ( modulo_modulo @ nat @ M @ N2 )
= R2 ) ) ) ) ).
% mod_nat_eqI
thf(fact_1753_exp__div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N2: nat] :
( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
!= ( zero_zero @ A ) )
& ( ord_less_eq @ nat @ N2 @ M ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ) ).
% exp_div_exp_eq
thf(fact_1754_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X3: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1755_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X3: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_1756_signed__take__bit__int__less__exp,axiom,
! [N2: nat,K: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% signed_take_bit_int_less_exp
thf(fact_1757_power__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,N2: nat,M: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N2 ) )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ) ).
% power_diff
thf(fact_1758_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
= ( plus_plus @ int @ K @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_1759_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N2 )
= ( minus_minus @ nat @ M @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1760_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( N2
= ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).
% Suc_pred'
thf(fact_1761_div__geq,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( ord_less @ nat @ M @ N2 )
=> ( ( divide_divide @ nat @ M @ N2 )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% div_geq
thf(fact_1762_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M2: nat,N: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M2 @ N )
| ( N
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ) ) ).
% div_if
thf(fact_1763_set__replicate__Suc,axiom,
! [A: $tType,N2: nat,X3: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N2 ) @ X3 ) )
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_1764_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M2: nat,N: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N ) ) ) ) ) ).
% add_eq_if
thf(fact_1765_set__replicate__conv__if,axiom,
! [A: $tType,N2: nat,X3: A] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N2 @ X3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N2 @ X3 ) )
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_1766_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1767_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_1768_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M2: nat,N: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N ) ) ) ) ) ).
% mult_eq_if
thf(fact_1769_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V2: A,R2: A,S: A] :
( ( ord_less_eq @ A @ U @ V2 )
=> ( ( 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 @ V2 @ U ) ) @ S ) ) @ V2 ) ) ) ) ) ).
% scaling_mono
thf(fact_1770_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N2: nat,M: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) )
!= ( zero_zero @ A ) ) ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_1771_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_1772_signed__take__bit__int__less__self__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_1773_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,N2: nat,M: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) )
= ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N2 @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1774_split__zmod,axiom,
! [P: int > $o,N2: int,K: int] :
( ( P @ ( modulo_modulo @ int @ N2 @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ N2 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I3: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_1775_int__mod__neg__eq,axiom,
! [A2: int,B2: int,Q4: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ 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_1776_int__mod__pos__eq,axiom,
! [A2: int,B2: int,Q4: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ 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_1777_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P6: A,M2: nat] :
( if @ A
@ ( M2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1778_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N2: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( times_times @ A @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ A2 )
= ( power_power @ A @ A2 @ N2 ) ) ) ) ).
% power_minus_mult
thf(fact_1779_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N2 ) @ ( minus_minus @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N2 ) ) ) ) ).
% diff_le_diff_pow
thf(fact_1780_le__div__geq,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( divide_divide @ nat @ M @ N2 )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% le_div_geq
thf(fact_1781_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A2: A] :
( ! [A5: A] :
( ( ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A5 )
=> ( P @ A5 ) )
=> ( ! [A5: A,B5: $o] :
( ( P @ A5 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A5 )
=> ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% bits_induct
thf(fact_1782_verit__le__mono__div__int,axiom,
! [A3: int,B3: int,N2: int] :
( ( ord_less @ int @ A3 @ B3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ N2 )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B3 @ N2 )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B3 @ N2 ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_1783_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( P @ ( divide_divide @ int @ N2 @ K ) @ ( modulo_modulo @ int @ N2 @ K ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_1784_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N2: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ( P @ ( divide_divide @ int @ N2 @ K ) @ ( modulo_modulo @ int @ N2 @ K ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_1785_int__power__div__base,axiom,
! [M: nat,K: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( divide_divide @ int @ ( power_power @ int @ K @ M ) @ K )
= ( power_power @ int @ K @ ( minus_minus @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1786_set__decode__plus__power__2,axiom,
! [N2: nat,Z2: nat] :
( ~ ( member @ nat @ N2 @ ( nat_set_decode @ Z2 ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ Z2 ) )
= ( insert @ nat @ N2 @ ( nat_set_decode @ Z2 ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_1787_exp__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N2: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M @ N2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% exp_mod_exp
thf(fact_1788_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X3: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X3 @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( 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 ) ) @ X3 ) @ Y ) ) ) ) ).
% power2_diff
thf(fact_1789_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N2: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_1790_even__mod__4__div__2,axiom,
! [N2: nat] :
( ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( suc @ ( zero_zero @ nat ) ) )
=> ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% even_mod_4_div_2
thf(fact_1791_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q4: int,R2: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
= ( ( K
= ( plus_plus @ int @ ( times_times @ int @ L @ Q4 ) @ 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 ) )
=> ( Q4
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_1792_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_1793_even__mask__div__iff_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% even_mask_div_iff'
thf(fact_1794_even__mask__div__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
= ( zero_zero @ A ) )
| ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).
% even_mask_div_iff
thf(fact_1795_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R2: A,Q4: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R2 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ ( 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 @ Q4 @ R2 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ R2 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_1796_inrange,axiom,
! [T2: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( 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 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ) ).
% inrange
thf(fact_1797_Suc__if__eq,axiom,
! [A: $tType,F2: nat > A,H2: nat > A,G: A,N2: nat] :
( ! [N3: nat] :
( ( F2 @ ( suc @ N3 ) )
= ( H2 @ N3 ) )
=> ( ( ( F2 @ ( zero_zero @ nat ) )
= G )
=> ( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( F2 @ N2 )
= G ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( F2 @ N2 )
= ( H2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% Suc_if_eq
thf(fact_1798_Divides_Oadjust__div__eq,axiom,
! [Q4: int,R2: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
= ( plus_plus @ int @ Q4
@ ( zero_neq_one_of_bool @ int
@ ( R2
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_1799_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N: nat,A4: A] :
( if @ A
@ ( N
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_1800_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q4: int,R2: int] :
( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A2 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q4 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_1801_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ( minus_minus @ A @ X3 @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% diff_shunt_var
thf(fact_1802_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N: nat,A4: A] :
( if @ A
@ ( N
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_1803_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_1804_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_1805_Diff__cancel,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_1806_empty__Diff,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_1807_Diff__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Diff_empty
thf(fact_1808_finite__Diff2,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_Diff2
thf(fact_1809_finite__Diff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% finite_Diff
thf(fact_1810_Compl__subset__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1811_Compl__anti__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_1812_Diff__insert0,axiom,
! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_1813_insert__Diff1,axiom,
! [A: $tType,X3: A,B3: set @ A,A3: set @ A] :
( ( member @ A @ X3 @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_1814_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% compl_le_compl_iff
thf(fact_1815_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_1816_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_1817_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_1818_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_1819_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_1820_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_1821_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less @ A @ Y @ X3 ) ) ) ).
% compl_less_compl_iff
thf(fact_1822_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_1823_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_1824_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_1825_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_1826_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_1827_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_1828_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_1829_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_1830_dvd__minus__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X3: A,Y: A] :
( ( dvd_dvd @ A @ X3 @ ( uminus_uminus @ A @ Y ) )
= ( dvd_dvd @ A @ X3 @ Y ) ) ) ).
% dvd_minus_iff
thf(fact_1831_minus__dvd__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X3: A,Y: A] :
( ( dvd_dvd @ A @ ( uminus_uminus @ A @ X3 ) @ Y )
= ( dvd_dvd @ A @ X3 @ Y ) ) ) ).
% minus_dvd_iff
thf(fact_1832_Diff__eq__empty__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_1833_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_1834_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_1835_insert__Diff__single,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A2 @ A3 ) ) ).
% insert_Diff_single
thf(fact_1836_finite__Diff__insert,axiom,
! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% finite_Diff_insert
thf(fact_1837_take__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% take_bit_of_0
thf(fact_1838_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_1839_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_1840_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_1841_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_1842_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_1843_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_1844_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_1845_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_1846_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_1847_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_1848_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_1849_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_1850_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_1851_add__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_1852_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_1853_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_1854_divide__minus1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A] :
( ( divide_divide @ A @ X3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ X3 ) ) ) ).
% divide_minus1
thf(fact_1855_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_1856_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_1857_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_1858_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_1859_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_1860_subset__Compl__singleton,axiom,
! [A: $tType,A3: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_1861_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_1862_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_1863_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( B2 = C2 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_1864_take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_Suc_1
thf(fact_1865_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_1866_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_1867_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_1868_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_1869_take__bit__of__1__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( one_one @ A ) )
= ( zero_zero @ A ) )
= ( N2
= ( zero_zero @ nat ) ) ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_1870_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W2 ) ) ) @ Y ) ) ) ).
% semiring_norm(168)
thf(fact_1871_neg__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N2: num] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( ord_less_eq @ num @ N2 @ M ) ) ) ).
% neg_numeral_le_iff
thf(fact_1872_neg__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N2: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( ord_less @ num @ N2 @ M ) ) ) ).
% neg_numeral_less_iff
thf(fact_1873_zle__diff1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq @ int @ W2 @ ( minus_minus @ int @ Z2 @ ( one_one @ int ) ) )
= ( ord_less @ int @ W2 @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1874_take__bit__of__Suc__0,axiom,
! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_neq_one_of_bool @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% take_bit_of_Suc_0
thf(fact_1875_not__neg__one__le__neg__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) )
= ( M != one2 ) ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_1876_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_1877_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_1878_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_1879_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_1880_neg__numeral__less__neg__one__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( M != one2 ) ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_1881_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_1882_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_1883_take__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% take_bit_of_1
thf(fact_1884_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_1885_even__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) )
= ( ( N2
= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_take_bit_eq
thf(fact_1886_signed__take__bit__Suc__minus__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_1887_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_1888_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_1889_take__bit__of__exp,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N2 @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% take_bit_of_exp
thf(fact_1890_take__bit__of__2,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_of_2
thf(fact_1891_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_1892_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% compl_mono
thf(fact_1893_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X3 ) )
=> ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_1894_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_1895_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_1896_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_1897_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X3 ) )
=> ( ord_less @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_less_swap1
thf(fact_1898_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X3 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) ) ) ).
% compl_less_swap2
thf(fact_1899_Compl__insert,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_1900_take__bit__add,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ B2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% take_bit_add
thf(fact_1901_take__bit__tightened,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A2: A,B2: A,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ B2 ) )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ M @ B2 ) ) ) ) ) ).
% take_bit_tightened
thf(fact_1902_take__bit__nat__less__eq__self,axiom,
! [N2: nat,M: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) @ M ) ).
% take_bit_nat_less_eq_self
thf(fact_1903_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M @ Q4 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ Q4 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_1904_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_1905_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_1906_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_1907_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_1908_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_1909_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_1910_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_1911_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_1912_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_1913_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_1914_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K: A,A2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( uminus_uminus @ A @ A3 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_1915_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_1916_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_1917_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_1918_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_1919_Diff__infinite__finite,axiom,
! [A: $tType,T4: set @ A,S3: set @ A] :
( ( finite_finite2 @ A @ T4 )
=> ( ~ ( finite_finite2 @ A @ S3 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S3 @ T4 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1920_double__diff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 )
=> ( ( minus_minus @ ( set @ A ) @ B3 @ ( minus_minus @ ( set @ A ) @ C5 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_1921_Diff__subset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ A3 ) ).
% Diff_subset
thf(fact_1922_Diff__mono,axiom,
! [A: $tType,A3: set @ A,C5: set @ A,D6: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ D6 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ C5 @ D6 ) ) ) ) ).
% Diff_mono
thf(fact_1923_insert__Diff__if,axiom,
! [A: $tType,X3: A,B3: set @ A,A3: set @ A] :
( ( ( member @ A @ X3 @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) )
& ( ~ ( member @ A @ X3 @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 )
= ( insert @ A @ X3 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1924_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% minus_int_code(1)
thf(fact_1925_uminus__int__code_I1_J,axiom,
( ( uminus_uminus @ int @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% uminus_int_code(1)
thf(fact_1926_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_1927_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_1928_psubset__imp__ex__mem,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1929_zdvd__zdiffD,axiom,
! [K: int,M: int,N2: int] :
( ( dvd_dvd @ int @ K @ ( minus_minus @ int @ M @ N2 ) )
=> ( ( dvd_dvd @ int @ K @ N2 )
=> ( dvd_dvd @ int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_1930_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_1931_take__bit__minus__small__eq,axiom,
! [K: int,N2: nat] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ K ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_1932_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_1933_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_1934_take__bit__tightened__less__eq__int,axiom,
! [M: nat,N2: nat,K: int] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_1935_take__bit__int__greater__self__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% take_bit_int_greater_self_iff
thf(fact_1936_not__take__bit__negative,axiom,
! [N2: nat,K: int] :
~ ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) ).
% not_take_bit_negative
thf(fact_1937_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat,A2: A,B2: A] :
( ( ( bit_ri4674362597316999326ke_bit @ A @ N2 @ A2 )
= ( bit_ri4674362597316999326ke_bit @ A @ N2 @ B2 ) )
= ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ B2 ) ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
thf(fact_1938_signed__take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N2: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) )
= ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N2 @ M ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 ) @ ( bit_ri4674362597316999326ke_bit @ A @ M ) @ A2 ) ) ) ).
% signed_take_bit_take_bit
thf(fact_1939_ex__nat__less,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
& ( P @ M2 ) ) )
= ( ? [X: nat] :
( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less
thf(fact_1940_all__nat__less,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( P @ M2 ) ) )
= ( ! [X: nat] :
( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less
thf(fact_1941_neg__numeral__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% neg_numeral_le_numeral
thf(fact_1942_not__numeral__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N2: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_1943_zero__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N2: num] :
( ( zero_zero @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% zero_neq_neg_numeral
thf(fact_1944_not__numeral__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_1945_neg__numeral__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% neg_numeral_less_numeral
thf(fact_1946_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_1947_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_1948_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_1949_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_1950_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_1951_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_1952_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_1953_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_1954_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_1955_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_1956_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_1957_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_1958_square__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [X3: A] :
( ( ( times_times @ A @ X3 @ X3 )
= ( one_one @ A ) )
= ( ( X3
= ( one_one @ A ) )
| ( X3
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% square_eq_1_iff
thf(fact_1959_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B3: A,K: A,B2: A,A2: A] :
( ( B3
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( minus_minus @ A @ A2 @ B3 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_1960_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A4: A,B4: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1961_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A4: A,B4: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1962_take__bit__unset__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N2 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( bit_se2638667681897837118et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_1963_take__bit__set__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N2 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( bit_se5668285175392031749et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_1964_take__bit__flip__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N2 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( bit_se8732182000553998342ip_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_1965_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_1966_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_1967_subset__Compl__self__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_1968_Diff__insert__absorb,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1969_Diff__insert2,axiom,
! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_1970_insert__Diff,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1971_Diff__insert,axiom,
! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_1972_subset__Diff__insert,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X3: A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ ( insert @ A @ X3 @ C5 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ C5 ) )
& ~ ( member @ A @ X3 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1973_real__0__less__add__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X3 @ Y ) )
= ( ord_less @ real @ ( uminus_uminus @ real @ X3 ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1974_real__add__less__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( plus_plus @ real @ X3 @ Y ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ Y @ ( uminus_uminus @ real @ X3 ) ) ) ).
% real_add_less_0_iff
thf(fact_1975_zminus1__lemma,axiom,
! [A2: int,B2: int,Q4: int,R2: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
=> ( ( B2
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A2 ) @ B2
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R2
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q4 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q4 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B2 @ R2 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_1976_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq @ int @ I @ K )
=> ( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq @ int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1977_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less @ int @ I @ K )
=> ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I4: int] :
( ( ord_less @ int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1978_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N2: int] :
( ( ( times_times @ int @ M @ N2 )
= ( one_one @ int ) )
=> ( ( M
= ( one_one @ int ) )
| ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1979_zmult__eq__1__iff,axiom,
! [M: int,N2: int] :
( ( ( times_times @ int @ M @ N2 )
= ( one_one @ int ) )
= ( ( ( M
= ( one_one @ int ) )
& ( N2
= ( one_one @ int ) ) )
| ( ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
& ( N2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1980_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( plus_plus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_1981_take__bit__Suc__minus__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_1982_take__bit__int__less__eq,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_1983_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 )
& ( ( ord_less @ A @ C2 @ A2 )
| ( ord_less @ A @ B2 @ D2 ) ) ) )
& ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1984_take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N2: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N2 @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) ) ) ) ).
% take_bit_signed_take_bit
thf(fact_1985_not__zero__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num] :
~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_1986_neg__numeral__le__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_le_zero
thf(fact_1987_neg__numeral__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_less_zero
thf(fact_1988_not__zero__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num] :
~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_1989_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_1990_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_1991_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_1992_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_1993_neg__numeral__le__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_le_one
thf(fact_1994_neg__one__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).
% neg_one_le_numeral
thf(fact_1995_neg__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% neg_numeral_le_neg_one
thf(fact_1996_not__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_le_neg_one
thf(fact_1997_not__one__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% not_one_le_neg_numeral
thf(fact_1998_not__neg__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1999_not__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% not_one_less_neg_numeral
thf(fact_2000_not__numeral__less__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_less_neg_one
thf(fact_2001_neg__one__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).
% neg_one_less_numeral
thf(fact_2002_neg__numeral__less__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_less_one
thf(fact_2003_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
= ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ C2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_2004_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) )
= A2 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B2 )
= ( times_times @ A @ A2 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_2005_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= C2 )
= ( ( uminus_uminus @ A @ A2 )
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_2006_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( C2
= ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) )
= ( ( times_times @ A @ C2 @ B2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_2007_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_2008_finite__empty__induct,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ A3 )
=> ( ! [A5: A,A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( member @ A @ A5 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_2009_infinite__coinduct,axiom,
! [A: $tType,X8: ( set @ A ) > $o,A3: set @ A] :
( ( X8 @ A3 )
=> ( ! [A8: set @ A] :
( ( X8 @ A8 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A8 )
& ( ( X8 @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_2010_infinite__remove,axiom,
! [A: $tType,S3: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ S3 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_2011_Diff__single__insert,axiom,
! [A: $tType,A3: set @ A,X3: A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_2012_subset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X3: A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) )
= ( ( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) )
& ( ~ ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_2013_take__bit__Suc__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_2014_atLeast0__atMost__Suc,axiom,
! [N2: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) )
= ( insert @ nat @ ( suc @ N2 ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_2015_mod__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( modulo_modulo @ int @ K @ L )
= ( modulo_modulo @ int @ ( minus_minus @ int @ K @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_2016_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ N2 )
= ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_2017_atLeastAtMostSuc__conv,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) )
= ( insert @ nat @ ( suc @ N2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_2018_atLeastAtMost__insertL,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) )
= ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ).
% atLeastAtMost_insertL
thf(fact_2019_plusinfinity,axiom,
! [D2: int,P5: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X5: int,K2: int] :
( ( P5 @ X5 )
= ( P5 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less @ int @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ? [X_12: int] : ( P5 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_2020_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X5: int,K2: int] :
( ( P1 @ X5 )
= ( P1 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less @ int @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P1 @ X5 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_2021_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq @ int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( ! [I4: int] :
( ( ord_less_eq @ int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_2022_subset__eq__atLeast0__atMost__finite,axiom,
! [N7: set @ nat,N2: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_2023_take__bit__int__less__exp,axiom,
! [N2: nat,K: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% take_bit_int_less_exp
thf(fact_2024_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_2025_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_2026_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_2027_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_2028_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_2029_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_2030_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_2031_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_2032_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X3 @ Z2 ) ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X3 ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_2033_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A2: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_2034_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A2: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_2035_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A2: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_2036_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X3 @ Z2 ) ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X3 ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_2037_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B3: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B3 )
=> ( P @ B3 ) )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A8 @ B3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_2038_finite__remove__induct,axiom,
! [A: $tType,B3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A8 @ B3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_2039_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) )
= ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_2040_finite__induct__select,axiom,
! [A: $tType,S3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T5: set @ A] :
( ( ord_less @ ( set @ A ) @ T5 @ S3 )
=> ( ( P @ T5 )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( minus_minus @ ( set @ A ) @ S3 @ T5 ) )
& ( P @ ( insert @ A @ X2 @ T5 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ).
% finite_induct_select
thf(fact_2041_psubset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X3: A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) )
= ( ( ( member @ A @ X3 @ B3 )
=> ( ord_less @ ( set @ A ) @ A3 @ B3 ) )
& ( ~ ( member @ A @ X3 @ B3 )
=> ( ( ( member @ A @ X3 @ A3 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) )
& ( ~ ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_2042_decr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( minus_minus @ int @ X5 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( minus_minus @ int @ X2 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_2043_verit__less__mono__div__int2,axiom,
! [A3: int,B3: int,N2: int] :
( ( ord_less_eq @ int @ A3 @ B3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N2 ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B3 @ N2 ) @ ( divide_divide @ int @ A3 @ N2 ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_2044_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_2045_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_2046_take__bit__nat__eq__self,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M )
= M ) ) ).
% take_bit_nat_eq_self
thf(fact_2047_take__bit__nat__less__exp,axiom,
! [N2: nat,M: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% take_bit_nat_less_exp
thf(fact_2048_take__bit__nat__eq__self__iff,axiom,
! [N2: nat,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M )
= M )
= ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_2049_take__bit__int__less__self__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ).
% take_bit_int_less_self_iff
thf(fact_2050_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_2051_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_2052_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_2053_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_2054_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_2055_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_2056_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_2057_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_2058_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_2059_signed__take__bit__int__less__eq,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K )
=> ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_2060_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N2 @ K ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_2061_signed__take__bit__int__eq__self,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K )
= K ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_2062_signed__take__bit__int__eq__self__iff,axiom,
! [N2: nat,K: int] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K )
= K )
= ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_2063_take__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ A2 ) ) ) ).
% take_bit_eq_0_iff
thf(fact_2064_take__bit__nat__less__self__iff,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) @ M )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M ) ) ).
% take_bit_nat_less_self_iff
thf(fact_2065_take__bit__int__eq__self,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ K )
= K ) ) ) ).
% take_bit_int_eq_self
thf(fact_2066_take__bit__int__eq__self__iff,axiom,
! [N2: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ K )
= K )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_2067_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_2068_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_2069_square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% square_le_1
thf(fact_2070_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% div_pos_geq
thf(fact_2071_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ K @ L )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_2072_take__bit__int__greater__eq,axiom,
! [K: int,N2: nat] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ).
% take_bit_int_greater_eq
thf(fact_2073_power__minus1__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N2: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% power_minus1_odd
thf(fact_2074_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A2 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( 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_2075_signed__take__bit__int__greater__eq,axiom,
! [K: int,N2: nat] :
( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_2076_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N2: nat] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_2077_signed__take__bit__Suc__minus__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_2078_concat__bit__Suc,axiom,
! [N2: nat,K: int,L: int] :
( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L )
= ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_2079_Bolzano,axiom,
! [A2: real,B2: real,P: real > real > $o] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [A5: real,B5: real,C3: real] :
( ( P @ A5 @ B5 )
=> ( ( P @ B5 @ C3 )
=> ( ( ord_less_eq @ real @ A5 @ B5 )
=> ( ( ord_less_eq @ real @ B5 @ C3 )
=> ( P @ A5 @ C3 ) ) ) ) )
=> ( ! [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
=> ( ( ord_less_eq @ real @ X5 @ B2 )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [A5: real,B5: real] :
( ( ( ord_less_eq @ real @ A5 @ X5 )
& ( ord_less_eq @ real @ X5 @ B5 )
& ( ord_less @ real @ ( minus_minus @ real @ B5 @ A5 ) @ D4 ) )
=> ( P @ A5 @ B5 ) ) ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Bolzano
thf(fact_2080_signed__take__bit__Suc__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_2081_and__int_Oelims,axiom,
! [X3: int,Xa2: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X3 @ Xa2 )
= Y )
=> ( ( ( ( member @ int @ X3 @ ( 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 ) ) @ X3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member @ int @ X3 @ ( 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 ) ) @ X3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_2082_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_2083_Diff__idemp,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ).
% Diff_idemp
thf(fact_2084_Diff__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( ( member @ A @ C2 @ A3 )
& ~ ( member @ A @ C2 @ B3 ) ) ) ).
% Diff_iff
thf(fact_2085_DiffI,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ~ ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_2086_ComplI,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ~ ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% ComplI
thf(fact_2087_Compl__iff,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( ~ ( member @ A @ C2 @ A3 ) ) ) ).
% Compl_iff
thf(fact_2088_Compl__eq__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A3 )
= ( uminus_uminus @ ( set @ A ) @ B3 ) )
= ( A3 = B3 ) ) ).
% Compl_eq_Compl_iff
thf(fact_2089_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_2090_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_2091_bit_Oconj__zero__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ X3 )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_left
thf(fact_2092_bit_Oconj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se5824344872417868541ns_and @ A @ X3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_right
thf(fact_2093_concat__bit__0,axiom,
! [K: int,L: int] :
( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L )
= L ) ).
% concat_bit_0
thf(fact_2094_semiring__norm_I80_J,axiom,
! [M: num,N2: num] :
( ( ord_less @ num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( ord_less @ num @ M @ N2 ) ) ).
% semiring_norm(80)
thf(fact_2095_and__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% and_negative_int_iff
thf(fact_2096_semiring__norm_I81_J,axiom,
! [M: num,N2: num] :
( ( ord_less @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less @ num @ M @ N2 ) ) ).
% semiring_norm(81)
thf(fact_2097_semiring__norm_I77_J,axiom,
! [N2: num] : ( ord_less @ num @ one2 @ ( bit1 @ N2 ) ) ).
% semiring_norm(77)
thf(fact_2098_concat__bit__negative__iff,axiom,
! [N2: nat,K: int,L: int] :
( ( ord_less @ int @ ( bit_concat_bit @ N2 @ K @ L ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ).
% concat_bit_negative_iff
thf(fact_2099_semiring__norm_I79_J,axiom,
! [M: num,N2: num] :
( ( ord_less @ num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( ord_less_eq @ num @ M @ N2 ) ) ).
% semiring_norm(79)
thf(fact_2100_semiring__norm_I74_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less @ num @ M @ N2 ) ) ).
% semiring_norm(74)
thf(fact_2101_and__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(5)
thf(fact_2102_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_2103_div__Suc__eq__div__add3,axiom,
! [M: nat,N2: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
= ( divide_divide @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N2 ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_2104_Suc__div__eq__add3__div__numeral,axiom,
! [M: nat,V2: num] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_2105_mod__Suc__eq__mod__add3,axiom,
! [M: nat,N2: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
= ( modulo_modulo @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N2 ) ) ) ).
% mod_Suc_eq_mod_add3
thf(fact_2106_Suc__mod__eq__add3__mod__numeral,axiom,
! [M: nat,V2: num] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_2107_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( 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 @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_2108_DiffD2,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ( member @ A @ C2 @ B3 ) ) ).
% DiffD2
thf(fact_2109_DiffD1,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
=> ( member @ A @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_2110_DiffE,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% DiffE
thf(fact_2111_ComplD,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
=> ~ ( member @ A @ C2 @ A3 ) ) ).
% ComplD
thf(fact_2112_double__complement,axiom,
! [A: $tType,A3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= A3 ) ).
% double_complement
thf(fact_2113_concat__bit__assoc,axiom,
! [N2: nat,K: int,M: nat,L: int,R2: int] :
( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
= ( bit_concat_bit @ ( plus_plus @ nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R2 ) ) ).
% concat_bit_assoc
thf(fact_2114_xor__num_Ocases,axiom,
! [X3: product_prod @ num @ num] :
( ( X3
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N3: num] :
( X3
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) )
=> ( ! [N3: num] :
( X3
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) )
=> ( ! [M3: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ one2 ) )
=> ( ! [M3: num,N3: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) )
=> ( ! [M3: num,N3: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) )
=> ( ! [M3: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ one2 ) )
=> ( ! [M3: num,N3: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) )
=> ~ ! [M3: num,N3: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_2115_and__less__eq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ K ) ) ).
% and_less_eq
thf(fact_2116_AND__upper1_H_H,axiom,
! [Y: int,Z2: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z2 )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya ) @ Z2 ) ) ) ).
% AND_upper1''
thf(fact_2117_AND__upper2_H_H,axiom,
! [Y: int,Z2: int,X3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z2 )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X3 @ Y ) @ Z2 ) ) ) ).
% AND_upper2''
thf(fact_2118_numeral__Bit1,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( numeral_numeral @ A @ ( bit1 @ N2 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_Bit1
thf(fact_2119_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I3: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I3 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I3 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_2120_eval__nat__numeral_I3_J,axiom,
! [N2: num] :
( ( numeral_numeral @ nat @ ( bit1 @ N2 ) )
= ( suc @ ( numeral_numeral @ nat @ ( bit0 @ N2 ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_2121_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num,Q4: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
!= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(3)
thf(fact_2122_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_2123_Suc3__eq__add__3,axiom,
! [N2: nat] :
( ( suc @ ( suc @ ( suc @ N2 ) ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N2 ) ) ).
% Suc3_eq_add_3
thf(fact_2124_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_2125_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_2126_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q4: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q4 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(11)
thf(fact_2127_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q4: num,N2: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ Q4 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(7)
thf(fact_2128_periodic__finite__ex,axiom,
! [D2: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X5: int,K2: int] :
( ( P @ X5 )
= ( P @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
& ( P @ X ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_2129_bset_I3_J,axiom,
! [D6: int,T2: int,B3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B3 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X2 = T2 )
=> ( ( minus_minus @ int @ X2 @ D6 )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_2130_bset_I4_J,axiom,
! [D6: int,T2: int,B3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ T2 @ B3 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X2 != T2 )
=> ( ( minus_minus @ int @ X2 @ D6 )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_2131_bset_I5_J,axiom,
! [D6: int,B3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X2 @ T2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X2 @ D6 ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_2132_bset_I7_J,axiom,
! [D6: int,T2: int,B3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ T2 @ B3 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X2 )
=> ( ord_less @ int @ T2 @ ( minus_minus @ int @ X2 @ D6 ) ) ) ) ) ) ).
% bset(7)
thf(fact_2133_aset_I3_J,axiom,
! [D6: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X2 = T2 )
=> ( ( plus_plus @ int @ X2 @ D6 )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_2134_aset_I4_J,axiom,
! [D6: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ T2 @ A3 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X2 != T2 )
=> ( ( plus_plus @ int @ X2 @ D6 )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_2135_aset_I5_J,axiom,
! [D6: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ T2 @ A3 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X2 @ T2 )
=> ( ord_less @ int @ ( plus_plus @ int @ X2 @ D6 ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_2136_aset_I7_J,axiom,
! [D6: int,A3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X2 )
=> ( ord_less @ int @ T2 @ ( plus_plus @ int @ X2 @ D6 ) ) ) ) ) ).
% aset(7)
thf(fact_2137_Suc__div__eq__add3__div,axiom,
! [M: nat,N2: nat] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N2 ) ) ).
% Suc_div_eq_add3_div
thf(fact_2138_Suc__mod__eq__add3__mod,axiom,
! [M: nat,N2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N2 ) ) ).
% Suc_mod_eq_add3_mod
thf(fact_2139_aset_I8_J,axiom,
! [D6: int,A3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X2 )
=> ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X2 @ D6 ) ) ) ) ) ).
% aset(8)
thf(fact_2140_aset_I6_J,axiom,
! [D6: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X2 @ T2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X2 @ D6 ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_2141_bset_I8_J,axiom,
! [D6: int,T2: int,B3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B3 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X2 )
=> ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X2 @ D6 ) ) ) ) ) ) ).
% bset(8)
thf(fact_2142_bset_I6_J,axiom,
! [D6: int,B3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X2 @ T2 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X2 @ D6 ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_2143_cppi,axiom,
! [D6: int,P: int > $o,P5: int > $o,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less @ int @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ! [X5: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa ) ) ) )
=> ( ( P @ X5 )
=> ( P @ ( plus_plus @ int @ X5 @ D6 ) ) ) )
=> ( ! [X5: int,K2: int] :
( ( P5 @ X5 )
= ( P5 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D6 ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
& ( P5 @ X ) )
| ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
& ? [Y2: int] :
( ( member @ int @ Y2 @ A3 )
& ( P @ ( minus_minus @ int @ Y2 @ X ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_2144_cpmi,axiom,
! [D6: int,P: int > $o,P5: int > $o,B3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less @ int @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ! [X5: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa ) ) ) )
=> ( ( P @ X5 )
=> ( P @ ( minus_minus @ int @ X5 @ D6 ) ) ) )
=> ( ! [X5: int,K2: int] :
( ( P5 @ X5 )
= ( P5 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D6 ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
& ( P5 @ X ) )
| ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
& ? [Y2: int] :
( ( member @ int @ Y2 @ B3 )
& ( P @ ( plus_plus @ int @ Y2 @ X ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_2145_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_2146_odd__mod__4__div__2,axiom,
! [N2: nat] :
( ( ( modulo_modulo @ nat @ N2 @ ( 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 @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% odd_mod_4_div_2
thf(fact_2147_mod__exhaust__less__4,axiom,
! [M: nat] :
( ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ nat ) )
| ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( one_one @ nat ) )
| ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
| ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ).
% mod_exhaust_less_4
thf(fact_2148_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N2: num] :
( ( ( ord_less_eq @ num @ M @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N2 ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_2149_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N2: num] :
( ( ( ord_less @ num @ M @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less @ num @ M @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N2 ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_2150_take__bit__Suc__minus__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_2151_and__int_Opsimps,axiom,
! [K: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) )
=> ( ( ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_2152_and__int_Opelims,axiom,
! [X3: int,Xa2: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X3 @ Xa2 ) )
=> ~ ( ( ( ( ( member @ int @ X3 @ ( 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 ) ) @ X3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member @ int @ X3 @ ( 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 ) ) @ X3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X3 @ ( 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 @ X3 @ Xa2 ) ) ) ) ) ).
% and_int.pelims
thf(fact_2153_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% finite_atLeastAtMost_int
thf(fact_2154_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_2155_and__nat__numerals_I3_J,axiom,
! [X3: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(3)
thf(fact_2156_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_2157_and__nat__numerals_I4_J,axiom,
! [X3: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(4)
thf(fact_2158_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num] :
( ( unique8689654367752047608divmod @ A @ M @ one2 )
= ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M ) @ ( zero_zero @ A ) ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_2159_add__neg__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_2160_add__neg__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N2 ) ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_2161_and__Suc__0__eq,axiom,
! [N2: nat] :
( ( bit_se5824344872417868541ns_and @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% and_Suc_0_eq
thf(fact_2162_Suc__0__and__eq,axiom,
! [N2: nat] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Suc_0_and_eq
thf(fact_2163_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_2164_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_2165_numeral__inc,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X3: num] :
( ( numeral_numeral @ A @ ( inc @ X3 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X3 ) @ ( one_one @ A ) ) ) ) ).
% numeral_inc
thf(fact_2166_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M2: num,N: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% divmod_int_def
thf(fact_2167_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M2: num,N: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ) ).
% divmod_def
thf(fact_2168_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M2: num,N: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_2169_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M2: nat,N: nat] :
( if @ nat
@ ( ( M2
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_2170_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M2: nat,N: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_2171_and__int_Opinduct,axiom,
! [A0: int,A12: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) )
=> ( ! [K2: int,L4: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L4 ) )
=> ( ( ~ ( ( member @ int @ K2 @ ( 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 ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K2 @ L4 ) ) )
=> ( P @ A0 @ A12 ) ) ) ).
% and_int.pinduct
thf(fact_2172_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M2: num,N: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M2 @ N ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M2 ) ) @ ( unique1321980374590559556d_step @ A @ N @ ( unique8689654367752047608divmod @ A @ M2 @ ( bit0 @ N ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_2173_upto_Opinduct,axiom,
! [A0: int,A12: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) )
=> ( ! [I4: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I4 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I4 @ J2 )
=> ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J2 ) )
=> ( P @ I4 @ J2 ) ) )
=> ( P @ A0 @ A12 ) ) ) ).
% upto.pinduct
thf(fact_2174_signed__take__bit__eq__take__bit__minus,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K3 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_2175_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_bit1
thf(fact_2176_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X3: num,N2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_2177_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: num,N2: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) @ A2 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_2178_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X3: num,N2: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_2179_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: num,N2: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) @ A2 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_2180_of__int__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W2: int,Z2: int] :
( ( ( ring_1_of_int @ A @ W2 )
= ( ring_1_of_int @ A @ Z2 ) )
= ( W2 = Z2 ) ) ) ).
% of_int_eq_iff
thf(fact_2181_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_2182_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_2183_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z2: int] :
( ( ( zero_zero @ A )
= ( ring_1_of_int @ A @ Z2 ) )
= ( Z2
= ( zero_zero @ int ) ) ) ) ).
% of_int_0_eq_iff
thf(fact_2184_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z2: int] :
( ( ( ring_1_of_int @ A @ Z2 )
= ( zero_zero @ A ) )
= ( Z2
= ( zero_zero @ int ) ) ) ) ).
% of_int_eq_0_iff
thf(fact_2185_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ W2 @ Z2 ) ) ) ).
% of_int_le_iff
thf(fact_2186_of__int__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z2: int,N2: num] :
( ( ( ring_1_of_int @ A @ Z2 )
= ( numeral_numeral @ A @ N2 ) )
= ( Z2
= ( numeral_numeral @ int @ N2 ) ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_2187_of__int__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ A @ K ) ) ) ).
% of_int_numeral
thf(fact_2188_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z2: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ W2 @ Z2 ) ) ) ).
% of_int_less_iff
thf(fact_2189_of__int__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z2: int] :
( ( ( ring_1_of_int @ A @ Z2 )
= ( one_one @ A ) )
= ( Z2
= ( one_one @ int ) ) ) ) ).
% of_int_eq_1_iff
thf(fact_2190_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_2191_of__int__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z2: int] :
( ( ring_1_of_int @ A @ ( times_times @ int @ W2 @ Z2 ) )
= ( times_times @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_mult
thf(fact_2192_of__int__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z2: int] :
( ( ring_1_of_int @ A @ ( plus_plus @ int @ W2 @ Z2 ) )
= ( plus_plus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_add
thf(fact_2193_of__int__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: int] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z2 ) )
= ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_minus
thf(fact_2194_of__int__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z2: int] :
( ( ring_1_of_int @ A @ ( minus_minus @ int @ W2 @ Z2 ) )
= ( minus_minus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_diff
thf(fact_2195_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_2196_eq__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ( numeral_numeral @ nat @ K )
= ( suc @ N2 ) )
= ( ( pred_numeral @ K )
= N2 ) ) ).
% eq_numeral_Suc
thf(fact_2197_Suc__eq__numeral,axiom,
! [N2: nat,K: num] :
( ( ( suc @ N2 )
= ( numeral_numeral @ nat @ K ) )
= ( N2
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_2198_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X3: int,B2: int,W2: nat] :
( ( ( ring_1_of_int @ A @ X3 )
= ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( X3
= ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_2199_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [B2: int,W2: nat,X3: int] :
( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 )
= ( ring_1_of_int @ A @ X3 ) )
= ( ( power_power @ int @ B2 @ W2 )
= X3 ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_2200_of__int__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: int,N2: nat] :
( ( ring_1_of_int @ A @ ( power_power @ int @ Z2 @ N2 ) )
= ( power_power @ A @ ( ring_1_of_int @ A @ Z2 ) @ N2 ) ) ) ).
% of_int_power
thf(fact_2201_of__int__of__bool,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P: $o] :
( ( ring_1_of_int @ A @ ( zero_neq_one_of_bool @ int @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% of_int_of_bool
thf(fact_2202_bit__numeral__Bit0__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( suc @ N2 ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N2 ) ) ) ).
% bit_numeral_Bit0_Suc_iff
thf(fact_2203_bit__numeral__Bit1__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( suc @ N2 ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N2 ) ) ) ).
% bit_numeral_Bit1_Suc_iff
thf(fact_2204_less__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
= ( ord_less @ nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% less_numeral_Suc
thf(fact_2205_less__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( ord_less @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less @ nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% less_Suc_numeral
thf(fact_2206_le__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
= ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% le_numeral_Suc
thf(fact_2207_le__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less_eq @ nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% le_Suc_numeral
thf(fact_2208_diff__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( minus_minus @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
= ( minus_minus @ nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% diff_Suc_numeral
thf(fact_2209_diff__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
= ( minus_minus @ nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% diff_numeral_Suc
thf(fact_2210_signed__take__bit__negative__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ).
% signed_take_bit_negative_iff
thf(fact_2211_of__int__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ).
% of_int_0_le_iff
thf(fact_2212_of__int__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) ) ) ) ).
% of_int_le_0_iff
thf(fact_2213_of__int__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( zero_zero @ A ) )
= ( ord_less @ int @ Z2 @ ( zero_zero @ int ) ) ) ) ).
% of_int_less_0_iff
thf(fact_2214_of__int__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ).
% of_int_0_less_iff
thf(fact_2215_of__int__le__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int,N2: num] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less_eq @ int @ Z2 @ ( numeral_numeral @ int @ N2 ) ) ) ) ).
% of_int_le_numeral_iff
thf(fact_2216_of__int__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num,Z2: int] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ ( numeral_numeral @ int @ N2 ) @ Z2 ) ) ) ).
% of_int_numeral_le_iff
thf(fact_2217_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int,N2: num] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less @ int @ Z2 @ ( numeral_numeral @ int @ N2 ) ) ) ) ).
% of_int_less_numeral_iff
thf(fact_2218_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num,Z2: int] :
( ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ ( numeral_numeral @ int @ N2 ) @ Z2 ) ) ) ).
% of_int_numeral_less_iff
thf(fact_2219_of__int__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ int @ Z2 @ ( one_one @ int ) ) ) ) ).
% of_int_le_1_iff
thf(fact_2220_of__int__1__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% of_int_1_le_iff
thf(fact_2221_of__int__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) )
= ( ord_less @ int @ Z2 @ ( one_one @ int ) ) ) ) ).
% of_int_less_1_iff
thf(fact_2222_of__int__1__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% of_int_1_less_iff
thf(fact_2223_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W2: nat,X3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W2 ) @ X3 ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_2224_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: int,B2: int,W2: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( ord_less_eq @ int @ X3 @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_2225_of__int__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X3: num,N2: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_2226_numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X3: num,N2: nat,Y: int] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 )
= Y ) ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_2227_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W2: nat,X3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X3 ) )
= ( ord_less @ int @ ( power_power @ int @ B2 @ W2 ) @ X3 ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_2228_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: int,B2: int,W2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X3 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( ord_less @ int @ X3 @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_2229_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W2: num,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) ) @ ( suc @ N2 ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ N2 ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_2230_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W2: num,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) ) @ ( suc @ N2 ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ N2 ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_2231_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_2232_bit__mod__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 )
= ( ( N2
= ( zero_zero @ nat ) )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% bit_mod_2_iff
thf(fact_2233_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: num,N2: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) @ A2 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_2234_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X3: num,N2: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_2235_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: num,N2: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) @ A2 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_2236_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X3: num,N2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_2237_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X3: num,N2: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) )
= ( Y
= ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_2238_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X3: num,N2: nat,Y: int] :
( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 )
= Y ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_2239_bit__disjunctive__add__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,B2: A,N2: nat] :
( ! [N3: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N2 )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
| ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_2240_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [Z3: int] : ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_le_of_int
thf(fact_2241_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [Z3: int] : ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_less_of_int
thf(fact_2242_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [Z3: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X3 ) ) ).
% ex_of_int_less
thf(fact_2243_mult__of__int__commute,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: int,Y: A] :
( ( times_times @ A @ ( ring_1_of_int @ A @ X3 ) @ Y )
= ( times_times @ A @ Y @ ( ring_1_of_int @ A @ X3 ) ) ) ) ).
% mult_of_int_commute
thf(fact_2244_not__bit__1__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N2 ) ) ) ).
% not_bit_1_Suc
thf(fact_2245_bit__1__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N2 )
= ( N2
= ( zero_zero @ nat ) ) ) ) ).
% bit_1_iff
thf(fact_2246_bit__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) @ N2 )
= ( ( ord_less @ nat @ N2 @ M )
& ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ) ).
% bit_take_bit_iff
thf(fact_2247_bit__of__bool__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [B2: $o,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ N2 )
= ( B2
& ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% bit_of_bool_iff
thf(fact_2248_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_2249_of__int__nonneg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_nonneg
thf(fact_2250_of__int__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_pos
thf(fact_2251_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_2252_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [X5: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X5 ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X5 @ ( one_one @ int ) ) ) )
& ! [Y5: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y5 ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y5 @ ( one_one @ int ) ) ) ) )
=> ( Y5 = X5 ) ) ) ) ).
% floor_exists1
thf(fact_2253_of__int__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).
% of_int_neg_numeral
thf(fact_2254_bit__imp__take__bit__positive,axiom,
! [N2: nat,M: nat,K: int] :
( ( ord_less @ nat @ N2 @ M )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ N2 )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_2255_int__le__real__less,axiom,
( ( ord_less_eq @ int )
= ( ^ [N: int,M2: int] : ( ord_less @ real @ ( ring_1_of_int @ real @ N ) @ ( plus_plus @ real @ ( ring_1_of_int @ real @ M2 ) @ ( one_one @ real ) ) ) ) ) ).
% int_le_real_less
thf(fact_2256_int__less__real__le,axiom,
( ( ord_less @ int )
= ( ^ [N: int,M2: int] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ M2 ) ) ) ) ).
% int_less_real_le
thf(fact_2257_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L: int,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K @ L ) @ N2 )
= ( ( ( ord_less @ nat @ N2 @ M )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) )
| ( ( ord_less_eq @ nat @ M @ N2 )
& ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_2258_exp__eq__0__imp__not__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N2: nat,A2: A] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
= ( zero_zero @ A ) )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_2259_bit__Suc,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ N2 ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 ) ) ) ).
% bit_Suc
thf(fact_2260_int__bit__bound,axiom,
! [K: int] :
~ ! [N3: nat] :
( ! [M6: nat] :
( ( ord_less_eq @ nat @ N3 @ M6 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M6 )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) ) ) ).
% int_bit_bound
thf(fact_2261_and__exp__eq__0__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N2: nat] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( zero_zero @ A ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_2262_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ N2 )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_2263_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N2: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ N2 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_2264_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N2: 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 ) ) @ N2 )
= ( ( ( N2
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N2 ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_2265_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A4: A,N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_2266_take__bit__Suc__from__most,axiom,
! [N2: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ K )
= ( plus_plus @ int @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ).
% take_bit_Suc_from_most
thf(fact_2267_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X3 @ ( 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 @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X3 )
= Y ) ) ) ) ).
% round_unique
thf(fact_2268_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) ) ) ).
% of_int_round_gt
thf(fact_2269_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) ) ) ).
% of_int_round_ge
thf(fact_2270_of__int__round__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% of_int_round_le
thf(fact_2271_divmod__BitM__2__eq,axiom,
! [M: num] :
( ( unique8689654367752047608divmod @ int @ ( bitM @ M ) @ ( bit0 @ one2 ) )
= ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% divmod_BitM_2_eq
thf(fact_2272_mask__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: num] :
( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N2 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N2 ) ) ) ) ) ) ).
% mask_numeral
thf(fact_2273_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_2274_mask__nat__positive__iff,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% mask_nat_positive_iff
thf(fact_2275_ceiling__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_2276_mask__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( ( bit_se2239418461657761734s_mask @ A @ N2 )
= ( zero_zero @ A ) )
= ( N2
= ( zero_zero @ nat ) ) ) ) ).
% mask_eq_0_iff
thf(fact_2277_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_2278_round__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_2279_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_2280_ceiling__add__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z2 ) ) ) ).
% ceiling_add_of_int
thf(fact_2281_ceiling__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) ) ) ).
% ceiling_le_zero
thf(fact_2282_zero__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) ) ) ).
% zero_less_ceiling
thf(fact_2283_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% ceiling_le_numeral
thf(fact_2284_ceiling__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) ) ) ).
% ceiling_less_one
thf(fact_2285_one__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) ) ) ).
% one_le_ceiling
thf(fact_2286_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V2 ) @ X3 ) ) ) ).
% numeral_less_ceiling
thf(fact_2287_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X3 @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_2288_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ).
% one_less_ceiling
thf(fact_2289_ceiling__add__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_add_numeral
thf(fact_2290_ceiling__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) ) ) ) ).
% ceiling_add_one
thf(fact_2291_ceiling__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_zero
thf(fact_2292_zero__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X3 ) ) ) ).
% zero_le_ceiling
thf(fact_2293_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_2294_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% numeral_le_ceiling
thf(fact_2295_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_2296_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X3 ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_2297_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_2298_less__eq__mask,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) ).
% less_eq_mask
thf(fact_2299_bit__Suc__0__iff,axiom,
! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% bit_Suc_0_iff
thf(fact_2300_not__bit__Suc__0__Suc,axiom,
! [N2: nat] :
~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( suc @ N2 ) ) ).
% not_bit_Suc_0_Suc
thf(fact_2301_ceiling__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ).
% ceiling_mono
thf(fact_2302_le__of__int__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ).
% le_of_int_ceiling
thf(fact_2303_ceiling__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( archimedean_ceiling @ A @ Y ) )
=> ( ord_less @ A @ X3 @ Y ) ) ) ).
% ceiling_less_cancel
thf(fact_2304_not__mask__negative__int,axiom,
! [N2: nat] :
~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N2 ) @ ( zero_zero @ int ) ) ).
% not_mask_negative_int
thf(fact_2305_not__bit__Suc__0__numeral,axiom,
! [N2: num] :
~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ N2 ) ) ).
% not_bit_Suc_0_numeral
thf(fact_2306_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z2 )
= ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% ceiling_le_iff
thf(fact_2307_ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,A2: int] :
( ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ A2 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ A2 ) ) ) ).
% ceiling_le
thf(fact_2308_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( ord_less @ int @ Z2 @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 ) ) ) ).
% less_ceiling_iff
thf(fact_2309_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( archimedean_ceiling @ A @ Y ) ) ) ) ).
% ceiling_add_le
thf(fact_2310_round__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X3 ) @ ( archimedean_round @ A @ Y ) ) ) ) ).
% round_mono
thf(fact_2311_less__mask,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
=> ( ord_less @ nat @ N2 @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) ) ).
% less_mask
thf(fact_2312_eval__nat__numeral_I2_J,axiom,
! [N2: num] :
( ( numeral_numeral @ nat @ ( bit0 @ N2 ) )
= ( suc @ ( numeral_numeral @ nat @ ( bitM @ N2 ) ) ) ) ).
% eval_nat_numeral(2)
thf(fact_2313_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_2314_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_2315_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% ceiling_split
thf(fact_2316_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,A2: int] :
( ( ( archimedean_ceiling @ A @ X3 )
= A2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ A2 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_2317_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) )
=> ( ( archimedean_ceiling @ A @ X3 )
= Z2 ) ) ) ) ).
% ceiling_unique
thf(fact_2318_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) @ ( one_one @ A ) ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ) ).
% ceiling_correct
thf(fact_2319_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_2320_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z2 )
= ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_2321_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( ord_less_eq @ int @ Z2 @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% le_ceiling_iff
thf(fact_2322_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less_eq @ A @ P4 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_2323_Suc__mask__eq__exp,axiom,
! [N2: nat] :
( ( suc @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% Suc_mask_eq_exp
thf(fact_2324_mask__nat__less__exp,axiom,
! [N2: nat] : ( ord_less @ nat @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% mask_nat_less_exp
thf(fact_2325_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) )
= ( N2
= ( zero_zero @ nat ) ) ) ) ).
% semiring_bit_operations_class.even_mask_iff
thf(fact_2326_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) @ P4 ) ) ) ).
% ceiling_divide_lower
thf(fact_2327_ceiling__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N2: int,X3: A] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ N2 ) @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N2 ) @ ( one_one @ A ) ) )
=> ( ( archimedean_ceiling @ A @ X3 )
= ( plus_plus @ int @ N2 @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_eq
thf(fact_2328_Suc__0__xor__eq,axiom,
! [N2: nat] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% Suc_0_xor_eq
thf(fact_2329_xor__Suc__0__eq,axiom,
! [N2: nat] :
( ( bit_se5824344971392196577ns_xor @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% xor_Suc_0_eq
thf(fact_2330_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_2331_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_2332_the__elem__eq,axiom,
! [A: $tType,X3: A] :
( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ).
% the_elem_eq
thf(fact_2333_vebt__member_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X3 @ Xa2 )
= Y )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( Y
= ( ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> Y )
=> ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( 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] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> Y )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_2334_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% set_vebt'_def
thf(fact_2335_finite__Collect__conjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_2336_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_2337_bit_Oxor__self,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X3 @ X3 )
= ( zero_zero @ A ) ) ) ).
% bit.xor_self
thf(fact_2338_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_2339_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_2340_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_2341_finite__Collect__bounded__ex,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: B > A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
? [Y2: A] :
( ( P @ Y2 )
& ( Q @ X @ Y2 ) ) ) )
= ( ! [Y2: A] :
( ( P @ Y2 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] : ( Q @ X @ Y2 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_2342_finite__Collect__subsets,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_2343_singleton__conv2,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ A2 ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_2344_singleton__conv,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ^ [X: A] : X = A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_2345_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less @ nat @ N @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_2346_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less_eq @ nat @ N @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_2347_finite__interval__int1,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less_eq @ int @ A2 @ I3 )
& ( ord_less_eq @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_2348_finite__interval__int4,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less @ int @ A2 @ I3 )
& ( ord_less @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int4
thf(fact_2349_finite__interval__int2,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less_eq @ int @ A2 @ I3 )
& ( ord_less @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_2350_finite__interval__int3,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less @ int @ A2 @ I3 )
& ( ord_less_eq @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_2351_xor__nat__numerals_I4_J,axiom,
! [X3: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) ) ).
% xor_nat_numerals(4)
thf(fact_2352_xor__nat__numerals_I3_J,axiom,
! [X3: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) ) ).
% xor_nat_numerals(3)
thf(fact_2353_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_2354_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_2355_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( 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 @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_2356_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( 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 @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_2357_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X: A] : $false ) ) ).
% empty_def
thf(fact_2358_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( A2 = X )
& ( P @ X ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_2359_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( X = A2 )
& ( P @ X ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_2360_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R )
@ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S3 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% pred_subset_eq2
thf(fact_2361_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_2362_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R ) )
= ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S3 ) ) )
= ( R = S3 ) ) ).
% pred_equals_eq2
thf(fact_2363_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_2364_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A6 )
@ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ).
% less_set_def
thf(fact_2365_Collect__subset,axiom,
! [A: $tType,A3: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A3 )
& ( P @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_2366_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A6 )
@ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_2367_prop__restrict,axiom,
! [A: $tType,X3: A,Z5: set @ A,X8: set @ A,P: A > $o] :
( ( member @ A @ X3 @ Z5 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z5
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ X8 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_2368_Collect__restrict,axiom,
! [A: $tType,X8: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ X8 )
& ( P @ X ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_2369_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S3: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ R )
@ ^ [X: A] : ( member @ A @ X @ S3 ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S3 ) ) ).
% pred_subset_eq
thf(fact_2370_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X: A] : X )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_2371_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_2372_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A6: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A6 ) ) ) ) ) ).
% uminus_set_def
thf(fact_2373_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X: A] :
~ ( P @ X ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_2374_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A6: set @ A] :
( collect @ A
@ ^ [X: A] :
~ ( member @ A @ X @ A6 ) ) ) ) ).
% Compl_eq
thf(fact_2375_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A6 )
& ~ ( member @ A @ X @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_2376_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A4: A,B6: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( X = A4 )
| ( member @ A @ X @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_2377_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A6 )
@ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_2378_insert__Collect,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( insert @ A @ A2 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U2: A] :
( ( U2 != A2 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_2379_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ? [X_1: A] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_2380_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,R: A > B > $o] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ? [Xa: B] :
( ( member @ B @ Xa @ B3 )
& ( R @ X5 @ Xa ) ) )
=> ? [X5: B] :
( ( member @ B @ X5 @ B3 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A4: A] :
( ( member @ A @ A4 @ A3 )
& ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_2381_finite__image__set2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > $o,Q: B > $o,F2: A > B > C] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ B @ ( collect @ B @ Q ) )
=> ( finite_finite2 @ C
@ ( collect @ C
@ ^ [Uu3: C] :
? [X: A,Y2: B] :
( ( Uu3
= ( F2 @ X @ Y2 ) )
& ( P @ X )
& ( Q @ Y2 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_2382_finite__image__set,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: A > B] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Uu3: B] :
? [X: A] :
( ( Uu3
= ( F2 @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_2383_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_2384_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_2385_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F2 @ N3 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less_eq @ nat @ ( F2 @ N ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_2386_numeral__code_I2_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( numeral_numeral @ A @ ( bit0 @ N2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% numeral_code(2)
thf(fact_2387_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% set_vebt_def
thf(fact_2388_finite__divisors__int,axiom,
! [I: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( finite_finite2 @ int
@ ( collect @ int
@ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ I ) ) ) ) ).
% finite_divisors_int
thf(fact_2389_numeral__code_I3_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( numeral_numeral @ A @ ( bit1 @ N2 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_code(3)
thf(fact_2390_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_2391_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs2: list @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [I3: nat] :
( ( Uu3
= ( nth @ A @ Xs2 @ I3 ) )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_2392_finite__lists__length__eq,axiom,
! [A: $tType,A3: set @ A,N2: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= N2 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_2393_finite__lists__length__le,axiom,
! [A: $tType,A3: set @ A,N2: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N2 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_2394_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_2395_vebt__buildup_Oelims,axiom,
! [X3: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X3 )
= Y )
=> ( ( ( X3
= ( zero_zero @ nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X3
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va2: nat] :
( ( X3
= ( 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_2396_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList2: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList2 @ S ) @ X3 )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_2397_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X: A] : ( minus_minus @ A @ ( plus_plus @ A @ X @ X ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_2398_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V2: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ X3 )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_2399_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
= ( ( X3 != Mi )
=> ( ( X3 != Ma )
=> ( ~ ( ord_less @ nat @ X3 @ Mi )
& ( ~ ( ord_less @ nat @ X3 @ Mi )
=> ( ~ ( ord_less @ nat @ Ma @ X3 )
& ( ~ ( ord_less @ nat @ Ma @ X3 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_2400_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V2: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) @ X3 )
= ( ( X3 = Mi )
| ( X3 = Ma )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_2401_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
= Y )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( Y
= ( ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> Y )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_2402_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_2403_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_2404_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X3
= ( 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,TreeList3: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) )
=> ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_2405_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M2: nat,N: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N
@ ( if @ nat
@ ( N
= ( zero_zero @ nat ) )
@ M2
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_2406_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M2: nat,N: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_2407_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_2408_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_2409_vebt__member_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X3 @ Xa2 )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_2410_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
=> ( ! [Uu2: $o,Uv2: $o] :
( X3
!= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X3
= ( 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,TreeList3: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) )
=> ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_2411_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X3
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> Y )
=> ( ( ? [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> Y )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X3
= ( 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,TreeList3: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_2412_bit__horner__sum__bit__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Bs: list @ $o,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) @ N2 )
= ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ $o ) @ Bs ) )
& ( nth @ $o @ Bs @ N2 ) ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_2413_vebt__member_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
!= ( 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] :
( X3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_2414_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_2415_finite__nth__roots,axiom,
! [N2: nat,C2: complex] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( finite_finite2 @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= C2 ) ) ) ) ).
% finite_nth_roots
thf(fact_2416_finite__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( power_power @ A @ Z6 @ N2 )
= ( one_one @ A ) ) ) ) ) ) ).
% finite_roots_unity
thf(fact_2417_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,X3: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( X3 @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( Y @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( times_times @ A @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_2418_vebt__member_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( Y
= ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X3
= ( 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] :
( ( X3
= ( 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] :
( ( X3
= ( 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,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) )
=> ~ ( 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 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_2419_vebt__member_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X3
= ( 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] :
( ( X3
= ( 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] :
( ( X3
= ( 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,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ 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 ) ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_2420_xor__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
!= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% xor_negative_int_iff
thf(fact_2421_bot2E,axiom,
! [A: $tType,B: $tType,X3: A,Y: B] :
~ ( bot_bot @ ( A > B > $o ) @ X3 @ Y ) ).
% bot2E
thf(fact_2422_XOR__upper,axiom,
! [X3: int,N2: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less @ int @ X3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X3 @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% XOR_upper
thf(fact_2423_vebt__member_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ 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 ) ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_2424_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,X3: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( X3 @ I3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( Y @ I3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( plus_plus @ A @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_2425_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( Y
= ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X3
= ( 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,TreeList3: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_2426_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) @ Xa2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_2427_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X3
= ( 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,TreeList3: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) @ Xa2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_2428_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X3
= ( 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] :
( ( X3
= ( 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] :
( ( X3
= ( 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,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ 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 ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_2429_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X3
= ( 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] :
( ( X3
= ( 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] :
( ( X3
= ( 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,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) )
=> ~ ( 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 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_2430_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X3
= ( 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,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ 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 ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_2431_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_2432_arsinh__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% arsinh_0
thf(fact_2433_artanh__0,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% artanh_0
thf(fact_2434_vebt__buildup_Opelims,axiom,
! [X3: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X3 )
= Y )
=> ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X3 )
=> ( ( ( X3
= ( zero_zero @ nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X3
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va2: nat] :
( ( X3
= ( 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_2435_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N2: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M @ N2 ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_2436_artanh__def,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A )
= ( ^ [X: A] : ( divide_divide @ A @ ( ln_ln @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% artanh_def
thf(fact_2437_ln__inj__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( ln_ln @ real @ X3 )
= ( ln_ln @ real @ Y ) )
= ( X3 = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_2438_ln__less__cancel__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y ) )
= ( ord_less @ real @ X3 @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_2439_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > C > A,A2: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F2 @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_2440_ln__le__cancel__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y ) )
= ( ord_less_eq @ real @ X3 @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_2441_ln__less__zero__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ) ).
% ln_less_zero_iff
thf(fact_2442_ln__gt__zero__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) )
= ( ord_less @ real @ ( one_one @ real ) @ X3 ) ) ) ).
% ln_gt_zero_iff
thf(fact_2443_ln__eq__zero__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( ln_ln @ real @ X3 )
= ( zero_zero @ real ) )
= ( X3
= ( one_one @ real ) ) ) ) ).
% ln_eq_zero_iff
thf(fact_2444_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_2445_ln__le__zero__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X3 @ ( one_one @ real ) ) ) ) ).
% ln_le_zero_iff
thf(fact_2446_ln__ge__zero__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) ) ) ).
% ln_ge_zero_iff
thf(fact_2447_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N2: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N2 ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_2448_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H2: C > D,F2: A > B > C,Prod: product_prod @ A @ B] :
( ( H2 @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X15: A,X24: B] : ( H2 @ ( F2 @ X15 @ X24 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_2449_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F2: A > B > C,X1: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F2 @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_2450_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q4: product_prod @ A @ B,F2: A > B > C,G: A > B > C,P4: product_prod @ A @ B] :
( ! [X5: A,Y4: B] :
( ( ( product_Pair @ A @ B @ X5 @ Y4 )
= Q4 )
=> ( ( F2 @ X5 @ Y4 )
= ( G @ X5 @ Y4 ) ) )
=> ( ( P4 = Q4 )
=> ( ( product_case_prod @ A @ B @ C @ F2 @ P4 )
= ( product_case_prod @ A @ B @ C @ G @ Q4 ) ) ) ) ).
% split_cong
thf(fact_2451_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X5: A,Y4: B] :
( ( F2 @ X5 @ Y4 )
= ( G @ ( product_Pair @ A @ B @ X5 @ Y4 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F2 )
= G ) ) ).
% cond_case_prod_eta
thf(fact_2452_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X: A,Y2: B] : ( F2 @ ( product_Pair @ A @ B @ X @ Y2 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_2453_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z2: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) )
=> ~ ! [X5: B,Y4: C] :
( ( Z2
= ( product_Pair @ B @ C @ X5 @ Y4 ) )
=> ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% case_prodE2
thf(fact_2454_ln__less__self,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ X3 ) ) ).
% ln_less_self
thf(fact_2455_ln__bound,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ X3 ) ) ).
% ln_bound
thf(fact_2456_ln__gt__zero__imp__gt__one,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less @ real @ ( one_one @ real ) @ X3 ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_2457_ln__less__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% ln_less_zero
thf(fact_2458_ln__gt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) ) ) ).
% ln_gt_zero
thf(fact_2459_ln__div,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ln_ln @ real @ ( divide_divide @ real @ X3 @ Y ) )
= ( minus_minus @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_2460_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_2461_ln__ge__zero__imp__ge__one,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_2462_ln__diff__le,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ X3 @ Y ) @ Y ) ) ) ) ).
% ln_diff_le
thf(fact_2463_ln__mult,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ln_ln @ real @ ( times_times @ real @ X3 @ Y ) )
= ( plus_plus @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_2464_ln__eq__minus__one,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( ln_ln @ real @ X3 )
= ( minus_minus @ real @ X3 @ ( one_one @ real ) ) )
=> ( X3
= ( one_one @ real ) ) ) ) ).
% ln_eq_minus_one
thf(fact_2465_ln__le__minus__one,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( minus_minus @ real @ X3 @ ( one_one @ real ) ) ) ) ).
% ln_le_minus_one
thf(fact_2466_ln__add__one__self__le__self2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) ) @ X3 ) ) ).
% ln_add_one_self_le_self2
thf(fact_2467_ln__one__minus__pos__upper__bound,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X3 ) ) @ ( uminus_uminus @ real @ X3 ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_2468_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_2469_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L2: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q5: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L2 ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_2470_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_2471_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M2: nat,N: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M2 @ N ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M2 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q5 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ) ) ).
% divmod_nat_if
thf(fact_2472_tanh__ln__real,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( tanh @ real @ ( ln_ln @ real @ X3 ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% tanh_ln_real
thf(fact_2473_Sum__Icc__nat,axiom,
! [M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Icc_nat
thf(fact_2474_arith__series__nat,axiom,
! [A2: nat,D2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ A2 @ ( times_times @ nat @ I3 @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ nat @ N2 @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_2475_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P4 )
= P4 ) ).
% case_prod_Pair_iden
thf(fact_2476_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,N2: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ ( ring_1_of_int @ A @ N2 ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X3 )
= N2 ) ) ) ).
% round_unique'
thf(fact_2477_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_2478_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_2479_case__prodI,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A2: A,B2: B] :
( ( F2 @ A2 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_2480_case__prodI2,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B,C2: A > B > $o] :
( ! [A5: A,B5: B] :
( ( P4
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( C2 @ A5 @ B5 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P4 ) ) ).
% case_prodI2
thf(fact_2481_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A2: B,B2: C] :
( ( member @ A @ Z2 @ ( C2 @ A2 @ B2 ) )
=> ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_2482_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P4: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
( ! [A5: A,B5: B] :
( ( P4
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( member @ C @ Z2 @ ( C2 @ A5 @ B5 ) ) )
=> ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_2483_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P4: product_prod @ A @ B,C2: A > B > C > $o,X3: C] :
( ! [A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A5 @ B5 )
= P4 )
=> ( C2 @ A5 @ B5 @ X3 ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ X3 ) ) ).
% case_prodI2'
thf(fact_2484_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_2485_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_2486_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_2487_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_2488_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_2489_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_2490_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_2491_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_2492_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_2493_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_2494_dvd__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: A,K: A] :
( ( dvd_dvd @ A @ M @ ( abs_abs @ A @ K ) )
= ( dvd_dvd @ A @ M @ K ) ) ) ).
% dvd_abs_iff
thf(fact_2495_abs__dvd__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: A,K: A] :
( ( dvd_dvd @ A @ ( abs_abs @ A @ M ) @ K )
= ( dvd_dvd @ A @ M @ K ) ) ) ).
% abs_dvd_iff
thf(fact_2496_of__int__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: int] :
( ( ring_1_of_int @ A @ ( abs_abs @ int @ X3 ) )
= ( abs_abs @ A @ ( ring_1_of_int @ A @ X3 ) ) ) ) ).
% of_int_abs
thf(fact_2497_abs__bool__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: $o] :
( ( abs_abs @ A @ ( zero_neq_one_of_bool @ A @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% abs_bool_eq
thf(fact_2498_zdvd1__eq,axiom,
! [X3: int] :
( ( dvd_dvd @ int @ X3 @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X3 )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_2499_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_2500_tanh__real__less__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( tanh @ real @ X3 ) @ ( tanh @ real @ Y ) )
= ( ord_less @ real @ X3 @ Y ) ) ).
% tanh_real_less_iff
thf(fact_2501_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu3: B] : ( zero_zero @ A )
@ A3 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_2502_of__int__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ A )
=> ! [F2: B > int,A3: set @ B] :
( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( ring_1_of_int @ A @ ( F2 @ X ) )
@ A3 ) ) ) ).
% of_int_sum
thf(fact_2503_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_2504_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_2505_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_2506_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_2507_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A] :
( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty
thf(fact_2508_sum_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% sum.infinite
thf(fact_2509_sum__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [F5: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ F5 )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ F5 )
= ( zero_zero @ A ) )
= ( ! [X: B] :
( ( member @ B @ X @ F5 )
=> ( ( F2 @ X )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_2510_artanh__minus__real,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( artanh @ real @ ( uminus_uminus @ real @ X3 ) )
= ( uminus_uminus @ real @ ( artanh @ real @ X3 ) ) ) ) ).
% artanh_minus_real
thf(fact_2511_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less @ int @ ( abs_abs @ int @ Z2 ) @ ( one_one @ int ) )
= ( Z2
= ( zero_zero @ int ) ) ) ).
% zabs_less_one_iff
thf(fact_2512_tanh__real__pos__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X3 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% tanh_real_pos_iff
thf(fact_2513_tanh__real__neg__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( tanh @ real @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% tanh_real_neg_iff
thf(fact_2514_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_2515_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_2516_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A3: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( abs_abs @ B @ ( F2 @ I3 ) )
@ A3 ) ) ) ).
% sum_abs
thf(fact_2517_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_2518_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_2519_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_2520_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X3: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X3 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) )
= ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% sum.insert
thf(fact_2521_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A3: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( abs_abs @ B @ ( F2 @ I3 ) )
@ A3 ) ) ) ).
% sum_abs_ge_zero
thf(fact_2522_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N2 ) )
= ( ( A2
!= ( zero_zero @ A ) )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_2523_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N2: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N2 ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N2 ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_2524_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: set @ nat,C2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) )
@ A3 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_2525_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: set @ nat,C2: nat > A,D2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D2 @ I3 ) )
@ A3 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D2 @ I3 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_2526_sum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,A3: set @ B] :
( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
!= ( zero_zero @ A ) )
=> ~ ! [A5: B] :
( ( member @ B @ A5 @ A3 )
=> ( ( G @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_2527_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% sum.neutral
thf(fact_2528_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_2529_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_2530_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_2531_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_2532_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_2533_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_2534_abs__eq__iff,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X3: A,Y: A] :
( ( ( abs_abs @ A @ X3 )
= ( abs_abs @ A @ Y ) )
= ( ( X3 = Y )
| ( X3
= ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% abs_eq_iff
thf(fact_2535_dvd__if__abs__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [L: A,K: A] :
( ( ( abs_abs @ A @ L )
= ( abs_abs @ A @ K ) )
=> ( dvd_dvd @ A @ L @ K ) ) ) ).
% dvd_if_abs_eq
thf(fact_2536_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P4: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P4 ) )
=> ~ ! [X5: B,Y4: C] :
( ( P4
= ( product_Pair @ B @ C @ X5 @ Y4 ) )
=> ~ ( member @ A @ Z2 @ ( C2 @ X5 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_2537_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_2538_case__prodD,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A2: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( F2 @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_2539_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P4: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P4 )
=> ~ ! [X5: A,Y4: B] :
( ( P4
= ( product_Pair @ A @ B @ X5 @ Y4 ) )
=> ~ ( C2 @ X5 @ Y4 ) ) ) ).
% case_prodE
thf(fact_2540_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K5: set @ B,F2: B > A,G: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ K5 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ K5 ) ) ) ) ).
% sum_mono
thf(fact_2541_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,H2: B > A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( G @ X ) @ ( H2 @ X ) )
@ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ A3 ) ) ) ) ).
% sum.distrib
thf(fact_2542_sum_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B3: set @ C,G: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ C @ B3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] :
( groups7311177749621191930dd_sum @ C @ A @ ( G @ X )
@ ( collect @ C
@ ^ [Y2: C] :
( ( member @ C @ Y2 @ B3 )
& ( R @ X @ Y2 ) ) ) )
@ A3 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y2: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( G @ X @ Y2 )
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A3 )
& ( R @ X @ Y2 ) ) ) )
@ B3 ) ) ) ) ) ).
% sum.swap_restrict
thf(fact_2543_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A2: A,B2: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 )
=> ( R @ A2 @ B2 @ C2 ) ) ).
% case_prodD'
thf(fact_2544_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P4: product_prod @ A @ B,Z2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ Z2 )
=> ~ ! [X5: A,Y4: B] :
( ( P4
= ( product_Pair @ A @ B @ X5 @ Y4 ) )
=> ~ ( C2 @ X5 @ Y4 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_2545_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ).
% sum_nonneg
thf(fact_2546_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_2547_sum__mono__inv,axiom,
! [A: $tType,I7: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [F2: I7 > A,I6: set @ I7,G: I7 > A,I: I7] :
( ( ( groups7311177749621191930dd_sum @ I7 @ A @ F2 @ I6 )
= ( groups7311177749621191930dd_sum @ I7 @ A @ G @ I6 ) )
=> ( ! [I4: I7] :
( ( member @ I7 @ I4 @ I6 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member @ I7 @ I @ I6 )
=> ( ( finite_finite2 @ I7 @ I6 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_2548_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_2549_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_2550_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_2551_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_2552_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ C2 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D2 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).
% abs_mult_less
thf(fact_2553_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_2554_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_2555_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_2556_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_2557_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_2558_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_2559_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_2560_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_2561_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_2562_sum__cong__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ nat,F2: nat > A,G: nat > A] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( ! [X5: nat] :
( ( member @ nat @ ( suc @ X5 ) @ A3 )
=> ( ( F2 @ ( suc @ X5 ) )
= ( G @ ( suc @ X5 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A3 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ A3 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_2563_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A3: A > B > $o,B3: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B3 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_2564_abs__zmult__eq__1,axiom,
! [M: int,N2: int] :
( ( ( abs_abs @ int @ ( times_times @ int @ M @ N2 ) )
= ( one_one @ int ) )
=> ( ( abs_abs @ int @ M )
= ( one_one @ int ) ) ) ).
% abs_zmult_eq_1
thf(fact_2565_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A3 )
& ( P @ X ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( G @ X ) @ ( zero_zero @ A ) )
@ A3 ) ) ) ) ).
% sum.inter_filter
thf(fact_2566_sum__subtractf__nat,axiom,
! [A: $tType,A3: set @ A,G: A > nat,F2: A > nat] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ nat @ ( G @ X5 ) @ ( F2 @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X: A] : ( minus_minus @ nat @ ( F2 @ X ) @ ( G @ X ) )
@ A3 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A3 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_2567_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,X3: A > B,A2: A > B,B2: B,Delta: B] :
( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X3 @ I4 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X3 @ I6 )
= ( one_one @ B ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A2 @ I4 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( times_times @ B @ ( A2 @ I3 ) @ ( X3 @ I3 ) )
@ I6 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_2568_infinite__int__iff__unbounded__le,axiom,
! [S3: set @ int] :
( ( ~ ( finite_finite2 @ int @ S3 ) )
= ( ! [M2: int] :
? [N: int] :
( ( ord_less_eq @ int @ M2 @ ( abs_abs @ int @ N ) )
& ( member @ int @ N @ S3 ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_2569_infinite__int__iff__unbounded,axiom,
! [S3: set @ int] :
( ( ~ ( finite_finite2 @ int @ S3 ) )
= ( ! [M2: int] :
? [N: int] :
( ( ord_less @ int @ M2 @ ( abs_abs @ int @ N ) )
& ( member @ int @ N @ S3 ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_2570_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_2571_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_2572_tanh__real__lt__1,axiom,
! [X3: real] : ( ord_less @ real @ ( tanh @ real @ X3 ) @ ( one_one @ real ) ) ).
% tanh_real_lt_1
thf(fact_2573_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,T2: set @ C,G: C > A,I: C > B,F2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X5 ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S )
=> ? [Xa: C] :
( ( member @ C @ Xa @ T2 )
& ( ( I @ Xa )
= X5 )
& ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_2574_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 )
= ( zero_zero @ A ) )
= ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( ( F2 @ X )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_2575_sum__strict__mono__ex1,axiom,
! [A: $tType,I7: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: set @ I7,F2: I7 > A,G: I7 > A] :
( ( finite_finite2 @ I7 @ A3 )
=> ( ! [X5: I7] :
( ( member @ I7 @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ? [X2: I7] :
( ( member @ I7 @ X2 @ A3 )
& ( ord_less @ A @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I7 @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ I7 @ A @ G @ A3 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_2576_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R: A > A > $o,S3: set @ B,H2: B > A,G: B > A] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X16: A,Y15: A,X23: A,Y23: A] :
( ( ( R @ X16 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus @ A @ X16 @ Y15 ) @ ( plus_plus @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
=> ( R @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% sum.related
thf(fact_2577_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A] :
( ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ E ) )
=> ( X3
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_2578_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y ) @ X3 )
= ( abs_abs @ A @ ( times_times @ A @ Y @ X3 ) ) ) ) ) ).
% abs_mult_pos
thf(fact_2579_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_2580_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_2581_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_2582_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_2583_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_2584_zero__le__power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N2 ) ) ) ).
% zero_le_power_abs
thf(fact_2585_abs__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( divide_divide @ A @ ( abs_abs @ A @ X3 ) @ Y )
= ( abs_abs @ A @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% abs_div_pos
thf(fact_2586_abs__if__raw,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A4: A] : ( if @ A @ ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A4 ) @ A4 ) ) ) ) ).
% abs_if_raw
thf(fact_2587_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_2588_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A4: A] : ( if @ A @ ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A4 ) @ A4 ) ) ) ) ).
% abs_if
thf(fact_2589_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_2590_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_2591_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,A2: A,R2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ R2 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_2592_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X3: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X3 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) )
& ( ~ ( member @ B @ X3 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) )
= ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_2593_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,A2: A,R2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ R2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X3 )
& ( ord_less @ A @ X3 @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_2594_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S6: set @ B,T6: set @ C,S3: set @ B,I: C > B,J: B > C,T4: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) )
=> ( member @ C @ ( J @ A5 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ( J @ ( I @ B5 ) )
= B5 ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S6 )
=> ( ( G @ A5 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T6 )
=> ( ( H2 @ B5 )
= ( zero_zero @ A ) ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S3 )
=> ( ( H2 @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_2595_sum__eq__Suc0__iff,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X: A] :
( ( member @ A @ X @ A3 )
& ( ( F2 @ X )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y2: A] :
( ( member @ A @ Y2 @ A3 )
=> ( ( X != Y2 )
=> ( ( F2 @ Y2 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_2596_sum__SucD,axiom,
! [A: $tType,F2: A > nat,A3: set @ A,N2: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 )
= ( suc @ N2 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X5 ) ) ) ) ).
% sum_SucD
thf(fact_2597_sum__eq__1__iff,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 )
= ( one_one @ nat ) )
= ( ? [X: A] :
( ( member @ A @ X @ A3 )
& ( ( F2 @ X )
= ( one_one @ nat ) )
& ! [Y2: A] :
( ( member @ A @ Y2 @ A3 )
=> ( ( X != Y2 )
=> ( ( F2 @ Y2 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_2598_abs__real__def,axiom,
( ( abs_abs @ real )
= ( ^ [A4: real] : ( if @ real @ ( ord_less @ real @ A4 @ ( zero_zero @ real ) ) @ ( uminus_uminus @ real @ A4 ) @ A4 ) ) ) ).
% abs_real_def
thf(fact_2599_zabs__def,axiom,
( ( abs_abs @ int )
= ( ^ [I3: int] : ( if @ int @ ( ord_less @ int @ I3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I3 ) @ I3 ) ) ) ).
% zabs_def
thf(fact_2600_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F2: B > A,B3: A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S )
= B3 )
=> ( ( member @ B @ I @ S )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ B3 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_2601_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F2: B > A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S )
=> ( ( F2 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_2602_dvd__imp__le__int,axiom,
! [I: int,D2: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ D2 @ I )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ D2 ) @ ( abs_abs @ int @ I ) ) ) ) ).
% dvd_imp_le_int
thf(fact_2603_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( abs_abs @ int @ ( modulo_modulo @ int @ K @ L ) ) @ ( abs_abs @ int @ L ) ) ) ).
% abs_mod_less
thf(fact_2604_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [X: B] :
( ( G @ X )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_2605_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,M: nat,I6: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( power_power @ A @ X3 @ ( plus_plus @ nat @ M @ I3 ) )
@ I6 )
= ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ I6 ) ) ) ) ).
% sum_power_add
thf(fact_2606_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_2607_tanh__real__gt__neg1,axiom,
! [X3: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( tanh @ real @ X3 ) ) ).
% tanh_real_gt_neg1
thf(fact_2608_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,I: B,F2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( member @ B @ I @ I6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_2609_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ).
% sum_pos
thf(fact_2610_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X3 ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_2611_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_2612_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S3: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H2 @ X5 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_2613_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_2614_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_2615_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ C5 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
=> ( ( G @ A5 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B3 ) )
=> ( ( H2 @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B3 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_2616_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ C5 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
=> ( ( G @ A5 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B3 ) )
=> ( ( H2 @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B3 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_2617_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: set @ B,A3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B3 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_2618_sum__diff,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,B3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B3 ) ) ) ) ) ) ).
% sum_diff
thf(fact_2619_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: int,X3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N2 ) ) @ X3 )
=> ( ( N2
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ) ).
% of_int_leD
thf(fact_2620_sum__diff__nat,axiom,
! [A: $tType,B3: set @ A,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ B3 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_2621_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: int,X3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N2 ) ) @ X3 )
=> ( ( N2
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ) ).
% of_int_lessD
thf(fact_2622_sum__diff1__nat,axiom,
! [A: $tType,A2: A,A3: set @ A,F2: A > nat] :
( ( ( member @ A @ A2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( F2 @ A2 ) ) ) )
& ( ~ ( member @ A @ A2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) ) ) ) ).
% sum_diff1_nat
thf(fact_2623_sum__shift__lb__Suc0__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,K: nat] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_2624_sum_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_2625_sum_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_2626_sum_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G @ ( suc @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_2627_round__diff__minimal,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: A,M: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z2 @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z2 ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z2 @ ( ring_1_of_int @ A @ M ) ) ) ) ) ).
% round_diff_minimal
thf(fact_2628_zdvd__mult__cancel1,axiom,
! [M: int,N2: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ ( times_times @ int @ M @ N2 ) @ M )
= ( ( abs_abs @ int @ N2 )
= ( one_one @ int ) ) ) ) ).
% zdvd_mult_cancel1
thf(fact_2629_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G @ M )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_2630_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N2: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( minus_minus @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ M ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_2631_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F2: ( A > B ) > C,G: C] :
( ( F2
= ( ^ [X: A > B] : G ) )
=> ( ( F2
@ ^ [X: A] : ( zero_zero @ B ) )
= G ) ) ) ).
% fun_cong_unused_0
thf(fact_2632_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B3: set @ B,A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ B3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ B3 @ A3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B3 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_2633_abs__le__square__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ ( abs_abs @ A @ Y ) )
= ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_2634_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X3: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X3 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_2635_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,X3: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) )
= ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_2636_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,A2: B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ A2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( F2 @ A2 ) ) ) )
& ( ~ ( member @ B @ A2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_2637_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G: nat > A,P4: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N2 @ P4 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ P4 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_2638_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A2: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( plus_plus @ A @ ( B2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_2639_sum__count__set,axiom,
! [A: $tType,Xs: list @ A,X8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X8 )
=> ( ( finite_finite2 @ A @ X8 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs ) @ X8 )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_2640_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B3: set @ A,A3: set @ A,B2: A,F2: A > B] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F2 @ B2 ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B3 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X5 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ B3 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_2641_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A3: set @ C,F2: C > B] :
( ( member @ C @ I @ A3 )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ ( minus_minus @ ( set @ C ) @ A3 @ ( insert @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X5 ) ) )
=> ( ( finite_finite2 @ C @ A3 )
=> ( ord_less_eq @ B @ ( F2 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F2 @ A3 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_2642_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ Y ) ) ) ) ).
% power2_le_iff_abs_le
thf(fact_2643_abs__sqrt__wlog,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A > A > $o,X3: A] :
( ! [X5: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 )
=> ( P @ X5 @ ( power_power @ A @ X5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ ( abs_abs @ A @ X3 ) @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_2644_abs__square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ ( one_one @ A ) ) ) ) ).
% abs_square_le_1
thf(fact_2645_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X3 ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_2646_power__mono__even,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,A2: A,B2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ).
% power_mono_even
thf(fact_2647_nat__intermed__int__val,axiom,
! [M: nat,N2: nat,F2: nat > int,K: int] :
( ! [I4: nat] :
( ( ( ord_less_eq @ nat @ M @ I4 )
& ( ord_less @ nat @ I4 @ N2 ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I4 ) ) @ ( F2 @ I4 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less_eq @ int @ ( F2 @ M ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N2 ) )
=> ? [I4: nat] :
( ( ord_less_eq @ nat @ M @ I4 )
& ( ord_less_eq @ nat @ I4 @ N2 )
& ( ( F2 @ I4 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_2648_incr__lemma,axiom,
! [D2: int,Z2: int,X3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ Z2 @ ( plus_plus @ int @ X3 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X3 @ Z2 ) ) @ ( one_one @ int ) ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_2649_decr__lemma,axiom,
! [D2: int,X3: int,Z2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X3 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X3 @ Z2 ) ) @ ( one_one @ int ) ) @ D2 ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_2650_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N2: nat,F2: nat > A] :
( ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_2651_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N2: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) )
= ( minus_minus @ A @ ( F2 @ N2 ) @ ( F2 @ M ) ) ) ) ) ).
% sum_telescope''
thf(fact_2652_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N2: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N2 ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_2653_nat__ivt__aux,axiom,
! [N2: nat,F2: nat > int,K: int] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I4 ) ) @ ( F2 @ I4 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N2 ) )
=> ? [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N2 )
& ( ( F2 @ I4 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_2654_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M2: nat,N: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M2 @ N ) @ ( modulo_modulo @ nat @ M2 @ N ) ) ) ) ).
% divmod_nat_def
thf(fact_2655_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N2: nat,X3: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_2656_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% sum.in_pairs
thf(fact_2657_nat0__intermed__int__val,axiom,
! [N2: nat,F2: nat > int,K: int] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) @ ( F2 @ I4 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N2 ) )
=> ? [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N2 )
& ( ( F2 @ I4 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_2658_mask__eq__sum__exp__nat,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N2 ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_2659_gauss__sum__nat,axiom,
! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_2660_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) @ X3 ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_2661_listrel1p__def,axiom,
! [A: $tType] :
( ( listrel1p @ A )
= ( ^ [R5: A > A > $o,Xs2: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys3 ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% listrel1p_def
thf(fact_2662_lemma__interval,axiom,
! [A2: real,X3: real,B2: real] :
( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [Y5: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y5 ) ) @ D3 )
=> ( ( ord_less_eq @ real @ A2 @ Y5 )
& ( ord_less_eq @ real @ Y5 @ B2 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_2663_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N2: nat,M: nat,X3: A] :
( ( ( ord_less @ nat @ N2 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( ( X3
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ M ) ) ) )
& ( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_2664_gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_2665_lemma__interval__lt,axiom,
! [A2: real,X3: real,B2: real] :
( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [Y5: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y5 ) ) @ D3 )
=> ( ( ord_less @ real @ A2 @ Y5 )
& ( ord_less @ real @ Y5 @ B2 ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_2666_arctan__double,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ X3 ) )
= ( arctan @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arctan_double
thf(fact_2667_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: nat,N2: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( semiring_1_of_nat @ A @ N2 ) )
= ( M = N2 ) ) ) ).
% of_nat_eq_iff
thf(fact_2668_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A4: A,B4: B] :
( P
& ( Q @ A4 @ B4 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_2669_int__eq__iff__numeral,axiom,
! [M: nat,V2: num] :
( ( ( semiring_1_of_nat @ int @ M )
= ( numeral_numeral @ int @ V2 ) )
= ( M
= ( numeral_numeral @ nat @ V2 ) ) ) ).
% int_eq_iff_numeral
thf(fact_2670_abs__of__nat,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat] :
( ( abs_abs @ A @ ( semiring_1_of_nat @ A @ N2 ) )
= ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% abs_of_nat
thf(fact_2671_negative__eq__positive,axiom,
! [N2: nat,M: nat] :
( ( ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) )
= ( semiring_1_of_nat @ int @ M ) )
= ( ( N2
= ( zero_zero @ nat ) )
& ( M
= ( zero_zero @ nat ) ) ) ) ).
% negative_eq_positive
thf(fact_2672_of__int__of__nat__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N2: nat] :
( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N2 ) )
= ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% of_int_of_nat_eq
thf(fact_2673_negative__zle,axiom,
! [N2: nat,M: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zle
thf(fact_2674_int__dvd__int__iff,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N2 ) )
= ( dvd_dvd @ nat @ M @ N2 ) ) ).
% int_dvd_int_iff
thf(fact_2675_of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( zero_zero @ A ) )
= ( M
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_eq_0_iff
thf(fact_2676_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ( zero_zero @ A )
= ( semiring_1_of_nat @ A @ N2 ) )
= ( ( zero_zero @ nat )
= N2 ) ) ) ).
% of_nat_0_eq_iff
thf(fact_2677_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_2678_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ) ).
% of_nat_less_iff
thf(fact_2679_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% of_nat_le_iff
thf(fact_2680_of__nat__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N2: nat] :
( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N2 ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% of_nat_add
thf(fact_2681_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N2: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% of_nat_mult
thf(fact_2682_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_2683_of__nat__1__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ( one_one @ A )
= ( semiring_1_of_nat @ A @ N2 ) )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% of_nat_1_eq_iff
thf(fact_2684_of__nat__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ( semiring_1_of_nat @ A @ N2 )
= ( one_one @ A ) )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% of_nat_eq_1_iff
thf(fact_2685_negative__zless,axiom,
! [N2: nat,M: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zless
thf(fact_2686_arctan__less__zero__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( arctan @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% arctan_less_zero_iff
thf(fact_2687_zero__less__arctan__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( arctan @ X3 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% zero_less_arctan_iff
thf(fact_2688_of__nat__of__bool,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P: $o] :
( ( semiring_1_of_nat @ A @ ( zero_neq_one_of_bool @ nat @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% of_nat_of_bool
thf(fact_2689_of__nat__sum,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [F2: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( semiring_1_of_nat @ A @ ( F2 @ X ) )
@ A3 ) ) ) ).
% of_nat_sum
thf(fact_2690_of__nat__le__0__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
= ( M
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_le_0_iff
thf(fact_2691_of__nat__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ M ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) ) ) ).
% of_nat_Suc
thf(fact_2692_real__of__nat__less__numeral__iff,axiom,
! [N2: nat,W2: num] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( numeral_numeral @ real @ W2 ) )
= ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ W2 ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_2693_numeral__less__real__of__nat__iff,axiom,
! [W2: num,N2: nat] :
( ( ord_less @ real @ ( numeral_numeral @ real @ W2 ) @ ( semiring_1_of_nat @ real @ N2 ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ W2 ) @ N2 ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_2694_numeral__le__real__of__nat__iff,axiom,
! [N2: num,M: nat] :
( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N2 ) @ ( semiring_1_of_nat @ real @ M ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N2 ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_2695_of__nat__0__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% of_nat_0_less_iff
thf(fact_2696_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,B2: nat,W2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( ord_less @ nat @ X3 @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_2697_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W2: nat,X3: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X3 ) )
= ( ord_less @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X3 ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_2698_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,B2: nat,W2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( ord_less_eq @ nat @ X3 @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_2699_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W2: nat,X3: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X3 ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X3 ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_2700_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X3 ) @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_2701_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N2: nat,X3: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) @ ( semiring_1_of_nat @ A @ X3 ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) @ X3 ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_2702_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,I: num,N2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) )
= ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_2703_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,I: num,N2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) )
= ( ord_less_eq @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_2704_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N2: nat,X3: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) @ ( semiring_1_of_nat @ A @ X3 ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) @ X3 ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_2705_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_2706_int__sum,axiom,
! [B: $tType,F2: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ int
@ ^ [X: B] : ( semiring_1_of_nat @ int @ ( F2 @ X ) )
@ A3 ) ) ).
% int_sum
thf(fact_2707_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [N3: nat] : ( ord_less_eq @ A @ X3 @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% real_arch_simple
thf(fact_2708_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [N3: nat] : ( ord_less @ A @ X3 @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% reals_Archimedean2
thf(fact_2709_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X3: nat,Y: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X3 ) @ Y )
= ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X3 ) ) ) ) ).
% mult_of_nat_commute
thf(fact_2710_int__cases2,axiom,
! [Z2: int] :
( ! [N3: nat] :
( Z2
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( Z2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_2711_arctan__less__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( arctan @ X3 ) @ ( arctan @ Y ) )
= ( ord_less @ real @ X3 @ Y ) ) ).
% arctan_less_iff
thf(fact_2712_arctan__monotone,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ X3 @ Y )
=> ( ord_less @ real @ ( arctan @ X3 ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone
thf(fact_2713_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M3: nat,N3: nat] :
( Z2
!= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M3 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_2714_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,X3: int] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( ring_1_of_int @ A @ X3 ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ N2 ) @ X3 ) ) ) ).
% of_nat_less_of_int_iff
thf(fact_2715_of__nat__0__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% of_nat_0_le_iff
thf(fact_2716_of__nat__less__0__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat] :
~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% of_nat_less_0_iff
thf(fact_2717_of__nat__neq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ N2 ) )
!= ( zero_zero @ A ) ) ) ).
% of_nat_neq_0
thf(fact_2718_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) )
=> ( ord_less @ nat @ M @ N2 ) ) ) ).
% of_nat_less_imp_less
thf(fact_2719_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_2720_of__nat__mono,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% of_nat_mono
thf(fact_2721_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_2722_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A4: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_2723_int__of__nat__induct,axiom,
! [P: int > $o,Z2: int] :
( ! [N3: nat] : ( P @ ( semiring_1_of_nat @ int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) )
=> ( P @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_2724_int__cases,axiom,
! [Z2: int] :
( ! [N3: nat] :
( Z2
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( Z2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_2725_zle__int,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% zle_int
thf(fact_2726_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_2727_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ? [N3: nat] :
( K
= ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_2728_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_2729_zadd__int__left,axiom,
! [M: nat,N2: nat,Z2: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ Z2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N2 ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_2730_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_2731_int__plus,axiom,
! [N2: nat,M: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N2 @ M ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% int_plus
thf(fact_2732_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W3: int,Z6: int] :
? [N: nat] :
( Z6
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_2733_not__int__zless__negative,axiom,
! [N2: nat,M: nat] :
~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_2734_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A4: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_2735_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_2736_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ? [N3: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ X3 ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_2737_of__nat__diff,axiom,
! [A: $tType] :
( ( semiring_1_cancel @ A )
=> ! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N2 ) )
= ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).
% of_nat_diff
thf(fact_2738_reals__Archimedean3,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ! [Y5: real] :
? [N3: nat] : ( ord_less @ real @ Y5 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ X3 ) ) ) ).
% reals_Archimedean3
thf(fact_2739_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( M
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_2740_int__zle__neg,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) )
= ( ( N2
= ( zero_zero @ nat ) )
& ( M
= ( zero_zero @ nat ) ) ) ) ).
% int_zle_neg
thf(fact_2741_int__ops_I4_J,axiom,
! [A2: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ A2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( one_one @ int ) ) ) ).
% int_ops(4)
thf(fact_2742_int__Suc,axiom,
! [N2: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ N2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( one_one @ int ) ) ) ).
% int_Suc
thf(fact_2743_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W3: int,Z6: int] :
? [N: nat] :
( Z6
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_2744_negative__zle__0,axiom,
! [N2: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( zero_zero @ int ) ) ).
% negative_zle_0
thf(fact_2745_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_2746_sum__roots__unity,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X: complex] : X
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= ( one_one @ complex ) ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_roots_unity
thf(fact_2747_sum__nth__roots,axiom,
! [N2: nat,C2: complex] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X: complex] : X
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= C2 ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_nth_roots
thf(fact_2748_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,M: nat,N2: nat] :
( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) @ ( modulo_modulo @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_2749_pos__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% pos_int_cases
thf(fact_2750_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( K
= ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_2751_int__cases3,axiom,
! [K: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ! [N3: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% int_cases3
thf(fact_2752_nat__less__real__le,axiom,
( ( ord_less @ nat )
= ( ^ [N: nat,M2: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ).
% nat_less_real_le
thf(fact_2753_nat__le__real__less,axiom,
( ( ord_less_eq @ nat )
= ( ^ [N: nat,M2: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( one_one @ real ) ) ) ) ) ).
% nat_le_real_less
thf(fact_2754_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less @ int @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ I ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_2755_not__zle__0__negative,axiom,
! [N2: nat] :
~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ).
% not_zle_0_negative
thf(fact_2756_negative__zless__0,axiom,
! [N2: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) @ ( zero_zero @ int ) ) ).
% negative_zless_0
thf(fact_2757_negD,axiom,
! [X3: int] :
( ( ord_less @ int @ X3 @ ( zero_zero @ int ) )
=> ? [N3: nat] :
( X3
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_2758_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_2759_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_2760_of__nat__less__two__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat] : ( ord_less @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).
% of_nat_less_two_power
thf(fact_2761_inverse__of__nat__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( N2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_2762_real__archimedian__rdiv__eq__0,axiom,
! [X3: real,C2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ! [M3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M3 ) @ X3 ) @ C2 ) )
=> ( X3
= ( zero_zero @ real ) ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_2763_neg__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% neg_int_cases
thf(fact_2764_zdiff__int__split,axiom,
! [P: int > $o,X3: nat,Y: nat] :
( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X3 @ Y ) ) )
= ( ( ( ord_less_eq @ nat @ Y @ X3 )
=> ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X3 ) @ ( semiring_1_of_nat @ int @ Y ) ) ) )
& ( ( ord_less @ nat @ X3 @ Y )
=> ( P @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_2765_ln__realpow,axiom,
! [X3: real,N2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ln_ln @ real @ ( power_power @ real @ X3 @ N2 ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( ln_ln @ real @ X3 ) ) ) ) ).
% ln_realpow
thf(fact_2766_arctan__add,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( plus_plus @ real @ ( arctan @ X3 ) @ ( arctan @ Y ) )
= ( arctan @ ( divide_divide @ real @ ( plus_plus @ real @ X3 @ Y ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( times_times @ real @ X3 @ Y ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_2767_double__gauss__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N2: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum
thf(fact_2768_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,D2: A,N2: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ D2 ) ) ) ) ) ).
% double_arith_series
thf(fact_2769_gauss__sum,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum
thf(fact_2770_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,D2: A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_2771_double__gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N2: 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 ) ) @ N2 ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_2772_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,M: nat,N2: nat] :
( ( ( X3
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N2 ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) )
& ( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N2 ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_2773_of__nat__code__if,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N: nat] :
( if @ A
@ ( N
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [M2: nat,Q5: nat] :
( if @ A
@ ( Q5
= ( zero_zero @ nat ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( one_one @ A ) ) )
@ ( divmod_nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_2774_monoseq__arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( topological_monoseq @ real
@ ^ [N: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_2775_lemma__termdiff3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [H2: A,Z2: A,K5: real,N2: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K5 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z2 @ 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 @ Z2 @ H2 ) @ N2 ) @ ( power_power @ A @ Z2 @ N2 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) @ ( times_times @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( semiring_1_of_nat @ real @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( power_power @ real @ K5 @ ( minus_minus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_2776_ln__series,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ( ln_ln @ real @ X3 )
= ( suminf @ real
@ ^ [N: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X3 @ ( one_one @ real ) ) @ ( suc @ N ) ) ) ) ) ) ) ).
% ln_series
thf(fact_2777_lemma__termdiff2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [H2: A,Z2: A,N2: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H2 ) @ N2 ) @ ( power_power @ A @ Z2 @ N2 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ N2 @ ( 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 @ Z2 @ H2 ) @ Q5 ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ ( minus_minus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q5 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_2778_arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( arctan @ X3 )
= ( 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 @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_2779_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
= ( ord_less @ A @ I @ K ) ) ) ).
% lessThan_iff
thf(fact_2780_finite__lessThan,axiom,
! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_lessThan @ nat @ K ) ) ).
% finite_lessThan
thf(fact_2781_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X3 ) @ ( set_ord_lessThan @ A @ Y ) )
= ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% lessThan_subset_iff
thf(fact_2782_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_2783_single__Diff__lessThan,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A] :
( ( minus_minus @ ( set @ A ) @ ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K ) )
= ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% single_Diff_lessThan
thf(fact_2784_sum_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( G @ N2 ) ) ) ) ).
% sum.lessThan_Suc
thf(fact_2785_powser__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: nat > A] :
( ( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) )
= ( F2 @ ( zero_zero @ nat ) ) ) ) ).
% powser_zero
thf(fact_2786_int__int__eq,axiom,
! [M: nat,N2: nat] :
( ( ( semiring_1_of_nat @ int @ M )
= ( semiring_1_of_nat @ int @ N2 ) )
= ( M = N2 ) ) ).
% int_int_eq
thf(fact_2787_lessThan__non__empty,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X3: A] :
( ( set_ord_lessThan @ A @ X3 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% lessThan_non_empty
thf(fact_2788_infinite__Iio,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A2: A] :
~ ( finite_finite2 @ A @ ( set_ord_lessThan @ A @ A2 ) ) ) ).
% infinite_Iio
thf(fact_2789_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X: A] : ( ord_less @ A @ X @ U2 ) ) ) ) ) ).
% lessThan_def
thf(fact_2790_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( order_bot @ A ) )
=> ! [N2: A] :
( ( ( set_ord_lessThan @ A @ N2 )
= ( bot_bot @ ( set @ A ) ) )
= ( N2
= ( bot_bot @ A ) ) ) ) ).
% Iio_eq_empty_iff
thf(fact_2791_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N2: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N2 ) )
= ( ord_less @ A @ M @ N2 ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_2792_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_2793_lessThan__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan @ nat @ N2 )
= ( bot_bot @ ( set @ nat ) ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_2794_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_2795_finite__nat__bounded,axiom,
! [S3: set @ nat] :
( ( finite_finite2 @ nat @ S3 )
=> ? [K2: nat] : ( ord_less_eq @ ( set @ nat ) @ S3 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ).
% finite_nat_bounded
thf(fact_2796_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_2797_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P: A > nat,N2: A] :
( ! [X5: A] : ( ord_less_eq @ nat @ ( Q @ X5 ) @ ( P @ X5 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N2 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X: A] : ( minus_minus @ nat @ ( P @ X ) @ ( Q @ X ) )
@ ( set_ord_lessThan @ A @ N2 ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_2798_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X8: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ K6 ) ) )
= ( ? [N5: nat] :
! [N: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_2799_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X8: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ K6 ) ) )
= ( ? [N5: nat] :
! [N: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_2800_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_2801_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_2802_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ M ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_2803_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_2804_lemma__termdiff1,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [Z2: A,H2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H2 ) @ ( minus_minus @ nat @ M @ P6 ) ) @ ( power_power @ A @ Z2 @ P6 ) ) @ ( power_power @ A @ Z2 @ M ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ Z2 @ P6 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H2 ) @ ( minus_minus @ nat @ M @ P6 ) ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ M @ P6 ) ) ) )
@ ( set_ord_lessThan @ nat @ M ) ) ) ) ).
% lemma_termdiff1
thf(fact_2805_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N2: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) @ ( power_power @ A @ Y @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X3 @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X3 @ P6 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N2 @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_2806_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N2: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X3 @ N2 ) @ ( power_power @ A @ Y @ N2 ) )
= ( times_times @ A @ ( minus_minus @ A @ X3 @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power @ A @ X3 @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_2807_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,F2: nat > A,K5: A,K: nat] :
( ! [P7: nat] :
( ( ord_less @ nat @ P7 @ N2 )
=> ( ord_less_eq @ A @ ( F2 @ P7 ) @ K5 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K5 )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ K5 ) ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_2808_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N2: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ N2 ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( power_power @ A @ X3 @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_2809_sum__split__even__odd,axiom,
! [F2: nat > real,G: nat > real,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( F2 @ I3 ) @ ( G @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( F2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( one_one @ nat ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% sum_split_even_odd
thf(fact_2810_norm__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( zero_zero @ real ) )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% norm_le_zero_iff
thf(fact_2811_zero__less__norm__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
= ( X3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_norm_iff
thf(fact_2812_suminf__geometric,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) )
=> ( ( suminf @ A @ ( power_power @ A @ C2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C2 ) ) ) ) ) ).
% suminf_geometric
thf(fact_2813_norm__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( zero_zero @ A ) )
= ( zero_zero @ real ) ) ) ).
% norm_zero
thf(fact_2814_norm__eq__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A] :
( ( ( real_V7770717601297561774m_norm @ A @ X3 )
= ( zero_zero @ real ) )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% norm_eq_zero
thf(fact_2815_suminf__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A
@ ^ [N: nat] : ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% suminf_zero
thf(fact_2816_norm__not__less__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A] :
~ ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( zero_zero @ real ) ) ) ).
% norm_not_less_zero
thf(fact_2817_norm__uminus__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y: A] :
( ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) )
= ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ).
% norm_uminus_minus
thf(fact_2818_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_2819_power__eq__imp__eq__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W2: A,N2: nat,Z2: A] :
( ( ( power_power @ A @ W2 @ N2 )
= ( power_power @ A @ Z2 @ N2 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( real_V7770717601297561774m_norm @ A @ W2 )
= ( real_V7770717601297561774m_norm @ A @ Z2 ) ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_2820_norm__mult__less,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X3: A,R2: real,Y: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X3 @ Y ) ) @ ( times_times @ real @ R2 @ S ) ) ) ) ) ).
% norm_mult_less
thf(fact_2821_norm__triangle__lt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y: A,E3: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ E3 ) ) ) ).
% norm_triangle_lt
thf(fact_2822_norm__add__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,R2: real,Y: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ).
% norm_add_less
thf(fact_2823_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_2824_norm__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ).
% norm_triangle_ineq
thf(fact_2825_norm__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y: A,E3: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ E3 ) ) ) ).
% norm_triangle_le
thf(fact_2826_norm__add__leD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C2: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ C2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ C2 ) ) ) ) ).
% norm_add_leD
thf(fact_2827_norm__diff__triangle__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y: A,E1: real,Z2: A,E22: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ Y ) ) @ E1 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ Z2 ) ) @ E22 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ Z2 ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_2828_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_2829_suminf__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [N7: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N7 )
=> ( ( F2 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A @ F2 )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N7 ) ) ) ) ) ).
% suminf_finite
thf(fact_2830_power__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W2: A,N2: nat] :
( ( ( power_power @ A @ W2 @ N2 )
= ( one_one @ A ) )
=> ( ( ( real_V7770717601297561774m_norm @ A @ W2 )
= ( one_one @ real ) )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% power_eq_1_iff
thf(fact_2831_norm__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_2832_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( F2 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_2833_sumr__cos__zero__one,axiom,
! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ ( zero_zero @ real ) @ M2 ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( one_one @ real ) ) ).
% sumr_cos_zero_one
thf(fact_2834_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_2835_summable__arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( 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 @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_2836_ceiling__log__nat__eq__powr__iff,axiom,
! [B2: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( one_one @ int ) ) )
= ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_2837_geometric__deriv__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Z2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( one_one @ real ) )
=> ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( power_power @ A @ Z2 @ N ) )
@ ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( minus_minus @ A @ ( one_one @ A ) @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_2838_summable__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( summable @ A
@ ^ [N: nat] : ( zero_zero @ A ) ) ) ).
% summable_zero
thf(fact_2839_summable__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F2: nat > A] :
( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ).
% summable_single
thf(fact_2840_sums__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( sums @ A
@ ^ [N: nat] : ( zero_zero @ A )
@ ( zero_zero @ A ) ) ) ).
% sums_zero
thf(fact_2841_summable__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A
@ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_iff_shift
thf(fact_2842_cos__coeff__0,axiom,
( ( cos_coeff @ ( zero_zero @ nat ) )
= ( one_one @ real ) ) ).
% cos_coeff_0
thf(fact_2843_summable__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F2 ) ) ) ) ).
% summable_cmult_iff
thf(fact_2844_summable__divide__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A
@ ^ [N: nat] : ( divide_divide @ A @ ( F2 @ N ) @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F2 ) ) ) ) ).
% summable_divide_iff
thf(fact_2845_summable__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F2: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite
thf(fact_2846_summable__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A3: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ A3 )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A3 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite_set
thf(fact_2847_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_2848_log__less__cancel__iff,axiom,
! [A2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y ) )
= ( ord_less @ real @ X3 @ Y ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_2849_log__less__one__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( log @ A2 @ X3 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X3 @ A2 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_2850_one__less__log__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( log @ A2 @ X3 ) )
= ( ord_less @ real @ A2 @ X3 ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_2851_log__less__zero__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( log @ A2 @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_2852_zero__less__log__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X3 ) )
= ( ord_less @ real @ ( one_one @ real ) @ X3 ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_2853_log__le__cancel__iff,axiom,
! [A2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y ) )
= ( ord_less_eq @ real @ X3 @ Y ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_2854_log__le__one__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X3 ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X3 @ A2 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_2855_one__le__log__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( log @ A2 @ X3 ) )
= ( ord_less_eq @ real @ A2 @ X3 ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_2856_log__le__zero__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X3 @ ( one_one @ real ) ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_2857_zero__le__log__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X3 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_2858_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A,X3: A] :
( ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( A2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) )
@ X3 )
= ( ( A2 @ ( zero_zero @ nat ) )
= X3 ) ) ) ).
% powser_sums_zero_iff
thf(fact_2859_summable__geometric__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( summable @ A @ ( power_power @ A @ C2 ) )
= ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) ) ) ) ).
% summable_geometric_iff
thf(fact_2860_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_2861_sums__0,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A] :
( ! [N3: nat] :
( ( F2 @ N3 )
= ( zero_zero @ A ) )
=> ( sums @ A @ F2 @ ( zero_zero @ A ) ) ) ) ).
% sums_0
thf(fact_2862_sums__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,G: nat > A,S: A,T2: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ N3 ) )
=> ( ( sums @ A @ F2 @ S )
=> ( ( sums @ A @ G @ T2 )
=> ( ord_less_eq @ A @ S @ T2 ) ) ) ) ) ).
% sums_le
thf(fact_2863_sums__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F2: nat > A] :
( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) )
@ ( F2 @ I ) ) ) ).
% sums_single
thf(fact_2864_sums__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,A2: A,G: nat > A,B2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( ( sums @ A @ G @ B2 )
=> ( sums @ A
@ ^ [N: nat] : ( plus_plus @ A @ ( F2 @ N ) @ ( G @ N ) )
@ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% sums_add
thf(fact_2865_summable__comparison__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test
thf(fact_2866_summable__comparison__test_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G: nat > real,N7: nat,F2: nat > A] :
( ( summable @ real @ G )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test'
thf(fact_2867_summable__const__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C2: A] :
( ( summable @ A
@ ^ [Uu3: nat] : C2 )
= ( C2
= ( zero_zero @ A ) ) ) ) ).
% summable_const_iff
thf(fact_2868_summable__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( summable @ A
@ ^ [N: nat] : ( plus_plus @ A @ ( F2 @ N ) @ ( G @ N ) ) ) ) ) ) ).
% summable_add
thf(fact_2869_summable__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_Suc_iff
thf(fact_2870_summable__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_2871_powser__insidea,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,X3: A,Z2: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ X3 @ N ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( summable @ real
@ ^ [N: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) ) ) ) ) ).
% powser_insidea
thf(fact_2872_suminf__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ N3 ) )
=> ( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) ) ) ) ) ) ).
% suminf_le
thf(fact_2873_summable__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N7: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N7 )
=> ( ( F2 @ N3 )
= ( zero_zero @ A ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_finite
thf(fact_2874_sums__mult__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F2: nat > A,D2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) )
@ ( times_times @ A @ C2 @ D2 ) )
= ( sums @ A @ F2 @ D2 ) ) ) ) ).
% sums_mult_iff
thf(fact_2875_sums__mult2__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F2: nat > A,D2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ C2 )
@ ( times_times @ A @ D2 @ C2 ) )
= ( sums @ A @ F2 @ D2 ) ) ) ) ).
% sums_mult2_iff
thf(fact_2876_summable__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_mult_D
thf(fact_2877_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_2878_pi__not__less__zero,axiom,
~ ( ord_less @ real @ pi @ ( zero_zero @ real ) ) ).
% pi_not_less_zero
thf(fact_2879_pi__gt__zero,axiom,
ord_less @ real @ ( zero_zero @ real ) @ pi ).
% pi_gt_zero
thf(fact_2880_suminf__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( ( plus_plus @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) )
= ( suminf @ A
@ ^ [N: nat] : ( plus_plus @ A @ ( F2 @ N ) @ ( G @ N ) ) ) ) ) ) ) ).
% suminf_add
thf(fact_2881_suminf__eq__zero__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ( ( suminf @ A @ F2 )
= ( zero_zero @ A ) )
= ( ! [N: nat] :
( ( F2 @ N )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_2882_suminf__nonneg,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_nonneg
thf(fact_2883_suminf__pos,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_pos
thf(fact_2884_sums__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A,A2: A] :
( ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) )
@ A2 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( sums @ A @ F2 @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ).
% sums_mult_D
thf(fact_2885_sums__Suc__imp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) )
@ S )
=> ( sums @ A @ F2 @ S ) ) ) ) ).
% sums_Suc_imp
thf(fact_2886_sums__Suc,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,L: A] :
( ( sums @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) )
@ L )
=> ( sums @ A @ F2 @ ( plus_plus @ A @ L @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc
thf(fact_2887_sums__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A] :
( ( sums @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) )
@ S )
= ( sums @ A @ F2 @ ( plus_plus @ A @ S @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc_iff
thf(fact_2888_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N2: nat,F2: nat > A,S: A] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
=> ( ( F2 @ I4 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) )
@ S )
= ( sums @ A @ F2 @ S ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_2889_summable__0__powser,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A] :
( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) ) ) ).
% summable_0_powser
thf(fact_2890_summable__zero__power_H,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A] :
( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) ) ) ).
% summable_zero_power'
thf(fact_2891_summable__powser__split__head,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,Z2: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z2 @ N ) ) )
= ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) ) ) ).
% summable_powser_split_head
thf(fact_2892_powser__split__head_I3_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z2: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) )
=> ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z2 @ N ) ) ) ) ) ).
% powser_split_head(3)
thf(fact_2893_summable__powser__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,M: nat,Z2: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( plus_plus @ nat @ N @ M ) ) @ ( power_power @ A @ Z2 @ N ) ) )
= ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_2894_sums__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N7: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N7 )
=> ( ( F2 @ N3 )
= ( zero_zero @ A ) ) )
=> ( sums @ A @ F2 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N7 ) ) ) ) ) ).
% sums_finite
thf(fact_2895_sums__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F2: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( collect @ nat @ P ) ) ) ) ) ).
% sums_If_finite
thf(fact_2896_sums__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A3: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ A3 )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A3 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A3 ) ) ) ) ).
% sums_If_finite_set
thf(fact_2897_summable__norm__comparison__test,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ real
@ ^ [N: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N ) ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_2898_summable__rabs__comparison__test,axiom,
! [F2: nat > real,G: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ real
@ ^ [N: nat] : ( abs_abs @ real @ ( F2 @ N ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_2899_suminf__pos__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) )
= ( ? [I3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_2900_suminf__pos2,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I: nat] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ).
% suminf_pos2
thf(fact_2901_suminf__le__const,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X3: A] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X3 )
=> ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ X3 ) ) ) ) ).
% suminf_le_const
thf(fact_2902_powser__sums__if,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [M: nat,Z2: A] :
( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( if @ A @ ( N = M ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z2 @ N ) )
@ ( power_power @ A @ Z2 @ M ) ) ) ).
% powser_sums_if
thf(fact_2903_powser__sums__zero,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A] :
( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( A2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) )
@ ( A2 @ ( zero_zero @ nat ) ) ) ) ).
% powser_sums_zero
thf(fact_2904_log__base__change,axiom,
! [A2: real,B2: real,X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ B2 @ X3 )
= ( divide_divide @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ B2 ) ) ) ) ) ).
% log_base_change
thf(fact_2905_log__of__power__eq,axiom,
! [M: nat,B2: real,N2: nat] :
( ( ( semiring_1_of_nat @ real @ M )
= ( power_power @ real @ B2 @ N2 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( semiring_1_of_nat @ real @ N2 )
= ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ).
% log_of_power_eq
thf(fact_2906_less__log__of__power,axiom,
! [B2: real,N2: nat,M: real] :
( ( ord_less @ real @ ( power_power @ real @ B2 @ N2 ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ M ) ) ) ) ).
% less_log_of_power
thf(fact_2907_powser__inside,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,X3: A,Z2: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ X3 @ N ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) ) ) ) ).
% powser_inside
thf(fact_2908_sums__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,N2: nat,S: A] :
( ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) )
@ S )
= ( sums @ A @ F2 @ ( plus_plus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).
% sums_iff_shift
thf(fact_2909_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X3: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X3 )
=> ( summable @ A @ F2 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_2910_sums__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A,N2: nat] :
( ( sums @ A @ F2 @ S )
=> ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_2911_sums__iff__shift_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,N2: nat,S: A] :
( ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) ) )
= ( sums @ A @ F2 @ S ) ) ) ).
% sums_iff_shift'
thf(fact_2912_complete__algebra__summable__geometric,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ X3 ) ) ) ) ).
% complete_algebra_summable_geometric
thf(fact_2913_summable__geometric,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ C2 ) ) ) ) ).
% summable_geometric
thf(fact_2914_sums__If__finite__set_H,axiom,
! [A: $tType] :
( ( ( topolo1287966508704411220up_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G: nat > A,S3: A,A3: set @ nat,S6: A,F2: nat > A] :
( ( sums @ A @ G @ S3 )
=> ( ( finite_finite2 @ nat @ A3 )
=> ( ( S6
= ( plus_plus @ A @ S3
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ N ) @ ( G @ N ) )
@ A3 ) ) )
=> ( sums @ A
@ ^ [N: nat] : ( if @ A @ ( member @ nat @ N @ A3 ) @ ( F2 @ N ) @ ( G @ N ) )
@ S6 ) ) ) ) ) ).
% sums_If_finite_set'
thf(fact_2915_suminf__split__head,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% suminf_split_head
thf(fact_2916_pi__less__4,axiom,
ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ).
% pi_less_4
thf(fact_2917_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I6: set @ nat] :
( ( summable @ A @ F2 )
=> ( ( finite_finite2 @ nat @ I6 )
=> ( ! [N3: nat] :
( ( member @ nat @ N3 @ ( uminus_uminus @ ( set @ nat ) @ I6 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ I6 ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ).
% sum_le_suminf
thf(fact_2918_log__mult,axiom,
! [A2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( log @ A2 @ ( times_times @ real @ X3 @ Y ) )
= ( plus_plus @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ).
% log_mult
thf(fact_2919_log__divide,axiom,
! [A2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( log @ A2 @ ( divide_divide @ real @ X3 @ Y ) )
= ( minus_minus @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ).
% log_divide
thf(fact_2920_le__log__of__power,axiom,
! [B2: real,N2: nat,M: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ B2 @ N2 ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ M ) ) ) ) ).
% le_log_of_power
thf(fact_2921_log__base__pow,axiom,
! [A2: real,N2: nat,X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( log @ ( power_power @ real @ A2 @ N2 ) @ X3 )
= ( divide_divide @ real @ ( log @ A2 @ X3 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ).
% log_base_pow
thf(fact_2922_log__nat__power,axiom,
! [X3: real,B2: real,N2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( log @ B2 @ ( power_power @ real @ X3 @ N2 ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ X3 ) ) ) ) ).
% log_nat_power
thf(fact_2923_suminf__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A @ F2 )
= ( plus_plus @ A
@ ( suminf @ A
@ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).
% suminf_split_initial_segment
thf(fact_2924_suminf__minus__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).
% suminf_minus_initial_segment
thf(fact_2925_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_2926_sum__less__suminf,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N2: nat] :
( ( summable @ A @ F2 )
=> ( ! [M3: nat] :
( ( ord_less_eq @ nat @ N2 @ M3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ M3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% sum_less_suminf
thf(fact_2927_powser__split__head_I1_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z2: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) )
=> ( ( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) )
= ( plus_plus @ A @ ( F2 @ ( zero_zero @ nat ) )
@ ( times_times @ A
@ ( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z2 @ N ) ) )
@ Z2 ) ) ) ) ) ).
% powser_split_head(1)
thf(fact_2928_powser__split__head_I2_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z2: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) )
=> ( ( times_times @ A
@ ( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z2 @ N ) ) )
@ Z2 )
= ( minus_minus @ A
@ ( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) )
@ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% powser_split_head(2)
thf(fact_2929_log__of__power__less,axiom,
! [M: nat,B2: real,N2: nat] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N2 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).
% log_of_power_less
thf(fact_2930_log__eq__div__ln__mult__log,axiom,
! [A2: real,B2: real,X3: 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 ) @ X3 )
=> ( ( log @ A2 @ X3 )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( ln_ln @ real @ A2 ) ) @ ( log @ B2 @ X3 ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_2931_summable__partial__sum__bound,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,E3: real] :
( ( summable @ A @ F2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ~ ! [N9: nat] :
~ ! [M6: nat] :
( ( ord_less_eq @ nat @ N9 @ M6 )
=> ! [N6: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ M6 @ N6 ) ) ) @ E3 ) ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_2932_suminf__exist__split,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,F2: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( summable @ A @ F2 )
=> ? [N9: nat] :
! [N6: nat] :
( ( ord_less_eq @ nat @ N9 @ N6 )
=> ( ord_less @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( suminf @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N6 ) ) ) )
@ R2 ) ) ) ) ) ).
% suminf_exist_split
thf(fact_2933_summable__power__series,axiom,
! [F2: nat > real,Z2: real] :
( ! [I4: nat] : ( ord_less_eq @ real @ ( F2 @ I4 ) @ ( one_one @ real ) )
=> ( ! [I4: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ I4 ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Z2 )
=> ( ( ord_less @ real @ Z2 @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [I3: nat] : ( times_times @ real @ ( F2 @ I3 ) @ ( power_power @ real @ Z2 @ I3 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_2934_Abel__lemma,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,R0: real,A2: nat > A,M7: 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 ) ) @ M7 )
=> ( summable @ real
@ ^ [N: nat] : ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N ) ) @ ( power_power @ real @ R2 @ N ) ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_2935_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_2936_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_2937_summable__ratio__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [C2: real,N7: nat,F2: nat > A] :
( ( ord_less @ real @ C2 @ ( one_one @ real ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ ( suc @ N3 ) ) ) @ ( times_times @ real @ C2 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_ratio_test
thf(fact_2938_sum__less__suminf2,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N2: nat,I: nat] :
( ( summable @ A @ F2 )
=> ( ! [M3: nat] :
( ( ord_less_eq @ nat @ N2 @ M3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ M3 ) ) )
=> ( ( ord_less_eq @ nat @ N2 @ I )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_2939_arctan__ubound,axiom,
! [Y: real] : ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arctan_ubound
thf(fact_2940_geometric__sums,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) )
=> ( sums @ A @ ( power_power @ A @ C2 ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C2 ) ) ) ) ) ).
% geometric_sums
thf(fact_2941_power__half__series,axiom,
( sums @ real
@ ^ [N: nat] : ( power_power @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( suc @ N ) )
@ ( one_one @ real ) ) ).
% power_half_series
thf(fact_2942_log__of__power__le,axiom,
! [M: nat,B2: real,N2: nat] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N2 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less_eq @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).
% log_of_power_le
thf(fact_2943_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_2944_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_2945_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_2946_less__log2__of__power,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% less_log2_of_power
thf(fact_2947_le__log2__of__power,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% le_log2_of_power
thf(fact_2948_log2__of__power__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ).
% log2_of_power_less
thf(fact_2949_log2__of__power__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less_eq @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ).
% log2_of_power_le
thf(fact_2950_sum__pos__lt__pair,axiom,
! [F2: nat > real,K: nat] :
( ( summable @ real @ F2 )
=> ( ! [D3: nat] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( F2 @ ( plus_plus @ nat @ K @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D3 ) ) ) @ ( F2 @ ( plus_plus @ nat @ K @ ( plus_plus @ nat @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D3 ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ord_less @ real @ ( groups7311177749621191930dd_sum @ nat @ real @ F2 @ ( set_ord_lessThan @ nat @ K ) ) @ ( suminf @ real @ F2 ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_2951_ceiling__log__nat__eq__if,axiom,
! [B2: nat,N2: nat,K: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
=> ( ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) )
=> ( ( 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 @ K ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_2952_ceiling__log2__div2,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) )
= ( 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 @ N2 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% ceiling_log2_div2
thf(fact_2953_sin__cos__npi,axiom,
! [N2: nat] :
( ( sin @ real @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) ) ).
% sin_cos_npi
thf(fact_2954_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C2: nat > A,X3: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) )
=> ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( C2 @ N ) ) @ ( power_power @ A @ X3 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_2955_cos__pi__eq__zero,axiom,
! [M: nat] :
( ( cos @ real @ ( divide_divide @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_eq_zero
thf(fact_2956_ceiling__log__eq__powr__iff,axiom,
! [X3: real,B2: real,K: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archimedean_ceiling @ real @ ( log @ B2 @ X3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) ) )
= ( ( ord_less @ real @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ K ) ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_2957_floor__log__nat__eq__powr__iff,axiom,
! [B2: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( semiring_1_of_nat @ int @ N2 ) )
= ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_2958_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_2959_powr__eq__0__iff,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [W2: A,Z2: A] :
( ( ( powr @ A @ W2 @ Z2 )
= ( zero_zero @ A ) )
= ( W2
= ( zero_zero @ A ) ) ) ) ).
% powr_eq_0_iff
thf(fact_2960_powr__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [Z2: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z2 )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_2961_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_2962_powr__zero__eq__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [X3: A] :
( ( ( X3
= ( zero_zero @ A ) )
=> ( ( powr @ A @ X3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) )
& ( ( X3
!= ( zero_zero @ A ) )
=> ( ( powr @ A @ X3 @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% powr_zero_eq_one
thf(fact_2963_floor__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_2964_powr__gt__zero,axiom,
! [X3: real,A2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( powr @ real @ X3 @ A2 ) )
= ( X3
!= ( zero_zero @ real ) ) ) ).
% powr_gt_zero
thf(fact_2965_powr__less__cancel__iff,axiom,
! [X3: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) )
= ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel_iff
thf(fact_2966_powr__eq__one__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ( powr @ real @ A2 @ X3 )
= ( one_one @ real ) )
= ( X3
= ( zero_zero @ real ) ) ) ) ).
% powr_eq_one_iff
thf(fact_2967_powr__le__cancel__iff,axiom,
! [X3: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) )
= ( ord_less_eq @ real @ A2 @ B2 ) ) ) ).
% powr_le_cancel_iff
thf(fact_2968_zero__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ) ).
% zero_le_floor
thf(fact_2969_sin__cos__squared__add3,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ X3 ) ) @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ X3 ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add3
thf(fact_2970_floor__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X3 @ ( zero_zero @ A ) ) ) ) ).
% floor_less_zero
thf(fact_2971_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V2 ) @ X3 ) ) ) ).
% numeral_le_floor
thf(fact_2972_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ).
% zero_less_floor
thf(fact_2973_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_2974_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% floor_less_numeral
thf(fact_2975_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ).
% one_le_floor
thf(fact_2976_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) )
= ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_2977_powr__log__cancel,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( powr @ real @ A2 @ ( log @ A2 @ X3 ) )
= X3 ) ) ) ) ).
% powr_log_cancel
thf(fact_2978_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_2979_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% numeral_less_floor
thf(fact_2980_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_2981_one__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) ) ) ).
% one_less_floor
thf(fact_2982_floor__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) )
= ( ord_less @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% floor_le_one
thf(fact_2983_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X3 ) ) ) ).
% neg_numeral_le_floor
thf(fact_2984_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_2985_sin__cos__squared__add2,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add2
thf(fact_2986_sin__cos__squared__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add
thf(fact_2987_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% neg_numeral_less_floor
thf(fact_2988_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_2989_cos__one__sin__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cos @ A @ X3 )
= ( one_one @ A ) )
=> ( ( sin @ A @ X3 )
= ( zero_zero @ A ) ) ) ) ).
% cos_one_sin_zero
thf(fact_2990_sin__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( sin @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ) ).
% sin_add
thf(fact_2991_cos__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( cos @ A @ ( minus_minus @ A @ X3 @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ) ).
% cos_diff
thf(fact_2992_cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( minus_minus @ A @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ) ).
% cos_add
thf(fact_2993_sin__zero__norm__cos__one,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( sin @ A @ X3 )
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( cos @ A @ X3 ) )
= ( one_one @ real ) ) ) ) ).
% sin_zero_norm_cos_one
thf(fact_2994_sincos__principal__value,axiom,
! [X3: real] :
? [Y4: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ Y4 )
& ( ord_less_eq @ real @ Y4 @ pi )
& ( ( sin @ real @ Y4 )
= ( sin @ real @ X3 ) )
& ( ( cos @ real @ Y4 )
= ( cos @ real @ X3 ) ) ) ).
% sincos_principal_value
thf(fact_2995_floor__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ).
% floor_mono
thf(fact_2996_of__int__floor__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) @ X3 ) ) ).
% of_int_floor_le
thf(fact_2997_floor__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) )
=> ( ord_less @ A @ X3 @ Y ) ) ) ).
% floor_less_cancel
thf(fact_2998_powr__non__neg,axiom,
! [A2: real,X3: real] :
~ ( ord_less @ real @ ( powr @ real @ A2 @ X3 ) @ ( zero_zero @ real ) ) ).
% powr_non_neg
thf(fact_2999_powr__less__mono2__neg,axiom,
! [A2: real,X3: real,Y: real] :
( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ Y )
=> ( ord_less @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X3 @ A2 ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_3000_powr__less__cancel,axiom,
! [X3: real,A2: real,B2: real] :
( ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel
thf(fact_3001_powr__less__mono,axiom,
! [A2: real,B2: real,X3: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) ) ) ) ).
% powr_less_mono
thf(fact_3002_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( ord_less_eq @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 ) ) ) ).
% le_floor_iff
thf(fact_3003_floor__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z2 )
= ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% floor_less_iff
thf(fact_3004_powr__mono2_H,axiom,
! [A2: real,X3: real,Y: real] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X3 @ A2 ) ) ) ) ) ).
% powr_mono2'
thf(fact_3005_powr__less__mono2,axiom,
! [A2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ Y )
=> ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_3006_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ).
% le_floor_add
thf(fact_3007_gr__one__powr,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less @ real @ ( one_one @ real ) @ ( powr @ real @ X3 @ Y ) ) ) ) ).
% gr_one_powr
thf(fact_3008_powr__inj,axiom,
! [A2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ( powr @ real @ A2 @ X3 )
= ( powr @ real @ A2 @ Y ) )
= ( X3 = Y ) ) ) ) ).
% powr_inj
thf(fact_3009_int__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( plus_plus @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 ) ) ) ) ).
% int_add_floor
thf(fact_3010_floor__add__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z2 )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ) ).
% floor_add_int
thf(fact_3011_sin__gt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ pi )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) ) ) ) ).
% sin_gt_zero
thf(fact_3012_sin__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z2: A] :
( ( times_times @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( cos @ A @ ( minus_minus @ A @ W2 @ Z2 ) ) @ ( cos @ A @ ( plus_plus @ A @ W2 @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_sin
thf(fact_3013_sin__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z2: A] :
( ( times_times @ A @ ( sin @ A @ W2 ) @ ( cos @ A @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( sin @ A @ ( plus_plus @ A @ W2 @ Z2 ) ) @ ( sin @ A @ ( minus_minus @ A @ W2 @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_cos
thf(fact_3014_cos__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z2: A] :
( ( times_times @ A @ ( cos @ A @ W2 ) @ ( sin @ A @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( sin @ A @ ( plus_plus @ A @ W2 @ Z2 ) ) @ ( sin @ A @ ( minus_minus @ A @ W2 @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_sin
thf(fact_3015_sin__plus__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z2: A] :
( ( plus_plus @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_3016_sin__diff__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z2: A] :
( ( minus_minus @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_3017_cos__diff__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z2: A] :
( ( minus_minus @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ Z2 @ W2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_3018_powr__add,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X3: A,A2: A,B2: A] :
( ( powr @ A @ X3 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( powr @ A @ X3 @ A2 ) @ ( powr @ A @ X3 @ B2 ) ) ) ) ).
% powr_add
thf(fact_3019_one__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% one_add_floor
thf(fact_3020_floor__log__eq__powr__iff,axiom,
! [X3: real,B2: real,K: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ B2 @ X3 ) )
= K )
= ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ K ) ) @ X3 )
& ( ord_less @ real @ X3 @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_3021_powr__realpow,axiom,
! [X3: real,N2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( powr @ real @ X3 @ ( semiring_1_of_nat @ real @ N2 ) )
= ( power_power @ real @ X3 @ N2 ) ) ) ).
% powr_realpow
thf(fact_3022_less__log__iff,axiom,
! [B2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ Y @ ( log @ B2 @ X3 ) )
= ( ord_less @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) ) ) ) ).
% less_log_iff
thf(fact_3023_log__less__iff,axiom,
! [B2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( log @ B2 @ X3 ) @ Y )
= ( ord_less @ real @ X3 @ ( powr @ real @ B2 @ Y ) ) ) ) ) ).
% log_less_iff
thf(fact_3024_less__powr__iff,axiom,
! [B2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( powr @ real @ B2 @ Y ) )
= ( ord_less @ real @ ( log @ B2 @ X3 ) @ Y ) ) ) ) ).
% less_powr_iff
thf(fact_3025_powr__less__iff,axiom,
! [B2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( powr @ real @ B2 @ Y ) @ X3 )
= ( ord_less @ real @ Y @ ( log @ B2 @ X3 ) ) ) ) ) ).
% powr_less_iff
thf(fact_3026_cos__mono__less__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ord_less @ real @ ( cos @ real @ X3 ) @ ( cos @ real @ Y ) )
= ( ord_less @ real @ Y @ X3 ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_3027_cos__monotone__0__pi,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ Y @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ pi )
=> ( ord_less @ real @ ( cos @ real @ X3 ) @ ( cos @ real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_3028_sin__eq__0__pi,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X3 )
=> ( ( ord_less @ real @ X3 @ pi )
=> ( ( ( sin @ real @ X3 )
= ( zero_zero @ real ) )
=> ( X3
= ( zero_zero @ real ) ) ) ) ) ).
% sin_eq_0_pi
thf(fact_3029_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_3030_floor__eq,axiom,
! [N2: int,X3: real] :
( ( ord_less @ real @ ( ring_1_of_int @ real @ N2 ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N2 ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X3 )
= N2 ) ) ) ).
% floor_eq
thf(fact_3031_sin__zero__pi__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ pi )
=> ( ( ( sin @ real @ X3 )
= ( zero_zero @ real ) )
= ( X3
= ( zero_zero @ real ) ) ) ) ).
% sin_zero_pi_iff
thf(fact_3032_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_3033_diffs__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( diffs @ A )
= ( ^ [C4: nat > A,N: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( C4 @ ( suc @ N ) ) ) ) ) ) ).
% diffs_def
thf(fact_3034_sin__expansion__lemma,axiom,
! [X3: real,M: nat] :
( ( sin @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( cos @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_3035_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 )
=> ( ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X3 )
= Z2 ) ) ) ) ).
% floor_unique
thf(fact_3036_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,A2: int] :
( ( ( archim6421214686448440834_floor @ A @ X3 )
= A2 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ X3 )
& ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_3037_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I3 ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% floor_split
thf(fact_3038_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_3039_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( ord_less @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% less_floor_iff
thf(fact_3040_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z2 )
= ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_3041_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_3042_cos__expansion__lemma,axiom,
! [X3: real,M: nat] :
( ( cos @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( uminus_uminus @ real @ ( sin @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_3043_powr__neg__one,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( powr @ real @ X3 @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ X3 ) ) ) ).
% powr_neg_one
thf(fact_3044_sin__gt__zero__02,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) ) ) ) ).
% sin_gt_zero_02
thf(fact_3045_le__log__iff,axiom,
! [B2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ Y @ ( log @ B2 @ X3 ) )
= ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) ) ) ) ).
% le_log_iff
thf(fact_3046_log__le__iff,axiom,
! [B2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( log @ B2 @ X3 ) @ Y )
= ( ord_less_eq @ real @ X3 @ ( powr @ real @ B2 @ Y ) ) ) ) ) ).
% log_le_iff
thf(fact_3047_le__powr__iff,axiom,
! [B2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( powr @ real @ B2 @ Y ) )
= ( ord_less_eq @ real @ ( log @ B2 @ X3 ) @ Y ) ) ) ) ).
% le_powr_iff
thf(fact_3048_powr__le__iff,axiom,
! [B2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y ) @ X3 )
= ( ord_less_eq @ real @ Y @ ( log @ B2 @ X3 ) ) ) ) ) ).
% powr_le_iff
thf(fact_3049_cos__two__less__zero,axiom,
ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_less_zero
thf(fact_3050_floor__eq2,axiom,
! [N2: int,X3: real] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ N2 ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N2 ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X3 )
= N2 ) ) ) ).
% floor_eq2
thf(fact_3051_cos__monotone__minus__pi__0,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y )
=> ( ( ord_less @ real @ Y @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cos @ real @ Y ) @ ( cos @ real @ X3 ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_3052_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ) ).
% floor_divide_lower
thf(fact_3053_sincos__total__2pi,axiom,
! [X3: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X3 @ ( 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 ) )
=> ( ( X3
= ( cos @ real @ T7 ) )
=> ( Y
!= ( sin @ real @ T7 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_3054_ln__powr__bound,axiom,
! [X3: real,A2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( divide_divide @ real @ ( powr @ real @ X3 @ A2 ) @ A2 ) ) ) ) ).
% ln_powr_bound
thf(fact_3055_ln__powr__bound2,axiom,
! [X3: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X3 ) @ A2 ) @ ( times_times @ real @ ( powr @ real @ A2 @ A2 ) @ X3 ) ) ) ) ).
% ln_powr_bound2
thf(fact_3056_log__add__eq__powr,axiom,
! [B2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( plus_plus @ real @ ( log @ B2 @ X3 ) @ Y )
= ( log @ B2 @ ( times_times @ real @ X3 @ ( powr @ real @ B2 @ Y ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_3057_add__log__eq__powr,axiom,
! [B2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( plus_plus @ real @ Y @ ( log @ B2 @ X3 ) )
= ( log @ B2 @ ( times_times @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_3058_minus__log__eq__powr,axiom,
! [B2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( minus_minus @ real @ Y @ ( log @ B2 @ X3 ) )
= ( log @ B2 @ ( divide_divide @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_3059_sin__pi__divide__n__ge__0,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).
% sin_pi_divide_n_ge_0
thf(fact_3060_cos__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z2: A] :
( ( times_times @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( cos @ A @ ( minus_minus @ A @ W2 @ Z2 ) ) @ ( cos @ A @ ( plus_plus @ A @ W2 @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_cos
thf(fact_3061_cos__plus__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z2: A] :
( ( plus_plus @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_3062_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less @ A @ P4 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) ) ) ) ).
% floor_divide_upper
thf(fact_3063_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_3064_sin__gt__zero2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) ) ) ) ).
% sin_gt_zero2
thf(fact_3065_sin__lt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ pi @ X3 )
=> ( ( ord_less @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less @ real @ ( sin @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% sin_lt_zero
thf(fact_3066_cos__double__less__one,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) ) @ ( one_one @ real ) ) ) ) ).
% cos_double_less_one
thf(fact_3067_cos__gt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X3 ) ) ) ) ).
% cos_gt_zero
thf(fact_3068_log__minus__eq__powr,axiom,
! [B2: real,X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( minus_minus @ real @ ( log @ B2 @ X3 ) @ Y )
= ( log @ B2 @ ( times_times @ real @ X3 @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ Y ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_3069_termdiff__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,K5: real,C2: nat > A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K5 )
=> ( ! [X5: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K5 )
=> ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ X5 @ N ) ) ) )
=> ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) ) ) ) ).
% termdiff_converges
thf(fact_3070_powr__neg__numeral,axiom,
! [X3: real,N2: num] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( powr @ real @ X3 @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ N2 ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ N2 ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_3071_sin__le__zero,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ pi @ X3 )
=> ( ( ord_less @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less_eq @ real @ ( sin @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% sin_le_zero
thf(fact_3072_sin__less__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( sin @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% sin_less_zero
thf(fact_3073_sin__mono__less__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( 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 @ X3 ) @ ( sin @ real @ Y ) )
= ( ord_less @ real @ X3 @ Y ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_3074_sin__monotone__2pi,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X3 ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_3075_cos__gt__zero__pi,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X3 ) ) ) ) ).
% cos_gt_zero_pi
thf(fact_3076_sin__pi__divide__n__gt__0,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).
% sin_pi_divide_n_gt_0
thf(fact_3077_floor__log2__div2,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% floor_log2_div2
thf(fact_3078_floor__log__nat__eq__if,axiom,
! [B2: nat,N2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
=> ( ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) )
=> ( ( 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 @ K ) ) )
= ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_3079_Maclaurin__minus__cos__expansion,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ? [T7: real] :
( ( ord_less @ real @ X3 @ T7 )
& ( ord_less @ real @ T7 @ ( zero_zero @ real ) )
& ( ( cos @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ X3 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_3080_Maclaurin__cos__expansion2,axiom,
! [X3: real,N2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ X3 )
& ( ( cos @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ X3 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_3081_sin__paired,axiom,
! [X3: real] :
( sums @ real
@ ^ [N: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) )
@ ( sin @ real @ X3 ) ) ).
% sin_paired
thf(fact_3082_tan__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( tan @ A @ X3 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_3083_arcosh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A )
= ( ^ [X: A] : ( ln_ln @ A @ ( plus_plus @ A @ X @ ( powr @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ ( 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_3084_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_3085_of__real__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X3: real] :
( ( ( real_Vector_of_real @ A @ X3 )
= ( zero_zero @ A ) )
= ( X3
= ( zero_zero @ real ) ) ) ) ).
% of_real_eq_0_iff
thf(fact_3086_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_3087_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_3088_of__real__add,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X3: real,Y: real] :
( ( real_Vector_of_real @ A @ ( plus_plus @ real @ X3 @ Y ) )
= ( plus_plus @ A @ ( real_Vector_of_real @ A @ X3 ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ).
% of_real_add
thf(fact_3089_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_3090_fact__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( semiring_char_0_fact @ A @ ( suc @ N2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).
% fact_Suc
thf(fact_3091_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_3092_norm__of__real__add1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X3: real] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X3 ) @ ( one_one @ A ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X3 @ ( one_one @ real ) ) ) ) ) ).
% norm_of_real_add1
thf(fact_3093_norm__of__real__addn,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X3: real,B2: num] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X3 ) @ ( numeral_numeral @ A @ B2 ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X3 @ ( numeral_numeral @ real @ B2 ) ) ) ) ) ).
% norm_of_real_addn
thf(fact_3094_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_3095_fact__nonzero,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N2: nat] :
( ( semiring_char_0_fact @ A @ N2 )
!= ( zero_zero @ A ) ) ) ).
% fact_nonzero
thf(fact_3096_fact__ge__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ).
% fact_ge_zero
thf(fact_3097_fact__not__neg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] :
~ ( ord_less @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( zero_zero @ A ) ) ) ).
% fact_not_neg
thf(fact_3098_fact__gt__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ).
% fact_gt_zero
thf(fact_3099_fact__ge__1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ).
% fact_ge_1
thf(fact_3100_fact__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).
% fact_mono
thf(fact_3101_fact__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ M ) ) ) ) ).
% fact_dvd
thf(fact_3102_fact__less__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ) ).
% fact_less_mono
thf(fact_3103_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N2: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N2 ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N2 ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_3104_fact__mod,axiom,
! [A: $tType] :
( ( ( linordered_semidom @ A )
& ( semidom_modulo @ A ) )
=> ! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( modulo_modulo @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ M ) )
= ( zero_zero @ A ) ) ) ) ).
% fact_mod
thf(fact_3105_fact__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] : ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ N2 @ N2 ) ) ) ) ).
% fact_le_power
thf(fact_3106_norm__less__p1,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X3: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) @ ( one_one @ A ) ) ) ) ) ).
% norm_less_p1
thf(fact_3107_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).
% choose_dvd
thf(fact_3108_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M2: nat] :
( if @ A
@ ( M2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_3109_fact__reduce,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( semiring_char_0_fact @ A @ N2 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% fact_reduce
thf(fact_3110_tan__gt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( tan @ real @ X3 ) ) ) ) ).
% tan_gt_zero
thf(fact_3111_lemma__tan__total,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ? [X5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X5 )
& ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ord_less @ real @ Y @ ( tan @ real @ X5 ) ) ) ) ).
% lemma_tan_total
thf(fact_3112_lemma__tan__total1,axiom,
! [Y: 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 )
= Y ) ) ).
% lemma_tan_total1
thf(fact_3113_tan__mono__lt__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( 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 @ X3 ) @ ( tan @ real @ Y ) )
= ( ord_less @ real @ X3 @ Y ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_3114_tan__monotone_H,axiom,
! [Y: real,X3: 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 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y @ X3 )
= ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X3 ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_3115_tan__monotone,axiom,
! [Y: real,X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X3 ) ) ) ) ) ).
% tan_monotone
thf(fact_3116_tan__total,axiom,
! [Y: 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 )
= Y )
& ! [Y5: real] :
( ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y5 )
& ( ord_less @ real @ Y5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ Y5 )
= Y ) )
=> ( Y5 = X5 ) ) ) ).
% tan_total
thf(fact_3117_add__tan__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) )
= ( divide_divide @ A @ ( sin @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_3118_tan__total__pos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ? [X5: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X5 )
& ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X5 )
= Y ) ) ) ).
% tan_total_pos
thf(fact_3119_tan__pos__pi2__le,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tan @ real @ X3 ) ) ) ) ).
% tan_pos_pi2_le
thf(fact_3120_tan__less__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( tan @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% tan_less_zero
thf(fact_3121_tan__mono__le__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( 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 @ X3 ) @ ( tan @ real @ Y ) )
= ( ord_less_eq @ real @ X3 @ Y ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_3122_tan__mono__le,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( tan @ real @ X3 ) @ ( tan @ real @ Y ) ) ) ) ) ).
% tan_mono_le
thf(fact_3123_tan__bound__pi2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( abs_abs @ real @ ( tan @ real @ X3 ) ) @ ( one_one @ real ) ) ) ).
% tan_bound_pi2
thf(fact_3124_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_3125_arctan__tan,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arctan @ ( tan @ real @ X3 ) )
= X3 ) ) ) ).
% arctan_tan
thf(fact_3126_arctan__unique,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( tan @ real @ X3 )
= Y )
=> ( ( arctan @ Y )
= X3 ) ) ) ) ).
% arctan_unique
thf(fact_3127_Maclaurin__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X3: real,N2: nat,Diff: nat > A > real] :
( ( X3
= ( zero_zero @ real ) )
=> ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X3 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_3128_Maclaurin__lemma,axiom,
! [H2: real,F2: real > real,J: nat > real,N2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ? [B10: real] :
( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J @ M2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ B10 @ ( divide_divide @ real @ ( power_power @ real @ H2 @ N2 ) @ ( semiring_char_0_fact @ real @ N2 ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_3129_tan__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_3130_tan__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( minus_minus @ A @ X3 @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( minus_minus @ A @ X3 @ Y ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_3131_lemma__tan__add1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) )
= ( divide_divide @ A @ ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_3132_minus__sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( uminus_uminus @ A @ ( sin @ A @ X3 ) )
= ( cos @ A @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% minus_sin_cos_eq
thf(fact_3133_tan__total__pi4,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( 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 )
= X3 ) ) ) ).
% tan_total_pi4
thf(fact_3134_tan__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X: A] : ( divide_divide @ A @ ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) @ ( plus_plus @ A @ ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% tan_half
thf(fact_3135_arsinh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A )
= ( ^ [X: A] : ( ln_ln @ A @ ( plus_plus @ A @ X @ ( powr @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( 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_3136_Maclaurin__sin__expansion3,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ X3 )
& ( ( sin @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( sin_coeff @ M2 ) @ ( power_power @ real @ X3 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_3137_Maclaurin__sin__expansion4,axiom,
! [X3: real,N2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ X3 )
& ( ( sin @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( sin_coeff @ M2 ) @ ( power_power @ real @ X3 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_3138_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ).
% sin_coeff_def
thf(fact_3139_Maclaurin__exp__lt,axiom,
! [X3: real,N2: nat] :
( ( X3
!= ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T7 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) )
& ( ( exp @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( divide_divide @ real @ ( power_power @ real @ X3 @ M2 ) @ ( semiring_char_0_fact @ real @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_3140_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X ) ) @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ) ).
% round_altdef
thf(fact_3141_exp__less__mono,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ X3 @ Y )
=> ( ord_less @ real @ ( exp @ real @ X3 ) @ ( exp @ real @ Y ) ) ) ).
% exp_less_mono
thf(fact_3142_exp__less__cancel__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( exp @ real @ X3 ) @ ( exp @ real @ Y ) )
= ( ord_less @ real @ X3 @ Y ) ) ).
% exp_less_cancel_iff
thf(fact_3143_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_3144_frac__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int] :
( ( archimedean_frac @ A @ ( ring_1_of_int @ A @ Z2 ) )
= ( zero_zero @ A ) ) ) ).
% frac_of_int
thf(fact_3145_sin__coeff__0,axiom,
( ( sin_coeff @ ( zero_zero @ nat ) )
= ( zero_zero @ real ) ) ).
% sin_coeff_0
thf(fact_3146_one__less__exp__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X3 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% one_less_exp_iff
thf(fact_3147_exp__less__one__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( exp @ real @ X3 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% exp_less_one_iff
thf(fact_3148_exp__ln__iff,axiom,
! [X3: real] :
( ( ( exp @ real @ ( ln_ln @ real @ X3 ) )
= X3 )
= ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% exp_ln_iff
thf(fact_3149_exp__ln,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( exp @ real @ ( ln_ln @ real @ X3 ) )
= X3 ) ) ).
% exp_ln
thf(fact_3150_fact__mono__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ).
% fact_mono_nat
thf(fact_3151_fact__ge__self,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( semiring_char_0_fact @ nat @ N2 ) ) ).
% fact_ge_self
thf(fact_3152_exp__less__cancel,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( exp @ real @ X3 ) @ ( exp @ real @ Y ) )
=> ( ord_less @ real @ X3 @ Y ) ) ).
% exp_less_cancel
thf(fact_3153_exp__not__eq__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( ( exp @ A @ X3 )
!= ( zero_zero @ A ) ) ) ).
% exp_not_eq_zero
thf(fact_3154_fact__less__mono__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ) ).
% fact_less_mono_nat
thf(fact_3155_exp__total,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ? [X5: real] :
( ( exp @ real @ X5 )
= Y ) ) ).
% exp_total
thf(fact_3156_exp__gt__zero,axiom,
! [X3: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( exp @ real @ X3 ) ) ).
% exp_gt_zero
thf(fact_3157_not__exp__less__zero,axiom,
! [X3: real] :
~ ( ord_less @ real @ ( exp @ real @ X3 ) @ ( zero_zero @ real ) ) ).
% not_exp_less_zero
thf(fact_3158_mult__exp__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( times_times @ A @ ( exp @ A @ X3 ) @ ( exp @ A @ Y ) )
= ( exp @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ).
% mult_exp_exp
thf(fact_3159_exp__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A,Y: A] :
( ( ( times_times @ A @ X3 @ Y )
= ( times_times @ A @ Y @ X3 ) )
=> ( ( exp @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( times_times @ A @ ( exp @ A @ X3 ) @ ( exp @ A @ Y ) ) ) ) ) ).
% exp_add_commuting
thf(fact_3160_frac__ge__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X3 ) ) ) ).
% frac_ge_0
thf(fact_3161_frac__lt__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X3 ) @ ( one_one @ A ) ) ) ).
% frac_lt_1
thf(fact_3162_frac__1__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) )
= ( archimedean_frac @ A @ X3 ) ) ) ).
% frac_1_eq
thf(fact_3163_fact__ge__Suc__0__nat,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ).
% fact_ge_Suc_0_nat
thf(fact_3164_exp__gt__one,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X3 ) ) ) ).
% exp_gt_one
thf(fact_3165_dvd__fact,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( dvd_dvd @ nat @ M @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ) ).
% dvd_fact
thf(fact_3166_fact__diff__Suc,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ N2 @ ( suc @ M ) )
=> ( ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N2 ) )
= ( times_times @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_3167_ln__ge__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ Y @ ( ln_ln @ real @ X3 ) )
= ( ord_less_eq @ real @ ( exp @ real @ Y ) @ X3 ) ) ) ).
% ln_ge_iff
thf(fact_3168_fact__div__fact__le__pow,axiom,
! [R2: nat,N2: nat] :
( ( ord_less_eq @ nat @ R2 @ N2 )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ R2 ) ) ) @ ( power_power @ nat @ N2 @ R2 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_3169_powr__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( powr @ A )
= ( ^ [X: A,A4: A] :
( if @ A
@ ( X
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( exp @ A @ ( times_times @ A @ A4 @ ( ln_ln @ A @ X ) ) ) ) ) ) ) ).
% powr_def
thf(fact_3170_exp__divide__power__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N2: nat,X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X3 @ ( semiring_1_of_nat @ A @ N2 ) ) ) @ N2 )
= ( exp @ A @ X3 ) ) ) ) ).
% exp_divide_power_eq
thf(fact_3171_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ( archimedean_frac @ A @ X3 )
= X3 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
& ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_3172_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_3173_tanh__altdef,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A )
= ( ^ [X: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_3174_sin__coeff__Suc,axiom,
! [N2: nat] :
( ( sin_coeff @ ( suc @ N2 ) )
= ( divide_divide @ real @ ( cos_coeff @ N2 ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) ) ).
% sin_coeff_Suc
thf(fact_3175_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_3176_cos__coeff__Suc,axiom,
! [N2: nat] :
( ( cos_coeff @ ( suc @ N2 ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( sin_coeff @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) ) ).
% cos_coeff_Suc
thf(fact_3177_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N2: nat,X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N2 ) ) @ X3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X3 @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ N2 ) @ ( exp @ real @ X3 ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_3178_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X3: real,N2: nat] :
( ( ord_less_eq @ real @ X3 @ ( semiring_1_of_nat @ real @ N2 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X3 @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ N2 ) @ ( exp @ real @ ( uminus_uminus @ real @ X3 ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_3179_log__base__10__eq2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X3 )
= ( times_times @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X3 ) ) ) ) ).
% log_base_10_eq2
thf(fact_3180_log__base__10__eq1,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X3 )
= ( 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 @ X3 ) ) ) ) ).
% log_base_10_eq1
thf(fact_3181_sin__tan,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( sin @ real @ X3 )
= ( divide_divide @ real @ ( tan @ real @ X3 ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_tan
thf(fact_3182_cos__tan,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( cos @ real @ X3 )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_tan
thf(fact_3183_monoseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X6: nat > A] :
( ! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ A @ ( X6 @ M2 ) @ ( X6 @ N ) ) )
| ! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ A @ ( X6 @ N ) @ ( X6 @ M2 ) ) ) ) ) ) ) ).
% monoseq_def
thf(fact_3184_monoI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ M3 ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% monoI2
thf(fact_3185_monoI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% monoI1
thf(fact_3186_real__sqrt__less__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( sqrt @ X3 ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ X3 @ Y ) ) ).
% real_sqrt_less_iff
thf(fact_3187_real__sqrt__lt__0__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( sqrt @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% real_sqrt_lt_0_iff
thf(fact_3188_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_3189_real__sqrt__lt__1__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( sqrt @ X3 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ).
% real_sqrt_lt_1_iff
thf(fact_3190_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_3191_real__sqrt__less__mono,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ X3 @ Y )
=> ( ord_less @ real @ ( sqrt @ X3 ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_less_mono
thf(fact_3192_real__sqrt__gt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ X3 ) ) ) ).
% real_sqrt_gt_zero
thf(fact_3193_sqrt2__less__2,axiom,
ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% sqrt2_less_2
thf(fact_3194_real__less__rsqrt,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ord_less @ real @ X3 @ ( sqrt @ Y ) ) ) ).
% real_less_rsqrt
thf(fact_3195_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_3196_real__less__lsqrt,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X3 @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sqrt @ X3 ) @ Y ) ) ) ) ).
% real_less_lsqrt
thf(fact_3197_ln__sqrt,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ln_ln @ real @ ( sqrt @ X3 ) )
= ( divide_divide @ real @ ( ln_ln @ real @ X3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% ln_sqrt
thf(fact_3198_arsinh__real__aux,axiom,
! [X3: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X3 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% arsinh_real_aux
thf(fact_3199_real__sqrt__sum__squares__less,axiom,
! [X3: real,U: real,Y: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( 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 @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_3200_sqrt__sum__squares__half__less,axiom,
! [X3: real,U: real,Y: real] :
( ( ord_less @ real @ X3 @ ( 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 ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_3201_monoseq__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X6: nat > A] :
( ! [N: nat] : ( ord_less_eq @ A @ ( X6 @ N ) @ ( X6 @ ( suc @ N ) ) )
| ! [N: nat] : ( ord_less_eq @ A @ ( X6 @ ( suc @ N ) ) @ ( X6 @ N ) ) ) ) ) ) ).
% monoseq_Suc
thf(fact_3202_mono__SucI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% mono_SucI2
thf(fact_3203_mono__SucI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% mono_SucI1
thf(fact_3204_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
= ( 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 ) ) @ N2 ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N2 ) ) ) ) ).
% pochhammer_double
thf(fact_3205_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I3: A] : ( plus_plus @ A @ I3 @ ( one_one @ A ) )
@ N
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_3206_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R2: A,M: 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 ) @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R2 @ ( suc @ M ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_3207_central__binomial__lower__bound,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ N2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ ( semiring_1_of_nat @ real @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ N2 ) ) ) ) ).
% central_binomial_lower_bound
thf(fact_3208_complex__unimodular__polar,axiom,
! [Z2: complex] :
( ( ( real_V7770717601297561774m_norm @ complex @ Z2 )
= ( 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 ) )
=> ( Z2
!= ( complex2 @ ( cos @ real @ T7 ) @ ( sin @ real @ T7 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_3209_binomial__Suc__n,axiom,
! [N2: nat] :
( ( binomial @ ( suc @ N2 ) @ N2 )
= ( suc @ N2 ) ) ).
% binomial_Suc_n
thf(fact_3210_gbinomial__0_I2_J,axiom,
! [B: $tType] :
( ( ( semiring_char_0 @ B )
& ( semidom_divide @ B ) )
=> ! [K: nat] :
( ( gbinomial @ B @ ( zero_zero @ B ) @ ( suc @ K ) )
= ( zero_zero @ B ) ) ) ).
% gbinomial_0(2)
thf(fact_3211_binomial__1,axiom,
! [N2: nat] :
( ( binomial @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= N2 ) ).
% binomial_1
thf(fact_3212_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_3213_binomial__eq__0__iff,axiom,
! [N2: nat,K: nat] :
( ( ( binomial @ N2 @ K )
= ( zero_zero @ nat ) )
= ( ord_less @ nat @ N2 @ K ) ) ).
% binomial_eq_0_iff
thf(fact_3214_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_3215_binomial__Suc__Suc,axiom,
! [N2: nat,K: nat] :
( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_3216_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_3217_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_3218_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% pochhammer_Suc0
thf(fact_3219_binomial__n__0,axiom,
! [N2: nat] :
( ( binomial @ N2 @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_3220_zero__less__binomial__iff,axiom,
! [N2: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N2 @ K ) )
= ( ord_less_eq @ nat @ K @ N2 ) ) ).
% zero_less_binomial_iff
thf(fact_3221_binomial__eq__0,axiom,
! [N2: nat,K: nat] :
( ( ord_less @ nat @ N2 @ K )
=> ( ( binomial @ N2 @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_3222_Suc__times__binomial,axiom,
! [K: nat,N2: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) )
= ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ).
% Suc_times_binomial
thf(fact_3223_Suc__times__binomial__eq,axiom,
! [N2: nat,K: nat] :
( ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) )
= ( times_times @ nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_3224_binomial__symmetric,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( binomial @ N2 @ K )
= ( binomial @ N2 @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_3225_choose__mult__lemma,axiom,
! [M: nat,R2: nat,K: nat] :
( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R2 ) @ K ) @ ( plus_plus @ nat @ M @ K ) ) @ ( binomial @ ( plus_plus @ nat @ M @ K ) @ K ) )
= ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus @ nat @ M @ R2 ) @ M ) ) ) ).
% choose_mult_lemma
thf(fact_3226_binomial__le__pow,axiom,
! [R2: nat,N2: nat] :
( ( ord_less_eq @ nat @ R2 @ N2 )
=> ( ord_less_eq @ nat @ ( binomial @ N2 @ R2 ) @ ( power_power @ nat @ N2 @ R2 ) ) ) ).
% binomial_le_pow
thf(fact_3227_pochhammer__pos,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X3 @ N2 ) ) ) ) ).
% pochhammer_pos
thf(fact_3228_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat,N2: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ M )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A2 @ N2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_3229_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N2: nat,M: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N2 )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A2 @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_3230_zero__less__binomial,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N2 @ K ) ) ) ).
% zero_less_binomial
thf(fact_3231_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_3232_choose__mult,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( times_times @ nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
= ( times_times @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus @ nat @ N2 @ K ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_3233_binomial__Suc__Suc__eq__times,axiom,
! [N2: nat,K: nat] :
( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_3234_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K3: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_3235_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A2 @ K ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_3236_gbinomial__of__nat__symmetric,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N2 ) @ K )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_3237_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X3 @ N2 ) ) ) ) ).
% pochhammer_nonneg
thf(fact_3238_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,N2: nat,I: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N2 ) @ I )
= ( semiri8178284476397505188at_aux @ A @ Inc @ N2 @ ( Inc @ I ) ) ) ) ).
% of_nat_aux.simps(2)
thf(fact_3239_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,I: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
= I ) ) ).
% of_nat_aux.simps(1)
thf(fact_3240_pochhammer__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N2 )
= ( one_one @ A ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N2 )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_0_left
thf(fact_3241_binomial__absorption,axiom,
! [K: nat,N2: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) )
= ( times_times @ nat @ N2 @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorption
thf(fact_3242_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ A2 @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_3243_gbinomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,A2: A] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_3244_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ A2 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_3245_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ A2 @ ( gbinomial @ A @ A2 @ K ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_3246_binomial__fact__lemma,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
= ( semiring_char_0_fact @ nat @ N2 ) ) ) ).
% binomial_fact_lemma
thf(fact_3247_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) )
= ( times_times @ A @ A2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ N2 ) ) ) ) ).
% pochhammer_rec
thf(fact_3248_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N2 ) )
= ( times_times @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) ) ) ) ).
% pochhammer_rec'
thf(fact_3249_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A2 @ N2 ) @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_3250_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N2: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N2 )
= ( zero_zero @ A ) )
= ( ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N2 )
& ( A2
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_3251_pochhammer__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [N2: nat,K: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K )
= ( zero_zero @ A ) )
= ( ord_less @ nat @ N2 @ K ) ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_3252_pochhammer__of__nat__eq__0__lemma,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N2: nat,K: nat] :
( ( ord_less @ nat @ N2 @ K )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K )
= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_3253_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K )
!= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_3254_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N2: nat,M: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( plus_plus @ nat @ N2 @ M ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N2 ) ) @ M ) ) ) ) ).
% pochhammer_product'
thf(fact_3255_binomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_3256_binomial__maximum_H,axiom,
! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ N2 ) ) ).
% binomial_maximum'
thf(fact_3257_binomial__mono,axiom,
! [K: nat,K7: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N2 )
=> ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K7 ) ) ) ) ).
% binomial_mono
thf(fact_3258_binomial__antimono,axiom,
! [K: nat,K7: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K )
=> ( ( ord_less_eq @ nat @ K7 @ N2 )
=> ( ord_less_eq @ nat @ ( binomial @ N2 @ K7 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).
% binomial_antimono
thf(fact_3259_binomial__maximum,axiom,
! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% binomial_maximum
thf(fact_3260_binomial__le__pow2,axiom,
! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% binomial_le_pow2
thf(fact_3261_choose__reduce__nat,axiom,
! [N2: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( binomial @ N2 @ K )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_3262_times__binomial__minus1__eq,axiom,
! [K: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( times_times @ nat @ K @ ( binomial @ N2 @ K ) )
= ( times_times @ nat @ N2 @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_3263_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_3264_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) )
= ( times_times @ A @ A2 @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorption
thf(fact_3265_binomial__altdef__nat,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( binomial @ N2 @ K )
= ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N2 ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_3266_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,M: nat,A2: A] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( times_times @ A @ ( gbinomial @ A @ A2 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_3267_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N2: nat,Z2: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_3268_binomial__strict__antimono,axiom,
! [K: nat,K7: nat,N2: nat] :
( ( ord_less @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ N2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) )
=> ( ( ord_less_eq @ nat @ K7 @ N2 )
=> ( ord_less @ nat @ ( binomial @ N2 @ K7 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_3269_binomial__strict__mono,axiom,
! [K: nat,K7: nat,N2: nat] :
( ( ord_less @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N2 )
=> ( ord_less @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K7 ) ) ) ) ).
% binomial_strict_mono
thf(fact_3270_binomial__less__binomial__Suc,axiom,
! [K: nat,N2: nat] :
( ( ord_less @ nat @ K @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_3271_central__binomial__odd,axiom,
! [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( binomial @ N2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( binomial @ N2 @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% central_binomial_odd
thf(fact_3272_binomial__addition__formula,axiom,
! [N2: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( binomial @ N2 @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_3273_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_3274_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).
% fact_binomial
thf(fact_3275_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_3276_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).
% gbinomial_factors
thf(fact_3277_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R2: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R2 ) @ K ) )
= ( times_times @ A @ R2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R2 ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_3278_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_minus
thf(fact_3279_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A2 @ K )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_3280_pochhammer__minus,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_minus
thf(fact_3281_pochhammer__minus_H,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) ) ) ) ).
% pochhammer_minus'
thf(fact_3282_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_3283_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A2 @ K )
= ( times_times @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_3284_binomial__code,axiom,
( binomial
= ( ^ [N: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N @ ( minus_minus @ nat @ N @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N @ K3 ) @ ( one_one @ nat ) ) @ N @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_3285_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: 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 @ A4 @ ( 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_3286_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N2: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_3287_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A4: A,N: nat] :
( if @ A
@ ( N
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_3288_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A2 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_3289_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) )
= ( ord_less_eq @ A @ I @ K ) ) ) ).
% atMost_iff
thf(fact_3290_finite__atMost,axiom,
! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_atMost @ nat @ K ) ) ).
% finite_atMost
thf(fact_3291_of__nat__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F2: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( semiring_1_of_nat @ A @ ( F2 @ X ) )
@ A3 ) ) ) ).
% of_nat_prod
thf(fact_3292_of__int__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_ring_1 @ A )
=> ! [F2: B > int,A3: set @ B] :
( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( ring_1_of_int @ A @ ( F2 @ X ) )
@ A3 ) ) ) ).
% of_int_prod
thf(fact_3293_prod__zero__iff,axiom,
! [A: $tType,B: $tType] :
( ( semidom @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 )
= ( zero_zero @ A ) )
= ( ? [X: B] :
( ( member @ B @ X @ A3 )
& ( ( F2 @ X )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% prod_zero_iff
thf(fact_3294_prod_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A] :
( ( groups7121269368397514597t_prod @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty
thf(fact_3295_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_3296_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X3 ) @ ( set_ord_atMost @ A @ Y ) )
= ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% atMost_subset_iff
thf(fact_3297_dvd__prod__eqI,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: set @ B,A2: B,B2: A,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ A2 @ A3 )
=> ( ( B2
= ( F2 @ A2 ) )
=> ( dvd_dvd @ A @ B2 @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_3298_dvd__prodI,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: set @ B,A2: B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ A2 @ A3 )
=> ( dvd_dvd @ A @ ( F2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% dvd_prodI
thf(fact_3299_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_3300_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_3301_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X3: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X3 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) )
= ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% prod.insert
thf(fact_3302_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_3303_sum_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ).
% sum.atMost_Suc
thf(fact_3304_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( G @ N2 ) ) ) ) ).
% prod.lessThan_Suc
thf(fact_3305_prod_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ).
% prod.atMost_Suc
thf(fact_3306_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_3307_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N2: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N2 ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N2 ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_3308_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_3309_infinite__Iic,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A2: A] :
~ ( finite_finite2 @ A @ ( set_ord_atMost @ A @ A2 ) ) ) ).
% infinite_Iic
thf(fact_3310_prod_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B3: set @ C,G: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ C @ B3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] :
( groups7121269368397514597t_prod @ C @ A @ ( G @ X )
@ ( collect @ C
@ ^ [Y2: C] :
( ( member @ C @ Y2 @ B3 )
& ( R @ X @ Y2 ) ) ) )
@ A3 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y2: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( G @ X @ Y2 )
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A3 )
& ( R @ X @ Y2 ) ) ) )
@ B3 ) ) ) ) ) ).
% prod.swap_restrict
thf(fact_3311_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_3312_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X: A] : ( ord_less_eq @ A @ X @ U2 ) ) ) ) ) ).
% atMost_def
thf(fact_3313_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% prod.nested_swap'
thf(fact_3314_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) )
& ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod_mono
thf(fact_3315_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ).
% prod_nonneg
thf(fact_3316_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ).
% prod_pos
thf(fact_3317_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F2 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ).
% prod_ge_1
thf(fact_3318_prod__zero,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ( F2 @ X2 )
= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% prod_zero
thf(fact_3319_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_3320_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_3321_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( set_ord_atMost @ nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_3322_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K ) )
= ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% atMost_Suc
thf(fact_3323_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A3 )
& ( P @ X ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( G @ X ) @ ( one_one @ A ) )
@ A3 ) ) ) ) ).
% prod.inter_filter
thf(fact_3324_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_3325_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_3326_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_3327_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_3328_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) )
& ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_3329_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R: A > A > $o,S3: set @ B,H2: B > A,G: B > A] :
( ( R @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X16: A,Y15: A,X23: A,Y23: A] :
( ( ( R @ X16 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times @ A @ X16 @ Y15 ) @ ( times_times @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
=> ( R @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% prod.related
thf(fact_3330_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X3: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X3 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) )
& ( ~ ( member @ B @ X3 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) )
= ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_3331_prod__dvd__prod__subset,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B3: set @ B,A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ B3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B3 ) ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_3332_prod__dvd__prod__subset2,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B3: set @ B,A3: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ B3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ A3 )
=> ( dvd_dvd @ A @ ( F2 @ A5 ) @ ( G @ A5 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B3 ) ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_3333_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S6: set @ B,T6: set @ C,S3: set @ B,I: C > B,J: B > C,T4: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) )
=> ( member @ C @ ( J @ A5 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ( J @ ( I @ B5 ) )
= B5 ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S6 )
=> ( ( G @ A5 )
= ( one_one @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T6 )
=> ( ( H2 @ B5 )
= ( one_one @ A ) ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S3 )
=> ( ( H2 @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_3334_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% prod.in_pairs_0
thf(fact_3335_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [X: B] :
( ( G @ X )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_3336_exp__sum,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ B )
& ( real_Vector_banach @ B )
& ( real_V2822296259951069270ebra_1 @ B ) )
=> ! [I6: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( exp @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ I6 ) )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X: A] : ( exp @ B @ ( F2 @ X ) )
@ I6 ) ) ) ) ).
% exp_sum
thf(fact_3337_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_3338_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_3339_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_3340_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P4: nat,K: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K @ P4 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P4 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_3341_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,I: A,F2: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( member @ A @ I @ I6 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ I4 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_3342_sum__choose__upper,axiom,
! [M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ K3 @ M )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_3343_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I4 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_3344_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B3: set @ B,A3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B3 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_3345_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_3346_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S3: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H2 @ X5 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_3347_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_3348_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ T4 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_3349_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ C5 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
=> ( ( G @ A5 )
= ( one_one @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B3 ) )
=> ( ( H2 @ B5 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B3 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_3350_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ C5 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
=> ( ( G @ A5 )
= ( one_one @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B3 ) )
=> ( ( H2 @ B5 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B3 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_3351_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_3352_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_3353_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ ( suc @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_3354_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_3355_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ M )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_3356_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_3357_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_3358_polyfun__eq__coeffs,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N2: nat,D2: nat > A] :
( ( ! [X: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( D2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N2 )
=> ( ( C2 @ I3 )
= ( D2 @ I3 ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_3359_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_3360_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A2: nat > A,B3: 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 ) ) @ B3 )
=> ( summable @ A @ A2 ) ) ) ) ).
% bounded_imp_summable
thf(fact_3361_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% sum.nested_swap'
thf(fact_3362_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) )
& ( ord_less @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_3363_even__prod__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) )
= ( ? [X: B] :
( ( member @ B @ X @ A3 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_3364_sum__choose__lower,axiom,
! [R2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( plus_plus @ nat @ R2 @ K3 ) @ K3 )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R2 @ N2 ) ) @ N2 ) ) ).
% sum_choose_lower
thf(fact_3365_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X3: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X3 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_3366_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,X3: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) )
= ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_3367_choose__rising__sum_I1_J,axiom,
! [N2: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N2 @ J3 ) @ N2 )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).
% choose_rising_sum(1)
thf(fact_3368_choose__rising__sum_I2_J,axiom,
! [N2: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N2 @ J3 ) @ N2 )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ ( one_one @ nat ) ) @ M ) ) ).
% choose_rising_sum(2)
thf(fact_3369_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G: nat > A,P4: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N2 @ P4 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ P4 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_3370_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( times_times @ A @ ( B2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_3371_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F3: nat > A > A,A4: nat,B4: nat,Acc2: A] : ( if @ A @ ( ord_less @ nat @ B4 @ A4 ) @ Acc2 @ ( set_fo6178422350223883121st_nat @ A @ F3 @ ( plus_plus @ nat @ A4 @ ( one_one @ nat ) ) @ B4 @ ( F3 @ A4 @ Acc2 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_3372_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X3: nat > A > A,Xa2: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X3 @ Xa2 @ Xb @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X3 @ ( plus_plus @ nat @ Xa2 @ ( one_one @ nat ) ) @ Xb @ ( X3 @ Xa2 @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_3373_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N2: nat] :
( ( ! [X: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N2 )
=> ( ( C2 @ I3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_0
thf(fact_3374_zero__polynom__imp__zero__coeffs,axiom,
! [A: $tType] :
( ( ( ab_semigroup_mult @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [C2: nat > A,N2: nat,K: nat] :
( ! [W: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ W @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( C2 @ K )
= ( zero_zero @ A ) ) ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_3375_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_3376_sum__up__index__split,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ M ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ) ).
% sum_up_index_split
thf(fact_3377_atLeast1__atMost__eq__remove0,axiom,
! [N2: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N2 ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_3378_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K3 ) ) @ K3 )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( one_one @ A ) ) @ N2 ) ) ) ).
% gbinomial_parallel_sum
thf(fact_3379_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_3380_fact__eq__fact__times,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( semiring_char_0_fact @ nat @ M )
= ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N2 )
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_3381_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B3: set @ A,A3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ B5 ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A3 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ A5 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ B3 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_3382_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A3: set @ B,F2: B > A,A2: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A2 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( F2 @ A2 ) ) ) )
& ( ~ ( member @ B @ A2 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_3383_sum__choose__diagonal,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( minus_minus @ nat @ N2 @ K3 ) @ ( minus_minus @ nat @ M @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( suc @ N2 ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_3384_vandermonde,axiom,
! [M: nat,N2: nat,R2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( binomial @ M @ K3 ) @ ( binomial @ N2 @ ( minus_minus @ nat @ R2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R2 ) )
= ( binomial @ ( plus_plus @ nat @ M @ N2 ) @ R2 ) ) ).
% vandermonde
thf(fact_3385_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N2: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) ) ) ) ).
% sum_gp_basic
thf(fact_3386_polyfun__finite__roots,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N2: nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) )
= ( ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N2 )
& ( ( C2 @ I3 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_finite_roots
thf(fact_3387_polyfun__roots__finite,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N2: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_3388_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_3389_polyfun__linear__factor__root,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,A2: A,N2: nat] :
( ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ A2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) )
=> ~ ! [B5: nat > A] :
~ ! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( B5 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_3390_polyfun__linear__factor,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,N2: nat,A2: A] :
? [B5: nat > A] :
! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( plus_plus @ A
@ ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( B5 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ A2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ).
% polyfun_linear_factor
thf(fact_3391_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A4: A,N: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_3392_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N2: nat,X3: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_3393_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% sum.triangle_reindex
thf(fact_3394_fact__div__fact,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_3395_binomial,axiom,
! [A2: nat,B2: nat,N2: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ N2 )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power @ nat @ A2 @ K3 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ).
% binomial
thf(fact_3396_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% prod.in_pairs
thf(fact_3397_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,A2: nat,B2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A4: nat] : ( plus_plus @ A @ ( F2 @ A4 ) )
@ A2
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_3398_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% sum.in_pairs_0
thf(fact_3399_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M: nat,A2: nat > A,N2: nat,B2: nat > A,X3: A] :
( ! [I4: nat] :
( ( ord_less @ nat @ M @ I4 )
=> ( ( A2 @ I4 )
= ( zero_zero @ A ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N2 @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X3 @ I3 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( times_times @ A @ ( B2 @ J3 ) @ ( power_power @ A @ X3 @ J3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( 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 @ X3 @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_3400_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_3401_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N2: nat,K: A] :
( ( ! [X: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= K ) )
= ( ( ( C2 @ ( zero_zero @ nat ) )
= K )
& ! [X: nat] :
( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) )
=> ( ( C2 @ X )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_const
thf(fact_3402_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,B2: A,N2: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N2 )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K3 ) ) @ ( power_power @ A @ A2 @ K3 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% binomial_ring
thf(fact_3403_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A2: A,B2: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N2 )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ A2 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_3404_polynomial__product__nat,axiom,
! [M: nat,A2: nat > nat,N2: nat,B2: nat > nat,X3: nat] :
( ! [I4: nat] :
( ( ord_less @ nat @ M @ I4 )
=> ( ( A2 @ I4 )
= ( zero_zero @ nat ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N2 @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( times_times @ nat @ ( A2 @ I3 ) @ ( power_power @ nat @ X3 @ I3 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( times_times @ nat @ ( B2 @ J3 ) @ ( power_power @ nat @ X3 @ J3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( 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 @ X3 @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_3405_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ A2 @ ( suc @ K ) )
= ( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_3406_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P4: nat,K: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K @ P4 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P4 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_3407_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X3: 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 @ M ) @ A2 ) @ K3 ) @ ( power_power @ A @ X3 @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_3408_root__polyfun,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N2: nat,Z2: A,A2: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 )
=> ( ( ( power_power @ A @ Z2 @ N2 )
= A2 )
= ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] :
( times_times @ A
@ ( if @ A
@ ( I3
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ A2 )
@ ( if @ A @ ( I3 = N2 ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) )
@ ( power_power @ A @ Z2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_3409_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N2: nat] :
( ( ( X3
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N2 ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) )
& ( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ).
% sum_gp0
thf(fact_3410_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( N2
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_3411_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K3 ) ) @ K3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_3412_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X3: 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 @ M ) @ A2 ) @ K3 ) @ ( power_power @ A @ X3 @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A2 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X3 @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_3413_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N2: nat,A2: nat > A,X3: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X3 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X3 @ 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 @ X3 @ J3 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ J3 ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_3414_binomial__r__part__sum,axiom,
! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% binomial_r_part_sum
thf(fact_3415_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_3416_polyfun__extremal__lemma,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [E3: real,C2: nat > A,N2: 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
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
@ ( times_times @ real @ E3 @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z4 ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_3417_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N2: nat,A2: nat > A,X3: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X3 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X3 @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ J3 ) @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) )
@ ( power_power @ A @ X3 @ J3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_3418_gbinomial__r__part__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% gbinomial_r_part_sum
thf(fact_3419_choose__odd__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% choose_odd_sum
thf(fact_3420_choose__even__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% choose_even_sum
thf(fact_3421_sin__x__sin__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( sums @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
@ ( 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 @ N ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) ) @ ( power_power @ A @ X3 @ N ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ).
% sin_x_sin_y
thf(fact_3422_sums__cos__x__plus__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( sums @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: 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 @ N ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X3 @ N ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_3423_cos__x__cos__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( sums @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 )
& ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
@ ( 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 @ N ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X3 @ N ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) ) ) ).
% cos_x_cos_y
thf(fact_3424_cot__less__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cot @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% cot_less_zero
thf(fact_3425_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z6: int] :
( ( ord_less_eq @ int @ D5 @ Z6 )
& ( ord_less @ int @ Z7 @ Z6 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_3426_of__nat__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( ^ [N: nat] : N ) ) ).
% of_nat_id
thf(fact_3427_scaleR__cancel__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X3: A,B2: real] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X3 )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) )
= ( ( A2 = B2 )
| ( X3
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_cancel_right
thf(fact_3428_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_3429_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_3430_scaleR__zero__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( zero_zero @ real ) @ X3 )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_left
thf(fact_3431_scaleR__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X3: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X3 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ real ) )
| ( X3
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_eq_0_iff
thf(fact_3432_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_3433_prod__eq__1__iff,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A3 )
= ( one_one @ nat ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ( F2 @ X )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_3434_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_3435_prod__pos__nat__iff,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A3 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_3436_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_3437_int__prod,axiom,
! [B: $tType,F2: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ int
@ ^ [X: B] : ( semiring_1_of_nat @ int @ ( F2 @ X ) )
@ A3 ) ) ).
% int_prod
thf(fact_3438_scaleR__right__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X3: A,Y: A] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ).
% scaleR_right_distrib
thf(fact_3439_scaleR__right__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,A2: real,B2: real] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X3 )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) )
=> ( A2 = B2 ) ) ) ) ).
% scaleR_right_imp_eq
thf(fact_3440_scaleR__left__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,B2: real,X3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X3 )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) ) ) ).
% scaleR_left_distrib
thf(fact_3441_scaleR__left_Oadd,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: real,Y: real,Xa2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ X3 @ Y ) @ Xa2 )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ X3 @ Xa2 ) @ ( real_V8093663219630862766scaleR @ A @ Y @ Xa2 ) ) ) ) ).
% scaleR_left.add
thf(fact_3442_scaleR__right__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B2: real,A2: real,C2: A] :
( ( ord_less_eq @ real @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ C2 ) ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_3443_scaleR__right__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X3: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) ) ) ) ).
% scaleR_right_mono
thf(fact_3444_scaleR__le__cancel__left,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_3445_scaleR__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_3446_scaleR__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_3447_scaleR__left__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B2: A,A2: A,C2: real] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ real @ C2 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_3448_scaleR__left__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X3: A,Y: A,A2: real] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ) ).
% scaleR_left_mono
thf(fact_3449_eq__vector__fraction__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,U: real,V2: real,A2: A] :
( ( X3
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V2 ) @ A2 ) )
= ( ( ( V2
= ( zero_zero @ real ) )
=> ( X3
= ( zero_zero @ A ) ) )
& ( ( V2
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ V2 @ X3 )
= ( real_V8093663219630862766scaleR @ A @ U @ A2 ) ) ) ) ) ) ).
% eq_vector_fraction_iff
thf(fact_3450_vector__fraction__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,V2: real,A2: A,X3: A] :
( ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V2 ) @ A2 )
= X3 )
= ( ( ( V2
= ( zero_zero @ real ) )
=> ( X3
= ( zero_zero @ A ) ) )
& ( ( V2
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ U @ A2 )
= ( real_V8093663219630862766scaleR @ A @ V2 @ X3 ) ) ) ) ) ) ).
% vector_fraction_eq_iff
thf(fact_3451_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E3: A,C2: A,B2: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E3 ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ E3 ) @ C2 ) @ D2 ) ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_3452_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E3: A,C2: A,B2: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E3 ) @ D2 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_3453_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_3454_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_3455_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_3456_scaleR__nonpos__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X3: A] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_3457_scaleR__nonneg__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X3: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_3458_scaleR__nonneg__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X3: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_3459_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_3460_split__scaleR__neg__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X3: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( zero_zero @ A ) ) ) ) ).
% split_scaleR_neg_le
thf(fact_3461_scaleR__mono_H,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,C2: A,D2: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_3462_scaleR__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X3: A,Y: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ Y ) ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_3463_scaleR__left__le__one__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X3: A,A2: real] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ real @ A2 @ ( one_one @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ X3 ) ) ) ) ).
% scaleR_left_le_one_le
thf(fact_3464_scaleR__2,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 )
= ( plus_plus @ A @ X3 @ X3 ) ) ) ).
% scaleR_2
thf(fact_3465_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X: int] : X
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_3466_ln__prod,axiom,
! [A: $tType,I6: set @ A,F2: A > real] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ I4 ) ) )
=> ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F2 @ I6 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X: A] : ( ln_ln @ real @ ( F2 @ X ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_3467_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N2 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_3468_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N2 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% prod.triangle_reindex
thf(fact_3469_cot__gt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cot @ real @ X3 ) ) ) ) ).
% cot_gt_zero
thf(fact_3470_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z6: int] :
( ( ord_less_eq @ int @ D5 @ Z7 )
& ( ord_less @ int @ Z7 @ Z6 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_3471_exp__first__two__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X: A] :
( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X )
@ ( suminf @ A
@ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ A @ X @ ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_3472_arctan__def,axiom,
( arctan
= ( ^ [Y2: real] :
( the @ real
@ ^ [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 ) ) ) )
& ( ( tan @ real @ X )
= Y2 ) ) ) ) ) ).
% arctan_def
thf(fact_3473_modulo__int__unfold,axiom,
! [L: int,K: int,N2: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N2
= ( zero_zero @ nat ) ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N2
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L )
@ ( minus_minus @ int
@ ( semiring_1_of_nat @ int
@ ( times_times @ nat @ N2
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_3474_powr__int,axiom,
! [X3: real,I: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ I ) )
= ( power_power @ real @ X3 @ ( nat2 @ I ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ I ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( nat2 @ ( uminus_uminus @ int @ I ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_3475_inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
= A2 ) ) ).
% inverse_inverse_eq
thf(fact_3476_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_3477_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_3478_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_3479_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_3480_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_3481_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_3482_inverse__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A] :
( ( ( inverse_inverse @ A @ X3 )
= ( one_one @ A ) )
= ( X3
= ( one_one @ A ) ) ) ) ).
% inverse_eq_1_iff
thf(fact_3483_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_3484_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_3485_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_3486_sgn__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_zero
thf(fact_3487_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_3488_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_3489_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_3490_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_3491_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_3492_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_3493_nat__int,axiom,
! [N2: nat] :
( ( nat2 @ ( semiring_1_of_nat @ int @ N2 ) )
= N2 ) ).
% nat_int
thf(fact_3494_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_3495_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_3496_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_3497_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_3498_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_3499_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_3500_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_3501_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_3502_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_3503_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_numeral
thf(fact_3504_nat__of__bool,axiom,
! [P: $o] :
( ( nat2 @ ( zero_neq_one_of_bool @ int @ P ) )
= ( zero_neq_one_of_bool @ nat @ P ) ) ).
% nat_of_bool
thf(fact_3505_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_3506_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_3507_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_3508_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_3509_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_3510_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_3511_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_3512_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_3513_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_3514_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z2 )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_3515_zless__nat__conj,axiom,
! [W2: int,Z2: int] :
( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 )
& ( ord_less @ int @ W2 @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_3516_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_3517_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_3518_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_3519_nat__zminus__int,axiom,
! [N2: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_3520_int__nat__eq,axiom,
! [Z2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) )
= Z2 ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) )
= ( zero_zero @ int ) ) ) ) ).
% int_nat_eq
thf(fact_3521_sgn__mult__dvd__iff,axiom,
! [R2: int,L: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ L ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R2
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_3522_mult__sgn__dvd__iff,axiom,
! [L: int,R2: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R2 ) ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R2
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_3523_dvd__sgn__mult__iff,axiom,
! [L: int,R2: int,K: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ K ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R2
= ( zero_zero @ int ) ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_3524_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R2: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ K @ ( sgn_sgn @ int @ R2 ) ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R2
= ( zero_zero @ int ) ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_3525_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_3526_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_3527_of__nat__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( semiring_1_of_nat @ A @ ( nat2 @ Z2 ) )
= ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_nat_nat
thf(fact_3528_sgn__of__nat,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat] :
( ( sgn_sgn @ A @ ( semiring_1_of_nat @ A @ N2 ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% sgn_of_nat
thf(fact_3529_diff__nat__numeral,axiom,
! [V2: num,V4: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( numeral_numeral @ nat @ V4 ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ V4 ) ) ) ) ).
% diff_nat_numeral
thf(fact_3530_numeral__power__eq__nat__cancel__iff,axiom,
! [X3: num,N2: nat,Y: int] :
( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 )
= ( nat2 @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 )
= Y ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_3531_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X3: num,N2: nat] :
( ( ( nat2 @ Y )
= ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_3532_nat__abs__dvd__iff,axiom,
! [K: int,N2: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N2 )
= ( dvd_dvd @ int @ K @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% nat_abs_dvd_iff
thf(fact_3533_dvd__nat__abs__iff,axiom,
! [N2: nat,K: int] :
( ( dvd_dvd @ nat @ N2 @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ).
% dvd_nat_abs_iff
thf(fact_3534_nat__ceiling__le__eq,axiom,
! [X3: real,A2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X3 ) ) @ A2 )
= ( ord_less_eq @ real @ X3 @ ( semiring_1_of_nat @ real @ A2 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_3535_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_3536_nat__numeral__diff__1,axiom,
! [V2: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( one_one @ int ) ) ) ) ).
% nat_numeral_diff_1
thf(fact_3537_nat__less__numeral__power__cancel__iff,axiom,
! [A2: int,X3: num,N2: nat] :
( ( ord_less @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_3538_numeral__power__less__nat__cancel__iff,axiom,
! [X3: num,N2: nat,A2: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) @ ( nat2 @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) @ A2 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_3539_nat__le__numeral__power__cancel__iff,axiom,
! [A2: int,X3: num,N2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_3540_numeral__power__le__nat__cancel__iff,axiom,
! [X3: num,N2: nat,A2: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) @ ( nat2 @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) @ A2 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_3541_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_3542_sgn__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A] :
( ( ( sgn_sgn @ A @ X3 )
= ( zero_zero @ A ) )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% sgn_zero_iff
thf(fact_3543_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_3544_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_3545_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_3546_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_3547_mult__commute__imp__mult__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Y: A,X3: A] :
( ( ( times_times @ A @ Y @ X3 )
= ( times_times @ A @ X3 @ Y ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ Y ) @ X3 )
= ( times_times @ A @ X3 @ ( inverse_inverse @ A @ Y ) ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_3548_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_3549_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_3550_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_3551_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_3552_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_3553_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_3554_nonzero__inverse__scaleR__distrib,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A2: real,X3: A] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( X3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ A2 ) @ ( inverse_inverse @ A @ X3 ) ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_3555_norm__inverse__le__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [R2: real,X3: A] :
( ( ord_less_eq @ real @ R2 @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ X3 ) ) @ ( inverse_inverse @ real @ R2 ) ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_3556_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_3557_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_3558_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_3559_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_3560_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_3561_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_3562_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_3563_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_3564_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_3565_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_3566_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_3567_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A4: A,B4: A] : ( times_times @ A @ ( inverse_inverse @ A @ B4 ) @ A4 ) ) ) ) ).
% divide_inverse_commute
thf(fact_3568_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A4: A,B4: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B4 ) ) ) ) ) ).
% divide_inverse
thf(fact_3569_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A4: A,B4: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B4 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_3570_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_3571_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: nat,N2: nat] :
( ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ N2 ) @ ( power_power @ A @ X3 @ M ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_3572_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: nat] :
( ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( inverse_inverse @ A @ X3 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X3 ) @ ( power_power @ A @ X3 @ M ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_3573_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa2: nat,X3: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa2 ) ) @ X3 )
= ( times_times @ A @ X3 @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa2 ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_3574_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_3575_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_3576_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa2: int,X3: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa2 ) ) @ X3 )
= ( times_times @ A @ X3 @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa2 ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_3577_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_3578_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_3579_mult__sgn__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ X3 ) @ ( abs_abs @ A @ X3 ) )
= X3 ) ) ).
% mult_sgn_abs
thf(fact_3580_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_3581_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_3582_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_3583_nat__mono,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq @ int @ X3 @ Y )
=> ( ord_less_eq @ nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_3584_int__sgnE,axiom,
! [K: int] :
~ ! [N3: nat,L4: int] :
( K
!= ( times_times @ int @ ( sgn_sgn @ int @ L4 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% int_sgnE
thf(fact_3585_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_3586_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% ex_nat
thf(fact_3587_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
! [X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% all_nat
thf(fact_3588_eq__nat__nat__iff,axiom,
! [Z2: int,Z8: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z8 ) )
= ( Z2 = Z8 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_3589_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_3590_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_3591_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_3592_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_3593_inverse__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X3 ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ) ).
% inverse_le_1_iff
thf(fact_3594_one__less__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X3 ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
& ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% one_less_inverse_iff
thf(fact_3595_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_3596_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_3597_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_3598_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_3599_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_3600_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_3601_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_3602_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_3603_nat__mono__iff,axiom,
! [Z2: int,W2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ W2 @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_3604_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_3605_zless__nat__eq__int__zless,axiom,
! [M: nat,Z2: int] :
( ( ord_less @ nat @ M @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_3606_nat__le__iff,axiom,
! [X3: int,N2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X3 ) @ N2 )
= ( ord_less_eq @ int @ X3 @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% nat_le_iff
thf(fact_3607_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_3608_int__eq__iff,axiom,
! [M: nat,Z2: int] :
( ( ( semiring_1_of_nat @ int @ M )
= Z2 )
= ( ( M
= ( nat2 @ Z2 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ).
% int_eq_iff
thf(fact_3609_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_3610_nat__abs__mult__distrib,axiom,
! [W2: int,Z2: int] :
( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W2 @ Z2 ) ) )
= ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W2 ) ) @ ( nat2 @ ( abs_abs @ int @ Z2 ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_3611_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A4: nat,B4: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_3612_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_3613_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_3614_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X3 ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_3615_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X3 ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ) ).
% inverse_less_1_iff
thf(fact_3616_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_3617_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_3618_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ? [N3: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ X3 ) ) ) ).
% reals_Archimedean
thf(fact_3619_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X: A] :
( if @ A
@ ( X
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_3620_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_3621_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_3622_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I3: int] :
( if @ int
@ ( I3
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I3 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_3623_powr__real__of__int,axiom,
! [X3: real,N2: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ N2 ) )
= ( power_power @ real @ X3 @ ( nat2 @ N2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ N2 ) )
= ( inverse_inverse @ real @ ( power_power @ real @ X3 @ ( nat2 @ ( uminus_uminus @ int @ N2 ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_3624_real__vector__eq__affinity,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,Y: A,X3: A,C2: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( Y
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X3 ) @ C2 ) )
= ( ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) )
= X3 ) ) ) ) ).
% real_vector_eq_affinity
thf(fact_3625_real__vector__affinity__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,X3: A,C2: A,Y: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X3 ) @ C2 )
= Y )
= ( X3
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_3626_pos__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_divideR_le_eq
thf(fact_3627_pos__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% pos_le_divideR_eq
thf(fact_3628_neg__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% neg_divideR_le_eq
thf(fact_3629_neg__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_le_divideR_eq
thf(fact_3630_pos__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_divideR_less_eq
thf(fact_3631_pos__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% pos_less_divideR_eq
thf(fact_3632_neg__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% neg_divideR_less_eq
thf(fact_3633_neg__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_less_divideR_eq
thf(fact_3634_nat__less__eq__zless,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ W2 @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_3635_nat__le__eq__zle,axiom,
! [W2: int,Z2: int] :
( ( ( ord_less @ int @ ( zero_zero @ int ) @ W2 )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) )
=> ( ( ord_less_eq @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq @ int @ W2 @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_3636_norm__sgn,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A] :
( ( ( X3
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X3 ) )
= ( zero_zero @ real ) ) )
& ( ( X3
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X3 ) )
= ( one_one @ real ) ) ) ) ) ).
% norm_sgn
thf(fact_3637_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff
thf(fact_3638_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff2
thf(fact_3639_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N: nat] :
( ( I
= ( semiring_1_of_nat @ int @ N ) )
=> ( P @ N ) )
& ( ( ord_less @ int @ I @ ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ) ).
% split_nat
thf(fact_3640_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_3641_le__nat__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ nat @ N2 @ ( nat2 @ K ) )
= ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ) ).
% le_nat_iff
thf(fact_3642_nat__add__distrib,axiom,
! [Z2: int,Z8: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 )
=> ( ( nat2 @ ( plus_plus @ int @ Z2 @ Z8 ) )
= ( plus_plus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_3643_nat__mult__distrib,axiom,
! [Z2: int,Z8: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( nat2 @ ( times_times @ int @ Z2 @ Z8 ) )
= ( times_times @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ).
% nat_mult_distrib
thf(fact_3644_Suc__as__int,axiom,
( suc
= ( ^ [A4: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( one_one @ int ) ) ) ) ) ).
% Suc_as_int
thf(fact_3645_forall__pos__mono__1,axiom,
! [P: real > $o,E3: real] :
( ! [D3: real,E: real] :
( ( ord_less @ real @ D3 @ E )
=> ( ( P @ D3 )
=> ( P @ E ) ) )
=> ( ! [N3: nat] : ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( P @ E3 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_3646_forall__pos__mono,axiom,
! [P: real > $o,E3: real] :
( ! [D3: real,E: real] :
( ( ord_less @ real @ D3 @ E )
=> ( ( P @ D3 )
=> ( P @ E ) ) )
=> ( ! [N3: nat] :
( ( N3
!= ( zero_zero @ nat ) )
=> ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( P @ E3 ) ) ) ) ).
% forall_pos_mono
thf(fact_3647_real__arch__inverse,axiom,
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
= ( ? [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N ) ) )
& ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N ) ) @ E3 ) ) ) ) ).
% real_arch_inverse
thf(fact_3648_nat__diff__distrib,axiom,
! [Z8: int,Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 )
=> ( ( ord_less_eq @ int @ Z8 @ Z2 )
=> ( ( nat2 @ ( minus_minus @ int @ Z2 @ Z8 ) )
= ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_3649_nat__diff__distrib_H,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( minus_minus @ int @ X3 @ Y ) )
= ( minus_minus @ nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_3650_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K @ L ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_3651_nat__floor__neg,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X3 ) )
= ( zero_zero @ nat ) ) ) ).
% nat_floor_neg
thf(fact_3652_nat__power__eq,axiom,
! [Z2: int,N2: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( nat2 @ ( power_power @ int @ Z2 @ N2 ) )
= ( power_power @ nat @ ( nat2 @ Z2 ) @ N2 ) ) ) ).
% nat_power_eq
thf(fact_3653_ln__inverse,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ln_ln @ real @ ( inverse_inverse @ real @ X3 ) )
= ( uminus_uminus @ real @ ( ln_ln @ real @ X3 ) ) ) ) ).
% ln_inverse
thf(fact_3654_floor__eq3,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X3 ) )
= N2 ) ) ) ).
% floor_eq3
thf(fact_3655_le__nat__floor,axiom,
! [X3: nat,A2: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X3 ) @ A2 )
=> ( ord_less_eq @ nat @ X3 @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A2 ) ) ) ) ).
% le_nat_floor
thf(fact_3656_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X6: set @ A] :
( the @ A
@ ^ [X: A] :
( X6
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_3657_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_3658_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N3 ) ) @ X3 ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_3659_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: nat,N2: nat] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( power_power @ A @ X3 @ ( minus_minus @ nat @ N2 @ M ) )
= ( times_times @ A @ ( power_power @ A @ X3 @ N2 ) @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ M ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_3660_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_3661_pos__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divideR_eq
thf(fact_3662_pos__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_minus_divideR_le_eq
thf(fact_3663_neg__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_le_minus_divideR_eq
thf(fact_3664_neg__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divideR_le_eq
thf(fact_3665_pos__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divideR_eq
thf(fact_3666_pos__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_minus_divideR_less_eq
thf(fact_3667_neg__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_less_minus_divideR_eq
thf(fact_3668_neg__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divideR_less_eq
thf(fact_3669_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( suc @ ( nat2 @ Z2 ) )
= ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_3670_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ M )
= ( ord_less @ int @ W2 @ ( semiring_1_of_nat @ int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_3671_nat__mult__distrib__neg,axiom,
! [Z2: int,Z8: int] :
( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z2 @ Z8 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z8 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_3672_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_3673_log__inverse,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( log @ A2 @ ( inverse_inverse @ real @ X3 ) )
= ( uminus_uminus @ real @ ( log @ A2 @ X3 ) ) ) ) ) ) ).
% log_inverse
thf(fact_3674_floor__eq4,axiom,
! [N2: nat,X3: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X3 ) )
= N2 ) ) ) ).
% floor_eq4
thf(fact_3675_diff__nat__eq__if,axiom,
! [Z8: int,Z2: int] :
( ( ( ord_less @ int @ Z8 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
= ( nat2 @ Z2 ) ) )
& ( ~ ( ord_less @ int @ Z8 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z2 @ Z8 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z2 @ Z8 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_3676_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_3677_nat__dvd__iff,axiom,
! [Z2: int,M: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ Z2 ) @ M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( dvd_dvd @ int @ Z2 @ ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_dvd_iff
thf(fact_3678_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K: int,Q4: int] :
( ( ( sgn_sgn @ int @ R2 )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ L ) )
=> ( ( K
= ( plus_plus @ int @ ( times_times @ int @ Q4 @ L ) @ R2 ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_3679_plus__inverse__ge__2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ X3 @ ( inverse_inverse @ real @ X3 ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_3680_real__inv__sqrt__pow2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( power_power @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( inverse_inverse @ real @ X3 ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_3681_eucl__rel__int_Ocases,axiom,
! [A12: int,A23: int,A32: product_prod @ int @ int] :
( ( eucl_rel_int @ A12 @ A23 @ A32 )
=> ( ( ( A23
= ( zero_zero @ int ) )
=> ( A32
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A12 ) ) )
=> ( ! [Q2: int] :
( ( A32
= ( product_Pair @ int @ int @ Q2 @ ( zero_zero @ int ) ) )
=> ( ( A23
!= ( zero_zero @ int ) )
=> ( A12
!= ( times_times @ int @ Q2 @ A23 ) ) ) )
=> ~ ! [R4: int,Q2: int] :
( ( A32
= ( product_Pair @ int @ int @ Q2 @ R4 ) )
=> ( ( ( sgn_sgn @ int @ R4 )
= ( sgn_sgn @ int @ A23 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R4 ) @ ( abs_abs @ int @ A23 ) )
=> ( A12
!= ( plus_plus @ int @ ( times_times @ int @ Q2 @ A23 ) @ R4 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_3682_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A1: int,A22: int,A33: product_prod @ int @ int] :
( ? [K3: int] :
( ( A1 = K3 )
& ( A22
= ( zero_zero @ int ) )
& ( A33
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
| ? [L2: int,K3: int,Q5: int] :
( ( A1 = K3 )
& ( A22 = 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] :
( ( A1 = K3 )
& ( A22 = 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_3683_exp__series__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A,Y: A,N2: nat] :
( ( ( times_times @ A @ X3 @ Y )
= ( times_times @ A @ Y @ X3 ) )
=> ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I3 ) ) @ ( power_power @ A @ X3 @ I3 ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N2 @ I3 ) ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N2 @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ).
% exp_series_add_commuting
thf(fact_3684_exp__first__term,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X: A] :
( plus_plus @ A @ ( one_one @ A )
@ ( suminf @ A
@ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N ) ) ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_3685_tan__sec,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ ( inverse_inverse @ A @ ( cos @ A @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tan_sec
thf(fact_3686_exp__first__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [K: nat] :
( ( exp @ A )
= ( ^ [X: A] :
( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X @ N ) )
@ ( set_ord_lessThan @ nat @ K ) )
@ ( suminf @ A
@ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N @ K ) ) ) @ ( power_power @ A @ X @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_3687_divide__int__unfold,axiom,
! [L: int,K: int,N2: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N2
= ( zero_zero @ nat ) ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N2
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N2 ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N2 )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_3688_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X ) ) ) ) ).
% old.rec_prod_def
thf(fact_3689_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X3 @ N ) ) )
@ ( sinh @ A @ X3 ) ) ) ).
% sinh_converges
thf(fact_3690_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X3 @ N ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X3 ) ) ) ).
% cosh_converges
thf(fact_3691_sinh__ln__real,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( sinh @ real @ ( ln_ln @ real @ X3 ) )
= ( divide_divide @ real @ ( minus_minus @ real @ X3 @ ( inverse_inverse @ real @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% sinh_ln_real
thf(fact_3692_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_3693_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_3694_sinh__real__less__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( sinh @ real @ X3 ) @ ( sinh @ real @ Y ) )
= ( ord_less @ real @ X3 @ Y ) ) ).
% sinh_real_less_iff
thf(fact_3695_The__split__eq,axiom,
! [A: $tType,B: $tType,X3: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y6: B] :
( ( X3 = X9 )
& ( Y = Y6 ) ) ) )
= ( product_Pair @ A @ B @ X3 @ Y ) ) ).
% The_split_eq
thf(fact_3696_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_3697_sinh__real__neg__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( sinh @ real @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% sinh_real_neg_iff
thf(fact_3698_sinh__real__pos__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X3 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% sinh_real_pos_iff
thf(fact_3699_sinh__plus__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( ( plus_plus @ A @ ( sinh @ A @ X3 ) @ ( cosh @ A @ X3 ) )
= ( exp @ A @ X3 ) ) ) ).
% sinh_plus_cosh
thf(fact_3700_cosh__plus__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( ( plus_plus @ A @ ( cosh @ A @ X3 ) @ ( sinh @ A @ X3 ) )
= ( exp @ A @ X3 ) ) ) ).
% cosh_plus_sinh
thf(fact_3701_sinh__less__cosh__real,axiom,
! [X3: real] : ( ord_less @ real @ ( sinh @ real @ X3 ) @ ( cosh @ real @ X3 ) ) ).
% sinh_less_cosh_real
thf(fact_3702_sinh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( sinh @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sinh @ A @ X3 ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( cosh @ A @ X3 ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% sinh_add
thf(fact_3703_cosh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( cosh @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cosh @ A @ X3 ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( sinh @ A @ X3 ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% cosh_add
thf(fact_3704_cosh__real__pos,axiom,
! [X3: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X3 ) ) ).
% cosh_real_pos
thf(fact_3705_cosh__real__strict__mono,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ Y )
=> ( ord_less @ real @ ( cosh @ real @ X3 ) @ ( cosh @ real @ Y ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_3706_cosh__real__nonneg__less__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( cosh @ real @ X3 ) @ ( cosh @ real @ Y ) )
= ( ord_less @ real @ X3 @ Y ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_3707_cosh__real__nonpos__less__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( cosh @ real @ X3 ) @ ( cosh @ real @ Y ) )
= ( ord_less @ real @ Y @ X3 ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_3708_cosh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( power_power @ A @ ( cosh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( power_power @ A @ ( sinh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% cosh_square_eq
thf(fact_3709_cosh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( cosh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) )
= ( plus_plus @ A @ ( power_power @ A @ ( cosh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_double
thf(fact_3710_sgn__real__def,axiom,
( ( sgn_sgn @ real )
= ( ^ [A4: real] :
( if @ real
@ ( A4
= ( zero_zero @ real ) )
@ ( zero_zero @ real )
@ ( if @ real @ ( ord_less @ real @ ( zero_zero @ real ) @ A4 ) @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ) ).
% sgn_real_def
thf(fact_3711_arcsin__less__arcsin,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arcsin @ X3 ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_3712_arcsin__less__mono,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arcsin @ X3 ) @ ( arcsin @ Y ) )
= ( ord_less @ real @ X3 @ Y ) ) ) ) ).
% arcsin_less_mono
thf(fact_3713_sgn__power__injE,axiom,
! [A2: real,N2: nat,X3: real,B2: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A2 ) @ ( power_power @ real @ ( abs_abs @ real @ A2 ) @ N2 ) )
= X3 )
=> ( ( X3
= ( times_times @ real @ ( sgn_sgn @ real @ B2 ) @ ( power_power @ real @ ( abs_abs @ real @ B2 ) @ N2 ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( A2 = B2 ) ) ) ) ).
% sgn_power_injE
thf(fact_3714_cos__arcsin__nonzero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X3 ) )
!= ( zero_zero @ real ) ) ) ) ).
% cos_arcsin_nonzero
thf(fact_3715_tanh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( ( cosh @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cosh @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( tanh @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tanh @ A @ X3 ) @ ( tanh @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tanh @ A @ X3 ) @ ( tanh @ A @ Y ) ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_3716_sinh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( sinh @ A @ X3 )
= ( zero_zero @ A ) )
= ( member @ A @ ( exp @ A @ X3 ) @ ( insert @ A @ ( one_one @ A ) @ ( insert @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% sinh_zero_iff
thf(fact_3717_floor__real__def,axiom,
( ( archim6421214686448440834_floor @ real )
= ( ^ [X: real] :
( the @ int
@ ^ [Z6: int] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ Z6 ) @ X )
& ( ord_less @ real @ X @ ( ring_1_of_int @ real @ ( plus_plus @ int @ Z6 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_3718_cosh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cosh @ A )
= ( ^ [Z6: A] : ( divide_divide @ A @ ( plus_plus @ A @ ( exp @ A @ Z6 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z6 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_field_def
thf(fact_3719_cosh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cosh @ A @ X3 )
= ( zero_zero @ A ) )
= ( ( power_power @ A @ ( exp @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% cosh_zero_iff
thf(fact_3720_cosh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cosh @ A )
= ( ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ) ).
% cosh_def
thf(fact_3721_cosh__ln__real,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( cosh @ real @ ( ln_ln @ real @ X3 ) )
= ( divide_divide @ real @ ( plus_plus @ real @ X3 @ ( inverse_inverse @ real @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% cosh_ln_real
thf(fact_3722_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X: rat] :
( the @ int
@ ^ [Z6: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z6 ) @ X )
& ( ord_less @ rat @ X @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z6 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_3723_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F5: set @ A,I6: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ I6 )
& ( ( F2 @ I3 )
!= ( zero_zero @ B ) ) ) )
@ F5 )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_3724_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_3725_sum__power2,axiom,
! [K: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) )
= ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ ( one_one @ nat ) ) ) ).
% sum_power2
thf(fact_3726_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_3727_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_3728_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_3729_ivl__subset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,J: A,M: A,N2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M @ N2 ) )
= ( ( ord_less_eq @ A @ J @ I )
| ( ( ord_less_eq @ A @ M @ I )
& ( ord_less_eq @ A @ J @ N2 ) ) ) ) ) ).
% ivl_subset
thf(fact_3730_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_3731_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_3732_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_3733_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,N2: A,M: A] :
( ( ord_less_eq @ A @ I @ N2 )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M ) @ ( set_or7035219750837199246ssThan @ A @ I @ N2 ) )
= ( set_or7035219750837199246ssThan @ A @ N2 @ M ) ) ) ) ).
% ivl_diff
thf(fact_3734_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P4: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_3735_sum_Oeq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,P4: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I6 )
= ( groups7311177749621191930dd_sum @ B @ A @ P4 @ I6 ) ) ) ) ).
% sum.eq_sum
thf(fact_3736_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
= ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_3737_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,P4: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I6 )
& ( ( P4 @ X )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I6 ) ) )
& ( ~ ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) )
= ( plus_plus @ A @ ( P4 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I6 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_3738_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N2: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N2 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_3739_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N2: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N2 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_3740_sgn__rat__def,axiom,
( ( sgn_sgn @ rat )
= ( ^ [A4: rat] :
( if @ rat
@ ( A4
= ( zero_zero @ rat ) )
@ ( zero_zero @ rat )
@ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A4 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ).
% sgn_rat_def
thf(fact_3741_abs__rat__def,axiom,
( ( abs_abs @ rat )
= ( ^ [A4: rat] : ( if @ rat @ ( ord_less @ rat @ A4 @ ( zero_zero @ rat ) ) @ ( uminus_uminus @ rat @ A4 ) @ A4 ) ) ) ).
% abs_rat_def
thf(fact_3742_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( A2 = C2 )
& ( B2 = D2 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_3743_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( A2 = C2 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_3744_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_3745_less__eq__rat__def,axiom,
( ( ord_less_eq @ rat )
= ( ^ [X: rat,Y2: rat] :
( ( ord_less @ rat @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% less_eq_rat_def
thf(fact_3746_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ~ ! [S2: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ S2 )
=> ! [T7: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ T7 )
=> ( R2
!= ( plus_plus @ rat @ S2 @ T7 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_3747_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,I6: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I6 )
& ( ( G @ X )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) ) ) ).
% sum.non_neutral'
thf(fact_3748_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_3749_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_3750_ex__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ( P @ M2 ) ) )
= ( ? [X: nat] :
( ( member @ nat @ X @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less_eq
thf(fact_3751_all__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( P @ M2 ) ) )
= ( ! [X: nat] :
( ( member @ nat @ X @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less_eq
thf(fact_3752_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
= ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_3753_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_3754_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_3755_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_3756_sum_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_3757_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_3758_prod_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_3759_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H2 @ I3 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_3760_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A2: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
( ( A2 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X5: B] :
( ( ord_less_eq @ B @ C2 @ X5 )
=> ( ( ord_less @ B @ X5 @ D2 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_3761_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A2: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
( ( A2 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X5: B] :
( ( ord_less_eq @ B @ C2 @ X5 )
=> ( ( ord_less @ B @ X5 @ D2 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_3762_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,P4: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ P4 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P4 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_3763_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N2: nat,P4: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ P4 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P4 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_3764_size__list__estimation,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: nat,F2: A > nat] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( ord_less @ nat @ Y @ ( F2 @ X3 ) )
=> ( ord_less @ nat @ Y @ ( size_list @ A @ F2 @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_3765_size__list__pointwise,axiom,
! [A: $tType,Xs: list @ A,F2: A > nat,G: A > nat] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F2 @ Xs ) @ ( size_list @ A @ G @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_3766_size__list__estimation_H,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: nat,F2: A > nat] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ Y @ ( F2 @ X3 ) )
=> ( ord_less_eq @ nat @ Y @ ( size_list @ A @ F2 @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_3767_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,P4: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ P4 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P4 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_3768_atLeast0__lessThan__Suc,axiom,
! [N2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) )
= ( insert @ nat @ N2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_3769_subset__eq__atLeast0__lessThan__finite,axiom,
! [N7: set @ nat,N2: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_3770_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ T4 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_3771_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_3772_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T4: set @ B,H2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H2 @ I4 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_3773_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T4: set @ B,G: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_3774_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_3775_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_3776_sum__shift__lb__Suc0__0__upt,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,K: nat] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_3777_sum_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ N2 ) ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_3778_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_3779_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_3780_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I6 )
& ( ( G @ X )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I6 )
& ( ( H2 @ X )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H2 @ I3 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_3781_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A4: A,B4: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A4 @ B4 ) @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_3782_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ N2 ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_3783_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_3784_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_3785_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
@ ^ [X: B] :
( ( member @ B @ X @ I8 )
& ( ( P6 @ X )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P6
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I8 )
& ( ( P6 @ X )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_3786_sum_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( plus_plus @ A @ ( G @ N2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ) ) ).
% sum.last_plus
thf(fact_3787_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( times_times @ A @ ( G @ N2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_3788_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N2: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
= ( minus_minus @ A @ ( F2 @ N2 ) @ ( F2 @ M ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_3789_atLeastLessThanSuc,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) )
= ( insert @ nat @ N2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N2 )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_3790_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( suc @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_3791_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% sum.nested_swap
thf(fact_3792_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( suc @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_3793_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% prod.nested_swap
thf(fact_3794_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N2 @ K ) ) ) ) ) ).
% sum.nat_group
thf(fact_3795_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,K: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N2 @ K ) ) ) ) ) ).
% prod.nat_group
thf(fact_3796_prod__Suc__Suc__fact,axiom,
! [N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
= ( semiring_char_0_fact @ nat @ N2 ) ) ).
% prod_Suc_Suc_fact
thf(fact_3797_prod__Suc__fact,axiom,
! [N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( semiring_char_0_fact @ nat @ N2 ) ) ).
% prod_Suc_fact
thf(fact_3798_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N2: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N2 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ).
% sum.head_if
thf(fact_3799_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N2: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N2 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ).
% prod.head_if
thf(fact_3800_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_3801_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_3802_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_3803_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A4: A,N: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% pochhammer_prod
thf(fact_3804_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K: num] :
( ( ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
= ( insert @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M @ ( pred_numeral @ K ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_3805_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_3806_summable__Cauchy,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ( ( summable @ A )
= ( ^ [F3: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N5: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ N5 @ M2 )
=> ! [N: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) @ E4 ) ) ) ) ) ) ).
% summable_Cauchy
thf(fact_3807_sums__group,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A,S: A,K: nat] :
( ( sums @ A @ F2 @ S )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( sums @ A
@ ^ [N: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K ) @ K ) ) )
@ S ) ) ) ) ).
% sums_group
thf(fact_3808_atLeast1__lessThan__eq__remove0,axiom,
! [N2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N2 ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_3809_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( semiring_char_0_fact @ A @ N2 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N2 @ K ) @ N2 ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).
% fact_split
thf(fact_3810_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_3811_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_3812_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_3813_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_3814_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K3: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_3815_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I6: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ I6 )
& ( ( F2 @ I3 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_3816_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F3: B > A,A4: A,Xs2: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( times_times @ A @ ( F3 @ ( nth @ B @ Xs2 @ N ) ) @ ( power_power @ A @ A4 @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_3817_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,A2: nat > A,B2: nat > A] :
( ! [I4: nat,J2: nat] :
( ( ord_less_eq @ nat @ I4 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N2 )
=> ( ord_less_eq @ A @ ( A2 @ I4 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I4: nat,J2: nat] :
( ( ord_less_eq @ nat @ I4 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N2 )
=> ( ord_less_eq @ A @ ( B2 @ J2 ) @ ( B2 @ I4 ) ) ) )
=> ( ord_less_eq @ A
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A2 @ K3 ) @ ( B2 @ K3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
@ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_3818_Chebyshev__sum__upper__nat,axiom,
! [N2: nat,A2: nat > nat,B2: nat > nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_eq @ nat @ I4 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N2 )
=> ( ord_less_eq @ nat @ ( A2 @ I4 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I4: nat,J2: nat] :
( ( ord_less_eq @ nat @ I4 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N2 )
=> ( ord_less_eq @ nat @ ( B2 @ J2 ) @ ( B2 @ I4 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N2
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( times_times @ nat @ ( A2 @ I3 ) @ ( B2 @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
@ ( times_times @ nat @ ( groups7311177749621191930dd_sum @ nat @ nat @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ nat @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_3819_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_3820_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% finite_atLeastLessThan_int
thf(fact_3821_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_3822_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_3823_rat__inverse__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A4: int,B4: int] :
( if @ ( product_prod @ int @ int )
@ ( A4
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A4 ) @ B4 ) @ ( abs_abs @ int @ A4 ) ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_inverse_code
thf(fact_3824_normalize__negative,axiom,
! [Q4: int,P4: int] :
( ( ord_less @ int @ Q4 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q4 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P4 ) @ ( uminus_uminus @ int @ Q4 ) ) ) ) ) ).
% normalize_negative
thf(fact_3825_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X6: nat > real] :
! [J3: nat] :
? [M9: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M9 @ M2 )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ M9 @ N )
=> ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X6 @ M2 ) @ ( X6 @ N ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_3826_vebt__mint_Opelims,axiom,
! [X3: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_mint @ X3 )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ X3 )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( ( B5
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B5
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( some @ nat @ Mi2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_mint.pelims
thf(fact_3827_quotient__of__number_I3_J,axiom,
! [K: num] :
( ( quotient_of @ ( numeral_numeral @ rat @ K ) )
= ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(3)
thf(fact_3828_normalize__denom__zero,axiom,
! [P4: int] :
( ( normalize @ ( product_Pair @ int @ int @ P4 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_3829_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_3830_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_3831_quotient__of__number_I5_J,axiom,
! [K: num] :
( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(5)
thf(fact_3832_quotient__of__number_I4_J,axiom,
( ( quotient_of @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(4)
thf(fact_3833_rat__divide__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P4 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A4: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A4 @ D5 ) @ ( times_times @ int @ C4 @ B4 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_divide_code
thf(fact_3834_rat__times__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( times_times @ rat @ P4 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A4: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A4 @ B4 ) @ ( times_times @ int @ C4 @ D5 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_times_code
thf(fact_3835_quotient__of__div,axiom,
! [R2: rat,N2: int,D2: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair @ int @ int @ N2 @ D2 ) )
=> ( R2
= ( divide_divide @ rat @ ( ring_1_of_int @ rat @ N2 ) @ ( ring_1_of_int @ rat @ D2 ) ) ) ) ).
% quotient_of_div
thf(fact_3836_rat__minus__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P4 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A4: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A4 @ D5 ) @ ( times_times @ int @ B4 @ C4 ) ) @ ( times_times @ int @ C4 @ D5 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_minus_code
thf(fact_3837_quotient__of__denom__pos,axiom,
! [R2: rat,P4: int,Q4: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% quotient_of_denom_pos
thf(fact_3838_rat__plus__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P4 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A4: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A4 @ D5 ) @ ( times_times @ int @ B4 @ C4 ) ) @ ( times_times @ int @ C4 @ D5 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_plus_code
thf(fact_3839_normalize__denom__pos,axiom,
! [R2: product_prod @ int @ int,P4: int,Q4: int] :
( ( ( normalize @ R2 )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% normalize_denom_pos
thf(fact_3840_normalize__crossproduct,axiom,
! [Q4: int,S: int,P4: int,R2: int] :
( ( Q4
!= ( zero_zero @ int ) )
=> ( ( S
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q4 ) )
= ( normalize @ ( product_Pair @ int @ int @ R2 @ S ) ) )
=> ( ( times_times @ int @ P4 @ S )
= ( times_times @ int @ R2 @ Q4 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_3841_rat__uminus__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A4: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_uminus_code
thf(fact_3842_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P6: rat,Q5: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A4: int,C4: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B4: int,D5: int] : ( ord_less @ int @ ( times_times @ int @ A4 @ D5 ) @ ( times_times @ int @ C4 @ B4 ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_code
thf(fact_3843_rat__abs__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A4: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_abs_code
thf(fact_3844_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X6: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M9: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M9 @ M2 )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ M9 @ N )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ M2 ) @ ( X6 @ N ) ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_3845_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M10 @ M3 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M10 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% CauchyI
thf(fact_3846_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,E3: real] :
( ( topolo3814608138187158403Cauchy @ A @ X8 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M8: nat] :
! [M6: nat] :
( ( ord_less_eq @ nat @ M8 @ M6 )
=> ! [N6: nat] :
( ( ord_less_eq @ nat @ M8 @ N6 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M6 ) @ ( X8 @ N6 ) ) ) @ E3 ) ) ) ) ) ) ).
% CauchyD
thf(fact_3847_vebt__maxt_Opelims,axiom,
! [X3: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_maxt @ X3 )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ X3 )
=> ( ! [A5: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( ( B5
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B5
=> ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( some @ nat @ Ma2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_maxt.pelims
thf(fact_3848_quotient__of__int,axiom,
! [A2: int] :
( ( quotient_of @ ( of_int @ A2 ) )
= ( product_Pair @ int @ int @ A2 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_3849_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X3 )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
=> ( ( ( X3
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv2: $o] :
( ( X3
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
=> ( ! [Uu2: $o] :
( ( X3
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
=> ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ( Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
thf(fact_3850_VEBT__internal_OminNull_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
=> ( ( ( X3
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
thf(fact_3851_VEBT__internal_OminNull_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
=> ( ! [Uv2: $o] :
( ( X3
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
=> ( ! [Uu2: $o] :
( ( X3
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(3)
thf(fact_3852_Frct__code__post_I5_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K ) ) )
= ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K ) ) ) ).
% Frct_code_post(5)
thf(fact_3853_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
( A6
= ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_3854_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_3855_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L2: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q5: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L2 ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L2 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_3856_is__singletonI,axiom,
! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_3857_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M2: num,N: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_3858_sgn__integer__code,axiom,
( ( sgn_sgn @ code_integer )
= ( ^ [K3: code_integer] :
( if @ code_integer
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).
% sgn_integer_code
thf(fact_3859_exhaustive__integer_H_Ocases,axiom,
! [X3: product_prod @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
~ ! [F4: code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: code_integer,I4: code_integer] :
( X3
!= ( product_Pair @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D3 @ I4 ) ) ) ).
% exhaustive_integer'.cases
thf(fact_3860_full__exhaustive__integer_H_Ocases,axiom,
! [X3: product_prod @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
~ ! [F4: ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: code_integer,I4: code_integer] :
( X3
!= ( product_Pair @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D3 @ I4 ) ) ) ).
% full_exhaustive_integer'.cases
thf(fact_3861_subseqs__refl,axiom,
! [A: $tType,Xs: list @ A] : ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs_refl
thf(fact_3862_is__singletonI_H,axiom,
! [A: $tType,A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( member @ A @ Y4 @ A3 )
=> ( X5 = Y4 ) ) )
=> ( is_singleton @ A @ A3 ) ) ) ).
% is_singletonI'
thf(fact_3863_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_3864_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
? [X: A] :
( A6
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_3865_is__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( is_singleton @ A @ A3 )
=> ~ ! [X5: A] :
( A3
!= ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_3866_Frct__code__post_I1_J,axiom,
! [A2: int] :
( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A2 ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(1)
thf(fact_3867_Frct__code__post_I2_J,axiom,
! [A2: int] :
( ( frct @ ( product_Pair @ int @ int @ A2 @ ( zero_zero @ int ) ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(2)
thf(fact_3868_Frct__code__post_I8_J,axiom,
! [A2: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ A2 @ ( uminus_uminus @ int @ B2 ) ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A2 @ B2 ) ) ) ) ).
% Frct_code_post(8)
thf(fact_3869_Frct__code__post_I7_J,axiom,
! [A2: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A2 ) @ B2 ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A2 @ B2 ) ) ) ) ).
% Frct_code_post(7)
thf(fact_3870_Frct__code__post_I3_J,axiom,
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) )
= ( one_one @ rat ) ) ).
% Frct_code_post(3)
thf(fact_3871_Frct__code__post_I4_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) )
= ( numeral_numeral @ rat @ K ) ) ).
% Frct_code_post(4)
thf(fact_3872_Frct__code__post_I6_J,axiom,
! [K: num,L: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( numeral_numeral @ int @ L ) ) )
= ( divide_divide @ rat @ ( numeral_numeral @ rat @ K ) @ ( numeral_numeral @ rat @ L ) ) ) ).
% Frct_code_post(6)
thf(fact_3873_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K3: int] :
( if @ code_integer @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K3 ) ) )
@ ( if @ code_integer
@ ( K3
= ( zero_zero @ int ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer
@ ( ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
@ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_3874_length__mul__elem,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N2: nat] :
( ! [X5: list @ A] :
( ( member @ ( list @ A ) @ X5 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ X5 )
= N2 ) )
=> ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs ) )
= ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) @ N2 ) ) ) ).
% length_mul_elem
thf(fact_3875_even__sum__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( finite_card @ B
@ ( collect @ B
@ ^ [A4: B] :
( ( member @ B @ A4 @ A3 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ A4 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_3876_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_3877_card__Collect__less__nat,axiom,
! [N2: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) )
= N2 ) ).
% card_Collect_less_nat
thf(fact_3878_card__atMost,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_3879_card__Collect__le__nat,axiom,
! [N2: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less_eq @ nat @ I3 @ N2 ) ) )
= ( suc @ N2 ) ) ).
% card_Collect_le_nat
thf(fact_3880_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_3881_card_Oinfinite,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_3882_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_3883_card__0__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_3884_card__insert__disjoint,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( finite_card @ A @ ( insert @ A @ X3 @ A3 ) )
= ( suc @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_3885_abs__integer__code,axiom,
( ( abs_abs @ code_integer )
= ( ^ [K3: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K3 ) @ K3 ) ) ) ).
% abs_integer_code
thf(fact_3886_less__integer__code_I1_J,axiom,
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(1)
thf(fact_3887_less__integer_Oabs__eq,axiom,
! [Xa2: int,X3: int] :
( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
= ( ord_less @ int @ Xa2 @ X3 ) ) ).
% less_integer.abs_eq
thf(fact_3888_n__subsets,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
& ( ( finite_card @ A @ B6 )
= K ) ) ) )
= ( binomial @ ( finite_card @ A @ A3 ) @ K ) ) ) ).
% n_subsets
thf(fact_3889_card__subset__eq,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ( finite_card @ A @ A3 )
= ( finite_card @ A @ B3 ) )
=> ( A3 = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_3890_infinite__arbitrarily__large,axiom,
! [A: $tType,A3: set @ A,N2: nat] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [B10: set @ A] :
( ( finite_finite2 @ A @ B10 )
& ( ( finite_card @ A @ B10 )
= N2 )
& ( ord_less_eq @ ( set @ A ) @ B10 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_3891_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B3: set @ A,A3: set @ B,R2: B > A > $o] :
( ( finite_finite2 @ A @ B3 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ A3 )
=> ? [B11: A] :
( ( member @ A @ B11 @ B3 )
& ( R2 @ A5 @ B11 ) ) )
=> ( ! [A13: B,A24: B,B5: A] :
( ( member @ B @ A13 @ A3 )
=> ( ( member @ B @ A24 @ A3 )
=> ( ( member @ A @ B5 @ B3 )
=> ( ( R2 @ A13 @ B5 )
=> ( ( R2 @ A24 @ B5 )
=> ( A13 = A24 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_3892_card__insert__le,axiom,
! [A: $tType,A3: set @ A,X3: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ ( insert @ A @ X3 @ A3 ) ) ) ).
% card_insert_le
thf(fact_3893_sum__multicount__gen,axiom,
! [A: $tType,B: $tType,S: set @ A,T2: set @ B,R: A > B > $o,K: B > nat] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T2 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ T2 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ S )
& ( R @ I3 @ X5 ) ) ) )
= ( K @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T2 )
& ( R @ I3 @ J3 ) ) ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ nat @ K @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_3894_card__lists__length__eq,axiom,
! [A: $tType,A3: set @ A,N2: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= N2 ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A3 ) @ N2 ) ) ) ).
% card_lists_length_eq
thf(fact_3895_card__eq__0__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% card_eq_0_iff
thf(fact_3896_card__ge__0__finite,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
=> ( finite_finite2 @ A @ A3 ) ) ).
% card_ge_0_finite
thf(fact_3897_card__insert__if,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( member @ A @ X3 @ A3 )
=> ( ( finite_card @ A @ ( insert @ A @ X3 @ A3 ) )
= ( finite_card @ A @ A3 ) ) )
& ( ~ ( member @ A @ X3 @ A3 )
=> ( ( finite_card @ A @ ( insert @ A @ X3 @ A3 ) )
= ( suc @ ( finite_card @ A @ A3 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_3898_card__Suc__eq__finite,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
= ( ? [B4: A,B6: set @ A] :
( ( A3
= ( insert @ A @ B4 @ B6 ) )
& ~ ( member @ A @ B4 @ B6 )
& ( ( finite_card @ A @ B6 )
= K )
& ( finite_finite2 @ A @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_3899_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_3900_obtain__subset__with__card__n,axiom,
! [A: $tType,N2: nat,S3: set @ A] :
( ( ord_less_eq @ nat @ N2 @ ( finite_card @ A @ S3 ) )
=> ~ ! [T5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T5 @ S3 )
=> ( ( ( finite_card @ A @ T5 )
= N2 )
=> ~ ( finite_finite2 @ A @ T5 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_3901_card__seteq,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B3 ) @ ( finite_card @ A @ A3 ) )
=> ( A3 = B3 ) ) ) ) ).
% card_seteq
thf(fact_3902_card__mono,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% card_mono
thf(fact_3903_card__less__sym__Diff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_3904_card__le__sym__Diff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_3905_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_3906_card__1__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( one_one @ nat ) )
=> ~ ! [X5: A] :
( A3
!= ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_3907_psubset__card__mono,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_3908_card__less,axiom,
! [M7: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_3909_card__less__Suc,axiom,
! [M7: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M7 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_3910_card__less__Suc2,axiom,
! [M7: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M7 )
& ( ord_less @ nat @ K3 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_3911_sum__Suc,axiom,
! [A: $tType,F2: A > nat,A3: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X: A] : ( suc @ ( F2 @ X ) )
@ A3 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( finite_card @ A @ A3 ) ) ) ).
% sum_Suc
thf(fact_3912_subset__card__intvl__is__intvl,axiom,
! [A3: set @ nat,K: nat] :
( ( ord_less_eq @ ( set @ nat ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A3 ) ) ) )
=> ( A3
= ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A3 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_3913_sum__multicount,axiom,
! [A: $tType,B: $tType,S3: set @ A,T4: set @ B,R: A > B > $o,K: nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ T4 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ S3 )
& ( R @ I3 @ X5 ) ) ) )
= K ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T4 )
& ( R @ I3 @ J3 ) ) ) )
@ S3 )
= ( times_times @ nat @ K @ ( finite_card @ B @ T4 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_3914_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,K5: A,F2: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ( ord_less_eq @ A @ K5 @ ( F2 @ I4 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ).
% sum_bounded_below
thf(fact_3915_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F2: B > A,K5: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ K5 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above
thf(fact_3916_card__gt__0__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
= ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A3 ) ) ) ).
% card_gt_0_iff
thf(fact_3917_card__Suc__eq,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
= ( ? [B4: A,B6: set @ A] :
( ( A3
= ( insert @ A @ B4 @ B6 ) )
& ~ ( member @ A @ B4 @ B6 )
& ( ( finite_card @ A @ B6 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B6
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_3918_card__eq__SucD,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
=> ? [B5: A,B10: set @ A] :
( ( A3
= ( insert @ A @ B5 @ B10 ) )
& ~ ( member @ A @ B5 @ B10 )
& ( ( finite_card @ A @ B10 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B10
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_3919_card__1__singleton__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X: A] :
( A3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_3920_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ A3 )
=> ( X = Y2 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_3921_card__le__Suc__iff,axiom,
! [A: $tType,N2: nat,A3: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( finite_card @ A @ A3 ) )
= ( ? [A4: A,B6: set @ A] :
( ( A3
= ( insert @ A @ A4 @ B6 ) )
& ~ ( member @ A @ A4 @ B6 )
& ( ord_less_eq @ nat @ N2 @ ( finite_card @ A @ B6 ) )
& ( finite_finite2 @ A @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_3922_card__Diff1__le,axiom,
! [A: $tType,A3: set @ A,X3: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ).
% card_Diff1_le
thf(fact_3923_card__Diff__subset,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_3924_card__psubset,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) )
=> ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% card_psubset
thf(fact_3925_diff__card__le__card__Diff,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_3926_card__lists__length__le,axiom,
! [A: $tType,A3: set @ A,N2: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A3 ) ) @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% card_lists_length_le
thf(fact_3927_card__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( power_power @ A @ Z6 @ N2 )
= ( one_one @ A ) ) ) )
@ N2 ) ) ) ).
% card_roots_unity
thf(fact_3928_subset__eq__atLeast0__lessThan__card,axiom,
! [N7: set @ nat,N2: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N7 ) @ N2 ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_3929_card__sum__le__nat__sum,axiom,
! [S3: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S3 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ S3 ) ) ).
% card_sum_le_nat_sum
thf(fact_3930_card__nth__roots,axiom,
! [C2: complex,N2: nat] :
( ( C2
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= C2 ) ) )
= N2 ) ) ) ).
% card_nth_roots
thf(fact_3931_card__roots__unity__eq,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= ( one_one @ complex ) ) ) )
= N2 ) ) ).
% card_roots_unity_eq
thf(fact_3932_card__2__iff,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X: A,Y2: A] :
( ( S3
= ( insert @ A @ X @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X != Y2 ) ) ) ) ).
% card_2_iff
thf(fact_3933_card__3__iff,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X: A,Y2: A,Z6: A] :
( ( S3
= ( insert @ A @ X @ ( insert @ A @ Y2 @ ( insert @ A @ Z6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X != Y2 )
& ( Y2 != Z6 )
& ( X != Z6 ) ) ) ) ).
% card_3_iff
thf(fact_3934_odd__card__imp__not__empty,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A3 ) )
=> ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_3935_card__Suc__Diff1,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_3936_card_Oinsert__remove,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ A @ ( insert @ A @ X3 @ A3 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_3937_card_Oremove,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( finite_card @ A @ A3 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_3938_card__Diff1__less,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ).
% card_Diff1_less
thf(fact_3939_card__Diff2__less,axiom,
! [A: $tType,A3: set @ A,X3: A,Y: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_3940_card__Diff1__less__iff,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) )
= ( ( finite_finite2 @ A @ A3 )
& ( member @ A @ X3 @ A3 ) ) ) ).
% card_Diff1_less_iff
thf(fact_3941_card__Diff__singleton__if,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( ( member @ A @ X3 @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X3 @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_3942_card__Diff__singleton,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( member @ A @ X3 @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_3943_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,N2: A,K: nat] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) )
& ( ord_less_eq @ A @ ( F2 @ I4 ) @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N2 )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( power_power @ A @ N2 @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_3944_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F2: B > A,K5: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ( ord_less @ A @ ( F2 @ I4 ) @ K5 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A3 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_3945_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A3: set @ B,F2: B > A,K5: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( divide_divide @ A @ K5 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_3946_card__insert__le__m1,axiom,
! [A: $tType,N2: nat,Y: set @ A,X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X3 @ Y ) ) @ N2 ) ) ) ).
% card_insert_le_m1
thf(fact_3947_polyfun__roots__card,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N2: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) )
@ N2 ) ) ) ) ).
% polyfun_roots_card
thf(fact_3948_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A,C2: A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ C2 )
@ S3 )
= ( times_times @ A @ ( B2 @ A2 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S3 ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ C2 )
@ S3 )
= ( power_power @ A @ C2 @ ( finite_card @ B @ S3 ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_3949_polyfun__rootbound,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N2: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) )
& ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) )
@ N2 ) ) ) ) ) ).
% polyfun_rootbound
thf(fact_3950_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_3951_card__lists__distinct__length__eq,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A3 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ( distinct @ A @ Xs2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_3952_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A3: set @ A] :
( ( ord_less @ nat @ K @ ( finite_card @ A @ A3 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ( distinct @ A @ Xs2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_3953_set__n__lists,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N2 @ Xs ) )
= ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys3 )
= N2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_3954_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K3: code_integer] :
( if @ int @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K3 ) ) )
@ ( if @ int
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ int )
@ ( product_case_prod @ code_integer @ code_integer @ int
@ ^ [L2: code_integer,J3: code_integer] :
( if @ int
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) )
@ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) ) @ ( one_one @ int ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_3955_distinct__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
= ( distinct @ A @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_3956_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A3: set @ A,N2: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= N2 )
& ( distinct @ A @ Xs2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_3957_distinct__enumerate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] : ( distinct @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) ) ).
% distinct_enumerate
thf(fact_3958_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_3959_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_3960_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_3961_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_3962_finite__distinct__list,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [Xs3: list @ A] :
( ( ( set2 @ A @ Xs3 )
= A3 )
& ( distinct @ A @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_3963_less__integer_Orep__eq,axiom,
( ( ord_less @ code_integer )
= ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_integer.rep_eq
thf(fact_3964_integer__less__iff,axiom,
( ( ord_less @ code_integer )
= ( ^ [K3: code_integer,L2: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% integer_less_iff
thf(fact_3965_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_3966_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_3967_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_3968_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs2: list @ A] :
! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( I3 != J3 )
=> ( ( nth @ A @ Xs2 @ I3 )
!= ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_3969_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I )
= ( nth @ A @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_3970_distinct__Ex1,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ? [X5: nat] :
( ( ord_less @ nat @ X5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ X5 )
= X3 )
& ! [Y5: nat] :
( ( ( ord_less @ nat @ Y5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ Y5 )
= X3 ) )
=> ( Y5 = X5 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_3971_length__n__lists__elem,axiom,
! [A: $tType,Ys: list @ A,N2: nat,Xs: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N2 @ Xs ) ) )
=> ( ( size_size @ ( list @ A ) @ Ys )
= N2 ) ) ).
% length_n_lists_elem
thf(fact_3972_distinct__list__update,axiom,
! [A: $tType,Xs: list @ A,A2: A,I: nat] :
( ( distinct @ A @ Xs )
=> ( ~ ( member @ A @ A2 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ I ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs @ I @ A2 ) ) ) ) ).
% distinct_list_update
thf(fact_3973_length__n__lists,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( n_lists @ A @ N2 @ Xs ) )
= ( power_power @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ).
% length_n_lists
thf(fact_3974_set__update__distinct,axiom,
! [A: $tType,Xs: list @ A,N2: nat,X3: A] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs @ N2 @ X3 ) )
= ( insert @ A @ X3 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ N2 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_3975_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K3 @ L2 ) @ ( modulo_modulo @ code_integer @ K3 @ L2 ) ) ) ) ).
% divmod_integer_def
thf(fact_3976_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K3: code_integer] :
( if @ nat @ ( ord_less_eq @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat )
@ ( product_case_prod @ code_integer @ code_integer @ nat
@ ^ [L2: code_integer,J3: code_integer] :
( if @ nat
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( one_one @ nat ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_3977_distinct__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
= ( distinct @ A @ Ys ) ) ).
% distinct_union
thf(fact_3978_bit__cut__integer__def,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
@ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K3 ) ) ) ) ).
% bit_cut_integer_def
thf(fact_3979_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K3: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) ) ) ) ).
% divmod_abs_def
thf(fact_3980_nat__of__integer__non__positive,axiom,
! [K: code_integer] :
( ( ord_less_eq @ code_integer @ K @ ( zero_zero @ code_integer ) )
=> ( ( code_nat_of_integer @ K )
= ( zero_zero @ nat ) ) ) ).
% nat_of_integer_non_positive
thf(fact_3981_nat__of__integer__code__post_I1_J,axiom,
( ( code_nat_of_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ nat ) ) ).
% nat_of_integer_code_post(1)
thf(fact_3982_divmod__abs__code_I6_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ ( zero_zero @ code_integer ) @ J )
= ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ) ).
% divmod_abs_code(6)
thf(fact_3983_divmod__abs__code_I5_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ J @ ( zero_zero @ code_integer ) )
= ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( abs_abs @ code_integer @ J ) ) ) ).
% divmod_abs_code(5)
thf(fact_3984_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( if @ ( product_prod @ code_integer @ $o )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o )
@ ^ [R5: code_integer,S8: code_integer] :
( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ R5 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ S8 ) )
@ ( S8
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_3985_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L2 )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S8: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S8
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L2 @ S8 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
@ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S8: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S8
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L2 ) @ S8 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_3986_card__Pow,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( set @ A ) @ ( pow2 @ A @ A3 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A3 ) ) ) ) ).
% card_Pow
thf(fact_3987_rec__nat__add__eq__if,axiom,
! [A: $tType,A2: A,F2: nat > A > A,V2: num,N2: nat] :
( ( rec_nat @ A @ A2 @ F2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N2 ) )
= ( F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N2 ) @ ( rec_nat @ A @ A2 @ F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N2 ) ) ) ) ).
% rec_nat_add_eq_if
thf(fact_3988_case__nat__add__eq__if,axiom,
! [A: $tType,A2: A,F2: nat > A,V2: num,N2: nat] :
( ( case_nat @ A @ A2 @ F2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N2 ) )
= ( F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N2 ) ) ) ).
% case_nat_add_eq_if
thf(fact_3989_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_3990_Pow__singleton__iff,axiom,
! [A: $tType,X8: set @ A,Y7: set @ A] :
( ( ( pow2 @ A @ X8 )
= ( insert @ ( set @ A ) @ Y7 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X8
= ( bot_bot @ ( set @ A ) ) )
& ( Y7
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_3991_PowI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B3 ) ) ) ).
% PowI
thf(fact_3992_Pow__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% Pow_iff
thf(fact_3993_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > B,X3: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F2 @ ( product_Pair @ A @ C @ X3 @ Y ) )
= ( product_Pair @ A @ B @ X3 @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_3994_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_3995_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_3996_finite__Pow__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ ( set @ A ) @ ( pow2 @ A @ A3 ) )
= ( finite_finite2 @ A @ A3 ) ) ).
% finite_Pow_iff
thf(fact_3997_Pow__bottom,axiom,
! [A: $tType,B3: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow2 @ A @ B3 ) ) ).
% Pow_bottom
thf(fact_3998_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 )
@ ^ [X: nat] : ( H2 @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_3999_Pow__not__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( pow2 @ A @ A3 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_4000_Pow__top,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ A3 ) ) ).
% Pow_top
thf(fact_4001_Pow__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B3 ) ) ) ).
% Pow_mono
thf(fact_4002_PowD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% PowD
thf(fact_4003_Pow__def,axiom,
! [A: $tType] :
( ( pow2 @ A )
= ( ^ [A6: set @ A] :
( collect @ ( set @ A )
@ ^ [B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% Pow_def
thf(fact_4004_Pow__Compl,axiom,
! [A: $tType,A3: set @ A] :
( ( pow2 @ A @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [B6: set @ A] :
( ( Uu3
= ( uminus_uminus @ ( set @ A ) @ B6 ) )
& ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B6 ) ) ) ) ) ).
% Pow_Compl
thf(fact_4005_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: nat > A,X22: nat] :
( ( case_nat @ A @ F1 @ F22 @ ( suc @ X22 ) )
= ( F22 @ X22 ) ) ).
% old.nat.simps(5)
thf(fact_4006_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_4007_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_4008_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_4009_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
= ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M ) @ N2 ) ) ).
% less_eq_nat.simps(2)
thf(fact_4010_diff__Suc,axiom,
! [M: nat,N2: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N2 ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [K3: nat] : K3
@ ( minus_minus @ nat @ M @ N2 ) ) ) ).
% diff_Suc
thf(fact_4011_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X: A,F3: nat > A,N: nat] :
( if @ A
@ ( N
= ( zero_zero @ nat ) )
@ X
@ ( F3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_4012_binomial__def,axiom,
( binomial
= ( ^ [N: 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 ) @ N ) ) )
& ( ( finite_card @ nat @ K6 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_4013_old_Orec__nat__def,axiom,
! [T: $tType] :
( ( rec_nat @ T )
= ( ^ [F12: T,F23: nat > T > T,X: nat] : ( the @ T @ ( rec_set_nat @ T @ F12 @ F23 @ X ) ) ) ) ).
% old.rec_nat_def
thf(fact_4014_rec__nat__0__imp,axiom,
! [A: $tType,F2: nat > A,F1: A,F22: nat > A > A] :
( ( F2
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F2 @ ( zero_zero @ nat ) )
= F1 ) ) ).
% rec_nat_0_imp
thf(fact_4015_rec__nat__Suc__imp,axiom,
! [A: $tType,F2: nat > A,F1: A,F22: nat > A > A,N2: nat] :
( ( F2
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F2 @ ( suc @ N2 ) )
= ( F22 @ N2 @ ( F2 @ N2 ) ) ) ) ).
% rec_nat_Suc_imp
thf(fact_4016_nat_Osplit__sels_I1_J,axiom,
! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ( ( Nat
= ( zero_zero @ nat ) )
=> ( P @ F1 ) )
& ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
=> ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_4017_pred__def,axiom,
( pred
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X24: nat] : X24 ) ) ).
% pred_def
thf(fact_4018_subset__Collect__iff,axiom,
! [A: $tType,B3: set @ A,A3: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A3 )
& ( P @ X ) ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ B3 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_4019_subset__CollectI,axiom,
! [A: $tType,B3: set @ A,A3: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B3 )
=> ( ( Q @ X5 )
=> ( P @ X5 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ B3 )
& ( Q @ X ) ) )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A3 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_4020_nat_Osplit__sels_I2_J,axiom,
! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ~ ( ( ( Nat
= ( zero_zero @ nat ) )
& ~ ( P @ F1 ) )
| ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
& ~ ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ).
% nat.split_sels(2)
thf(fact_4021_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N: nat,A4: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A4 ) @ N ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A4 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A4 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_4022_wmin__insertI,axiom,
! [X3: 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 ) @ X3 @ XS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ 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_4023_wmax__insertI,axiom,
! [Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X3: 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 ) @ X3 @ 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 ) @ X3 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_4024_bezw__0,axiom,
! [X3: nat] :
( ( bezw @ X3 @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_4025_push__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less @ int @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% push_bit_negative_int_iff
thf(fact_4026_push__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,A2: A] :
( ( ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% push_bit_eq_0_iff
thf(fact_4027_push__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% push_bit_of_0
thf(fact_4028_push__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N2: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ A2 ) ) ) ).
% push_bit_push_bit
thf(fact_4029_push__bit__Suc__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ K ) )
= ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_Suc_numeral
thf(fact_4030_push__bit__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_4031_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ A2 )
= ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_4032_push__bit__of__Suc__0,axiom,
! [N2: nat] :
( ( bit_se4730199178511100633sh_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% push_bit_of_Suc_0
thf(fact_4033_even__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) )
= ( ( N2
!= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_push_bit_iff
thf(fact_4034_wmin__emptyI,axiom,
! [X8: set @ ( product_prod @ nat @ nat )] : ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X8 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_4035_wmax__emptyI,axiom,
! [X8: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X8 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ X8 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_4036_push__bit__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ B2 ) ) ) ) ).
% push_bit_add
thf(fact_4037_push__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N2: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ ( bit_se4730199178511100633sh_bit @ A @ M @ A2 ) ) ) ) ).
% push_bit_take_bit
thf(fact_4038_bit__push__bit__iff__int,axiom,
! [M: nat,K: int,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M @ K ) @ N2 )
= ( ( ord_less_eq @ nat @ M @ N2 )
& ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_4039_bit__push__bit__iff__nat,axiom,
! [M: nat,Q4: nat,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M @ Q4 ) @ N2 )
= ( ( ord_less_eq @ nat @ M @ N2 )
& ( bit_se5641148757651400278ts_bit @ nat @ Q4 @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_4040_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A4: A,N: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A4 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_4041_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N: nat,A4: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A4 @ K3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% take_bit_sum
thf(fact_4042_smax__insertI,axiom,
! [Y: product_prod @ nat @ nat,Y7: set @ ( product_prod @ nat @ nat ),X3: product_prod @ nat @ nat,X8: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y @ Y7 )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ 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 ) ) @ X8 @ Y7 ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X3 @ X8 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_4043_smin__insertI,axiom,
! [X3: 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 ) @ X3 @ XS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ 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_4044_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K3: nat,M2: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M2 @ K3 ) @ ( product_Pair @ nat @ nat @ M2 @ ( minus_minus @ nat @ K3 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus @ nat @ M2 @ ( suc @ K3 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_4045_prod__decode__aux_Oelims,axiom,
! [X3: nat,Xa2: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
= Y )
=> ( ( ( ord_less_eq @ nat @ Xa2 @ X3 )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X3 @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa2 @ X3 )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_4046_smax__emptyI,axiom,
! [Y7: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y7 )
=> ( ( Y7
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ Y7 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_4047_smin__emptyI,axiom,
! [X8: set @ ( product_prod @ nat @ nat )] :
( ( X8
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X8 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_4048_Suc__0__div__numeral,axiom,
! [K: num] :
( ( divide_divide @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
= ( product_fst @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).
% Suc_0_div_numeral
thf(fact_4049_Suc__0__mod__numeral,axiom,
! [K: num] :
( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
= ( product_snd @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).
% Suc_0_mod_numeral
thf(fact_4050_finite__enumerate,axiom,
! [S3: set @ nat] :
( ( finite_finite2 @ nat @ S3 )
=> ? [R4: nat > nat] :
( ( strict_mono_on @ nat @ nat @ R4 @ ( set_ord_lessThan @ nat @ ( finite_card @ nat @ S3 ) ) )
& ! [N6: nat] :
( ( ord_less @ nat @ N6 @ ( finite_card @ nat @ S3 ) )
=> ( member @ nat @ ( R4 @ N6 ) @ S3 ) ) ) ) ).
% finite_enumerate
thf(fact_4051_card__UNION,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A3 )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A3 )
=> ( finite_finite2 @ A @ X5 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) )
= ( nat2
@ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
@ ^ [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 @ A3 )
& ( I8
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_4052_finite__Inter,axiom,
! [A: $tType,M7: set @ ( set @ A )] :
( ? [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ M7 )
& ( finite_finite2 @ A @ X2 ) )
=> ( finite_finite2 @ A @ ( complete_Inf_Inf @ ( set @ A ) @ M7 ) ) ) ).
% finite_Inter
thf(fact_4053_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,X3: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F2 @ X3 ) )
= ( product_fst @ A @ C @ X3 ) ) ).
% fst_apsnd
thf(fact_4054_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,X3: product_prod @ A @ C,G: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F2 @ X3 )
= ( product_apsnd @ C @ B @ A @ G @ X3 ) )
= ( ( F2 @ ( product_snd @ A @ C @ X3 ) )
= ( G @ ( product_snd @ A @ C @ X3 ) ) ) ) ).
% apsnd_eq_conv
thf(fact_4055_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F2: C > A,X3: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F2 @ X3 ) )
= ( F2 @ ( product_snd @ B @ C @ X3 ) ) ) ).
% snd_apsnd
thf(fact_4056_cSup__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X3 ) )
= X3 ) ) ) ).
% cSup_atLeastAtMost
thf(fact_4057_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= Y ) ) ) ).
% Sup_atLeastAtMost
thf(fact_4058_cSup__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% cSup_singleton
thf(fact_4059_cInf__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X3 ) )
= Y ) ) ) ).
% cInf_atLeastAtMost
thf(fact_4060_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= X3 ) ) ) ).
% Inf_atLeastAtMost
thf(fact_4061_cInf__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A] :
( ( complete_Inf_Inf @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% cInf_singleton
thf(fact_4062_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X3 @ Y ) )
= Y ) ) ) ).
% Sup_atLeastLessThan
thf(fact_4063_cSup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ X3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X3 ) )
= X3 ) ) ) ).
% cSup_atLeastLessThan
thf(fact_4064_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X3 @ Y ) )
= X3 ) ) ) ).
% Inf_atLeastLessThan
thf(fact_4065_cInf__atLeastLessThan,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X3 ) )
= Y ) ) ) ).
% cInf_atLeastLessThan
thf(fact_4066_Inf__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X3 ) )
= ( bot_bot @ A ) ) ) ).
% Inf_atMost
thf(fact_4067_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_4068_finite__Union,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A3 )
=> ( ! [M8: set @ A] :
( ( member @ ( set @ A ) @ M8 @ A3 )
=> ( finite_finite2 @ A @ M8 ) )
=> ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) ) ) ).
% finite_Union
thf(fact_4069_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F2: A > B > C,G: D > A] :
( ( ^ [U2: product_prod @ D @ B] : ( F2 @ ( G @ ( product_fst @ D @ B @ U2 ) ) @ ( product_snd @ D @ B @ U2 ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X: D] : ( F2 @ ( G @ X ) ) ) ) ).
% split_comp_eq
thf(fact_4070_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F3: A > B > C,X: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ) ).
% case_prod_beta'
thf(fact_4071_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_4072_The__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_4073_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z2: A,X8: set @ A] :
( ( member @ A @ Z2 @ X8 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ X5 @ Z2 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= Z2 ) ) ) ) ).
% cSup_eq_maximum
thf(fact_4074_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_bot @ A ) )
=> ! [X8: set @ A,A2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ X5 @ A2 ) )
=> ( ! [Y4: A] :
( ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ( ord_less_eq @ A @ X2 @ Y4 ) )
=> ( ord_less_eq @ A @ A2 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= A2 ) ) ) ) ).
% cSup_eq
thf(fact_4075_cInf__eq__minimum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z2: A,X8: set @ A] :
( ( member @ A @ Z2 @ X8 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ Z2 @ X5 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= Z2 ) ) ) ) ).
% cInf_eq_minimum
thf(fact_4076_cInf__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_top @ A ) )
=> ! [X8: set @ A,A2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ A2 @ X5 ) )
=> ( ! [Y4: A] :
( ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ( ord_less_eq @ A @ Y4 @ X2 ) )
=> ( ord_less_eq @ A @ Y4 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= A2 ) ) ) ) ).
% cInf_eq
thf(fact_4077_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X15: A,X24: B] : X24 ) ) ).
% snd_def
thf(fact_4078_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X15: A,X24: B] : X15 ) ) ).
% fst_def
thf(fact_4079_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y3: product_prod @ A @ B,Z: product_prod @ A @ B] : Y3 = Z )
= ( ^ [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_4080_prod__eqI,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B,Q4: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P4 )
= ( product_fst @ A @ B @ Q4 ) )
=> ( ( ( product_snd @ A @ B @ P4 )
= ( product_snd @ A @ B @ Q4 ) )
=> ( P4 = Q4 ) ) ) ).
% prod_eqI
thf(fact_4081_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_4082_split__beta,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F3: A > B > C,Prod3: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_4083_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F3: B > C > A,P6: product_prod @ B @ C] : ( F3 @ ( product_fst @ B @ C @ P6 ) @ ( product_snd @ B @ C @ P6 ) ) ) ) ).
% case_prod_beta
thf(fact_4084_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X3: product_prod @ A @ B,A3: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) )
=> ( A3 @ ( product_fst @ A @ B @ X3 ) @ ( product_snd @ A @ B @ X3 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_4085_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F2: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P @ ( F2 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_4086_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F2: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P @ ( F2 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_4087_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_4088_fst__eqD,axiom,
! [B: $tType,A: $tType,X3: A,Y: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y ) )
= A2 )
=> ( X3 = A2 ) ) ).
% fst_eqD
thf(fact_4089_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_4090_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_4091_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_4092_snd__eqD,axiom,
! [B: $tType,A: $tType,X3: B,Y: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_4093_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X3: A,Y: B,A2: product_prod @ A @ B] :
( ( P @ X3 @ Y )
=> ( ( A2
= ( product_Pair @ A @ B @ X3 @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_4094_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ A2 @ X5 ) )
=> ( ! [Y4: A] :
( ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ( ord_less_eq @ A @ Y4 @ X2 ) )
=> ( ord_less_eq @ A @ Y4 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= A2 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_4095_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,Z2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ Z2 @ X5 ) )
=> ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ).
% cInf_greatest
thf(fact_4096_cInf__le__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,X3: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X3 ) ) ) ) ).
% cInf_le_finite
thf(fact_4097_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Z2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Z2 )
=> ? [X5: A] :
( ( member @ A @ X5 @ X8 )
& ( ord_less @ A @ X5 @ Z2 ) ) ) ) ) ).
% cInf_lessD
thf(fact_4098_finite__imp__less__Inf,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,X3: A,A2: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X3 @ X8 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less @ A @ A2 @ X5 ) )
=> ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_4099_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ X5 @ A2 ) )
=> ( ! [Y4: A] :
( ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ( ord_less_eq @ A @ X2 @ Y4 ) )
=> ( ord_less_eq @ A @ A2 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= A2 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_4100_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,Z2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ X5 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X8 ) @ Z2 ) ) ) ) ).
% cSup_least
thf(fact_4101_le__cSup__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,X3: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% le_cSup_finite
thf(fact_4102_update__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,I: nat,Xy2: product_prod @ A @ B] :
( ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I @ Xy2 )
= ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ ( product_fst @ A @ B @ Xy2 ) ) @ ( list_update @ B @ Ys @ I @ ( product_snd @ A @ B @ Xy2 ) ) ) ) ).
% update_zip
thf(fact_4103_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y: A,X8: set @ A] :
( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X8 ) )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ~ ( ord_less @ A @ Y @ X5 ) ) ) ) ) ).
% less_cSupE
thf(fact_4104_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Z2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z2 @ ( complete_Sup_Sup @ A @ X8 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ X8 )
& ( ord_less @ A @ Z2 @ X5 ) ) ) ) ) ).
% less_cSupD
thf(fact_4105_finite__imp__Sup__less,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,X3: A,A2: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X3 @ X8 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less @ A @ X5 @ A2 ) )
=> ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A2 ) ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_4106_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_4107_finite__UnionD,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) )
=> ( finite_finite2 @ ( set @ A ) @ A3 ) ) ).
% finite_UnionD
thf(fact_4108_in__set__zip,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ P4 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( ? [N: nat] :
( ( ( nth @ A @ Xs @ N )
= ( product_fst @ A @ B @ P4 ) )
& ( ( nth @ B @ Ys @ N )
= ( product_snd @ A @ B @ P4 ) )
& ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ord_less @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ).
% in_set_zip
thf(fact_4109_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_4110_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,A2: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) )
= ( ! [X: A] :
( ( member @ A @ X @ X8 )
=> ( ord_less @ A @ A2 @ X ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_4111_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,A2: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A2 )
= ( ! [X: A] :
( ( member @ A @ X @ X8 )
=> ( ord_less @ A @ X @ A2 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_4112_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,A2: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X5 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S3 ) ) @ A2 ) ) ) ) ).
% cInf_abs_ge
thf(fact_4113_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,A2: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X5 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S3 ) ) @ A2 ) ) ) ) ).
% cSup_abs_le
thf(fact_4114_card__Union__le__sum__card__weak,axiom,
! [A: $tType,U3: set @ ( set @ A )] :
( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ U3 )
=> ( finite_finite2 @ A @ X5 ) )
=> ( 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_4115_in__set__enumerate__eq,axiom,
! [A: $tType,P4: product_prod @ nat @ A,N2: nat,Xs: list @ A] :
( ( member @ ( product_prod @ nat @ A ) @ P4 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) ) )
= ( ( ord_less_eq @ nat @ N2 @ ( product_fst @ nat @ A @ P4 ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P4 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) )
& ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P4 ) @ N2 ) )
= ( product_snd @ nat @ A @ P4 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_4116_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L: A,E3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X5 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S3 ) @ L ) ) @ E3 ) ) ) ) ).
% cInf_asclose
thf(fact_4117_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L: A,E3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X5 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S3 ) @ L ) ) @ E3 ) ) ) ) ).
% cSup_asclose
thf(fact_4118_finite__subset__Union,axiom,
! [A: $tType,A3: set @ A,B12: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ B12 ) )
=> ~ ! [F7: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F7 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F7 @ B12 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ F7 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_4119_ccpo__Sup__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X3: A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% ccpo_Sup_singleton
thf(fact_4120_Sup__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_4121_ccSup__empty,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% ccSup_empty
thf(fact_4122_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( ( complete_Inf_Inf @ A @ A3 )
= ( bot_bot @ A ) )
= ( ! [X: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ A3 )
& ( ord_less @ A @ Y2 @ X ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_4123_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( ( complete_Sup_Sup @ A @ A3 )
= ( bot_bot @ A ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( X
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_4124_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ A3 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( X
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_4125_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_4126_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_4127_quotient__of__denom__pos_H,axiom,
! [R2: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ ( quotient_of @ R2 ) ) ) ).
% quotient_of_denom_pos'
thf(fact_4128_bezw__non__0,axiom,
! [Y: nat,X3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y )
=> ( ( bezw @ X3 @ Y )
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_4129_bezw_Oelims,axiom,
! [X3: nat,Xa2: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X3 @ 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 @ X3 @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Xa2 ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_4130_bezw_Osimps,axiom,
( bezw
= ( ^ [X: nat,Y2: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y2
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y2 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_4131_rat__sgn__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( sgn_sgn @ rat @ P4 ) )
= ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P4 ) ) ) @ ( one_one @ int ) ) ) ).
% rat_sgn_code
thf(fact_4132_Sup__upper2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A3: set @ A,V2: A] :
( ( member @ A @ U @ A3 )
=> ( ( ord_less_eq @ A @ V2 @ U )
=> ( ord_less_eq @ A @ V2 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% Sup_upper2
thf(fact_4133_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B2: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ B2 )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ B2 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_4134_Sup__upper,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,A3: set @ A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Sup_upper
thf(fact_4135_Sup__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,Z2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ A @ X5 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ Z2 ) ) ) ).
% Sup_least
thf(fact_4136_Sup__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ! [A5: A] :
( ( member @ A @ A5 @ A3 )
=> ? [X2: A] :
( ( member @ A @ X2 @ B3 )
& ( ord_less_eq @ A @ A5 @ X2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ).
% Sup_mono
thf(fact_4137_Sup__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,X3: A] :
( ! [Y4: A] :
( ( member @ A @ Y4 @ A3 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ! [Y4: A] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ A3 )
=> ( ord_less_eq @ A @ Z4 @ Y4 ) )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ A3 )
= X3 ) ) ) ) ).
% Sup_eqI
thf(fact_4138_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: A,S3: set @ A] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ S3 ) )
= ( ? [X: A] :
( ( member @ A @ X @ S3 )
& ( ord_less @ A @ A2 @ X ) ) ) ) ) ).
% less_Sup_iff
thf(fact_4139_Inf__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,Z2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ A @ Z2 @ X5 ) )
=> ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ).
% Inf_greatest
thf(fact_4140_le__Inf__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: A,A3: set @ A] :
( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A3 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ B2 @ X ) ) ) ) ) ).
% le_Inf_iff
thf(fact_4141_Inf__lower2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A3: set @ A,V2: A] :
( ( member @ A @ U @ A3 )
=> ( ( ord_less_eq @ A @ U @ V2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ V2 ) ) ) ) ).
% Inf_lower2
thf(fact_4142_Inf__lower,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,A3: set @ A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ X3 ) ) ) ).
% Inf_lower
thf(fact_4143_Inf__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: set @ A,A3: set @ A] :
( ! [B5: A] :
( ( member @ A @ B5 @ B3 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less_eq @ A @ X2 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) ) ) ).
% Inf_mono
thf(fact_4144_Inf__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,X3: A] :
( ! [I4: A] :
( ( member @ A @ I4 @ A3 )
=> ( ord_less_eq @ A @ X3 @ I4 ) )
=> ( ! [Y4: A] :
( ! [I5: A] :
( ( member @ A @ I5 @ A3 )
=> ( ord_less_eq @ A @ Y4 @ I5 ) )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( complete_Inf_Inf @ A @ A3 )
= X3 ) ) ) ) ).
% Inf_eqI
thf(fact_4145_Inf__less__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [S3: set @ A,A2: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S3 ) @ A2 )
= ( ? [X: A] :
( ( member @ A @ X @ S3 )
& ( ord_less @ A @ X @ A2 ) ) ) ) ) ).
% Inf_less_iff
thf(fact_4146_Union__empty__conv,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ A3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A3 )
=> ( X
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_empty_conv
thf(fact_4147_empty__Union__conv,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ A3 ) )
= ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A3 )
=> ( X
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% empty_Union_conv
thf(fact_4148_Union__least,axiom,
! [A: $tType,A3: set @ ( set @ A ),C5: set @ A] :
( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ X10 @ C5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ C5 ) ) ).
% Union_least
thf(fact_4149_Union__upper,axiom,
! [A: $tType,B3: set @ A,A3: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) ) ).
% Union_upper
thf(fact_4150_Union__subsetI,axiom,
! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] :
( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A3 )
=> ? [Y5: set @ A] :
( ( member @ ( set @ A ) @ Y5 @ B3 )
& ( ord_less_eq @ ( set @ A ) @ X5 @ Y5 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ).
% Union_subsetI
thf(fact_4151_Inter__lower,axiom,
! [A: $tType,B3: set @ A,A3: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ B3 ) ) ).
% Inter_lower
thf(fact_4152_Inter__greatest,axiom,
! [A: $tType,A3: set @ ( set @ A ),C5: set @ A] :
( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ C5 @ X10 ) )
=> ( ord_less_eq @ ( set @ A ) @ C5 @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) ) ) ).
% Inter_greatest
thf(fact_4153_Union__SetCompr__eq,axiom,
! [B: $tType,A: $tType,F2: B > ( set @ A ),P: B > $o] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [X: B] :
( ( Uu3
= ( F2 @ X ) )
& ( P @ X ) ) ) )
= ( collect @ A
@ ^ [A4: A] :
? [X: B] :
( ( P @ X )
& ( member @ A @ A4 @ ( F2 @ X ) ) ) ) ) ).
% Union_SetCompr_eq
thf(fact_4154_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X3: A,A3: set @ A] :
( ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A3 ) )
= ( ! [Y2: A] :
( ( ord_less @ A @ Y2 @ X3 )
=> ? [X: A] :
( ( member @ A @ X @ A3 )
& ( ord_less @ A @ Y2 @ X ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_4155_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A,X3: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ X3 )
= ( ! [Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ? [X: A] :
( ( member @ A @ X @ A3 )
& ( ord_less @ A @ X @ Y2 ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_4156_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A3 )
=> ( ord_less_eq @ A @ U @ V3 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_4157_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ).
% Sup_subset_mono
thf(fact_4158_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A3 )
=> ( ord_less_eq @ A @ V3 @ U ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_4159_Inf__superset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) ) ) ).
% Inf_superset_mono
thf(fact_4160_Union__mono,axiom,
! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ).
% Union_mono
thf(fact_4161_Inter__anti__mono,axiom,
! [A: $tType,B3: set @ ( set @ A ),A3: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B3 ) ) ) ).
% Inter_anti_mono
thf(fact_4162_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_4163_Inter__subset,axiom,
! [A: $tType,A3: set @ ( set @ A ),B3: set @ A] :
( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ X10 @ B3 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ B3 ) ) ) ).
% Inter_subset
thf(fact_4164_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Inf_le_Sup
thf(fact_4165_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F3: A > nat,G2: B > nat,P6: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F3 @ ( product_fst @ A @ B @ P6 ) ) @ ( G2 @ ( product_snd @ A @ B @ P6 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_4166_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P4: A,Q: B > $o,Q4: B] :
( ( P @ P4 )
=> ( ( Q @ Q4 )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P4 @ Q4 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P4 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_4167_exI__realizer,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Y: A,X3: B] :
( ( P @ Y @ X3 )
=> ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_4168_bezw_Opelims,axiom,
! [X3: nat,Xa2: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X3 @ 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 @ X3 @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Xa2 ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) ) ) ) ).
% bezw.pelims
thf(fact_4169_prod__decode__aux_Opelims,axiom,
! [X3: nat,Xa2: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa2 @ X3 )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X3 @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa2 @ X3 )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_4170_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_4171_strict__mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( strict_mono_on @ A @ B )
= ( ^ [F3: A > B,A6: set @ A] :
! [R5: A,S8: A] :
( ( ( member @ A @ R5 @ A6 )
& ( member @ A @ S8 @ A6 )
& ( ord_less @ A @ R5 @ S8 ) )
=> ( ord_less @ B @ ( F3 @ R5 ) @ ( F3 @ S8 ) ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_4172_strict__mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [A3: set @ A,F2: A > B] :
( ! [R4: A,S2: A] :
( ( member @ A @ R4 @ A3 )
=> ( ( member @ A @ S2 @ A3 )
=> ( ( ord_less @ A @ R4 @ S2 )
=> ( ord_less @ B @ ( F2 @ R4 ) @ ( F2 @ S2 ) ) ) ) )
=> ( strict_mono_on @ A @ B @ F2 @ A3 ) ) ) ).
% strict_mono_onI
thf(fact_4173_gcd__left__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_gcd @ A @ A2 @ ( gcd_gcd @ A @ A2 @ B2 ) )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% gcd_left_idem
thf(fact_4174_gcd__right__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_gcd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ B2 )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% gcd_right_idem
thf(fact_4175_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_4176_gcd_Obottom__right__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A] :
( ( gcd_gcd @ A @ A2 @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.bottom_right_bottom
thf(fact_4177_gcd_Obottom__left__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A] :
( ( gcd_gcd @ A @ ( one_one @ A ) @ A2 )
= ( one_one @ A ) ) ) ).
% gcd.bottom_left_bottom
thf(fact_4178_gcd__add1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,N2: A] :
( ( gcd_gcd @ A @ ( plus_plus @ A @ M @ N2 ) @ N2 )
= ( gcd_gcd @ A @ M @ N2 ) ) ) ).
% gcd_add1
thf(fact_4179_gcd__add2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,N2: A] :
( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ M @ N2 ) )
= ( gcd_gcd @ A @ M @ N2 ) ) ) ).
% gcd_add2
thf(fact_4180_gcd__neg2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_gcd @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% gcd_neg2
thf(fact_4181_gcd__neg1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_gcd @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% gcd_neg1
thf(fact_4182_gcd__exp,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A2: A,N2: nat,B2: A] :
( ( gcd_gcd @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) )
= ( power_power @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ N2 ) ) ) ).
% gcd_exp
thf(fact_4183_gcd__dvd1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ A2 ) ) ).
% gcd_dvd1
thf(fact_4184_gcd__dvd2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ B2 ) ) ).
% gcd_dvd2
thf(fact_4185_gcd__greatest__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) )
= ( ( dvd_dvd @ A @ A2 @ B2 )
& ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% gcd_greatest_iff
thf(fact_4186_gcd__1__int,axiom,
! [M: int] :
( ( gcd_gcd @ int @ M @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% gcd_1_int
thf(fact_4187_gcd__neg2__int,axiom,
! [X3: int,Y: int] :
( ( gcd_gcd @ int @ X3 @ ( uminus_uminus @ int @ Y ) )
= ( gcd_gcd @ int @ X3 @ Y ) ) ).
% gcd_neg2_int
thf(fact_4188_gcd__neg1__int,axiom,
! [X3: int,Y: int] :
( ( gcd_gcd @ int @ ( uminus_uminus @ int @ X3 ) @ Y )
= ( gcd_gcd @ int @ X3 @ Y ) ) ).
% gcd_neg1_int
thf(fact_4189_abs__gcd__int,axiom,
! [X3: int,Y: int] :
( ( abs_abs @ int @ ( gcd_gcd @ int @ X3 @ Y ) )
= ( gcd_gcd @ int @ X3 @ Y ) ) ).
% abs_gcd_int
thf(fact_4190_gcd__abs1__int,axiom,
! [X3: int,Y: int] :
( ( gcd_gcd @ int @ ( abs_abs @ int @ X3 ) @ Y )
= ( gcd_gcd @ int @ X3 @ Y ) ) ).
% gcd_abs1_int
thf(fact_4191_gcd__abs2__int,axiom,
! [X3: int,Y: int] :
( ( gcd_gcd @ int @ X3 @ ( abs_abs @ int @ Y ) )
= ( gcd_gcd @ int @ X3 @ Y ) ) ).
% gcd_abs2_int
thf(fact_4192_gcd__neg__numeral__1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N2: num,A2: A] :
( ( gcd_gcd @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ A2 )
= ( gcd_gcd @ A @ ( numeral_numeral @ A @ N2 ) @ A2 ) ) ) ).
% gcd_neg_numeral_1
thf(fact_4193_gcd__neg__numeral__2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,N2: num] :
( ( gcd_gcd @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( gcd_gcd @ A @ A2 @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% gcd_neg_numeral_2
thf(fact_4194_is__unit__gcd__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( one_one @ A ) )
= ( ( gcd_gcd @ A @ A2 @ B2 )
= ( one_one @ A ) ) ) ) ).
% is_unit_gcd_iff
thf(fact_4195_gcd__pos__int,axiom,
! [M: int,N2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ M @ N2 ) )
= ( ( M
!= ( zero_zero @ int ) )
| ( N2
!= ( zero_zero @ int ) ) ) ) ).
% gcd_pos_int
thf(fact_4196_gcd__neg__numeral__1__int,axiom,
! [N2: num,X3: int] :
( ( gcd_gcd @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) @ X3 )
= ( gcd_gcd @ int @ ( numeral_numeral @ int @ N2 ) @ X3 ) ) ).
% gcd_neg_numeral_1_int
thf(fact_4197_gcd__neg__numeral__2__int,axiom,
! [X3: int,N2: num] :
( ( gcd_gcd @ int @ X3 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( gcd_gcd @ int @ X3 @ ( numeral_numeral @ int @ N2 ) ) ) ).
% gcd_neg_numeral_2_int
thf(fact_4198_gcd__0__int,axiom,
! [X3: int] :
( ( gcd_gcd @ int @ X3 @ ( zero_zero @ int ) )
= ( abs_abs @ int @ X3 ) ) ).
% gcd_0_int
thf(fact_4199_gcd__0__left__int,axiom,
! [X3: int] :
( ( gcd_gcd @ int @ ( zero_zero @ int ) @ X3 )
= ( abs_abs @ int @ X3 ) ) ).
% gcd_0_left_int
thf(fact_4200_gcd__proj1__if__dvd__int,axiom,
! [X3: int,Y: int] :
( ( dvd_dvd @ int @ X3 @ Y )
=> ( ( gcd_gcd @ int @ X3 @ Y )
= ( abs_abs @ int @ X3 ) ) ) ).
% gcd_proj1_if_dvd_int
thf(fact_4201_gcd__proj2__if__dvd__int,axiom,
! [Y: int,X3: int] :
( ( dvd_dvd @ int @ Y @ X3 )
=> ( ( gcd_gcd @ int @ X3 @ Y )
= ( abs_abs @ int @ Y ) ) ) ).
% gcd_proj2_if_dvd_int
thf(fact_4202_gcd__idem__int,axiom,
! [X3: int] :
( ( gcd_gcd @ int @ X3 @ X3 )
= ( abs_abs @ int @ X3 ) ) ).
% gcd_idem_int
thf(fact_4203_gcd__diff1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [M: A,N2: A] :
( ( gcd_gcd @ A @ ( minus_minus @ A @ M @ N2 ) @ N2 )
= ( gcd_gcd @ A @ M @ N2 ) ) ) ).
% gcd_diff1
thf(fact_4204_gcd__diff2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N2: A,M: A] :
( ( gcd_gcd @ A @ ( minus_minus @ A @ N2 @ M ) @ N2 )
= ( gcd_gcd @ A @ M @ N2 ) ) ) ).
% gcd_diff2
thf(fact_4205_gcd_Oassoc,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( gcd_gcd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ C2 )
= ( gcd_gcd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd.assoc
thf(fact_4206_gcd_Ocommute,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( gcd_gcd @ A )
= ( ^ [A4: A,B4: A] : ( gcd_gcd @ A @ B4 @ A4 ) ) ) ) ).
% gcd.commute
thf(fact_4207_gcd_Oleft__commute,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( gcd_gcd @ A @ B2 @ ( gcd_gcd @ A @ A2 @ C2 ) )
= ( gcd_gcd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd.left_commute
thf(fact_4208_gcd__red__int,axiom,
( ( gcd_gcd @ int )
= ( ^ [X: int,Y2: int] : ( gcd_gcd @ int @ Y2 @ ( modulo_modulo @ int @ X @ Y2 ) ) ) ) ).
% gcd_red_int
thf(fact_4209_gcd__mono,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ B2 @ D2 )
=> ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( gcd_gcd @ A @ C2 @ D2 ) ) ) ) ) ).
% gcd_mono
thf(fact_4210_dvd__gcdD1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) )
=> ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_gcdD1
thf(fact_4211_dvd__gcdD2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) )
=> ( dvd_dvd @ A @ A2 @ C2 ) ) ) ).
% dvd_gcdD2
thf(fact_4212_gcd__dvdI1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% gcd_dvdI1
thf(fact_4213_gcd__dvdI2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ C2 )
=> ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% gcd_dvdI2
thf(fact_4214_gcd__greatest,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ ( gcd_gcd @ A @ A2 @ B2 ) ) ) ) ) ).
% gcd_greatest
thf(fact_4215_gcd__dvd__antisym,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( gcd_gcd @ A @ C2 @ D2 ) )
=> ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ C2 @ D2 ) @ ( gcd_gcd @ A @ A2 @ B2 ) )
=> ( ( gcd_gcd @ A @ A2 @ B2 )
= ( gcd_gcd @ A @ C2 @ D2 ) ) ) ) ) ).
% gcd_dvd_antisym
thf(fact_4216_gcd__add__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,K: A,N2: A] :
( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ ( times_times @ A @ K @ M ) @ N2 ) )
= ( gcd_gcd @ A @ M @ N2 ) ) ) ).
% gcd_add_mult
thf(fact_4217_gcd__dvd__prod,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,K: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( times_times @ A @ K @ B2 ) ) ) ).
% gcd_dvd_prod
thf(fact_4218_gcd__ge__0__int,axiom,
! [X3: int,Y: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ X3 @ Y ) ) ).
% gcd_ge_0_int
thf(fact_4219_bezout__int,axiom,
! [X3: int,Y: int] :
? [U4: int,V3: int] :
( ( plus_plus @ int @ ( times_times @ int @ U4 @ X3 ) @ ( times_times @ int @ V3 @ Y ) )
= ( gcd_gcd @ int @ X3 @ Y ) ) ).
% bezout_int
thf(fact_4220_gcd__mult__distrib__int,axiom,
! [K: int,M: int,N2: int] :
( ( times_times @ int @ ( abs_abs @ int @ K ) @ ( gcd_gcd @ int @ M @ N2 ) )
= ( gcd_gcd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N2 ) ) ) ).
% gcd_mult_distrib_int
thf(fact_4221_gcd__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit2
thf(fact_4222_gcd__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit1
thf(fact_4223_gcd__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C2 @ A2 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit2
thf(fact_4224_gcd__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A2 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit1
thf(fact_4225_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_4226_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_4227_gcd__cases__int,axiom,
! [X3: int,Y: int,P: int > $o] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P @ ( gcd_gcd @ int @ X3 @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ X3 @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ X3 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X3 ) @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ X3 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X3 ) @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( P @ ( gcd_gcd @ int @ X3 @ Y ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_4228_gcd__unique__int,axiom,
! [D2: int,A2: int,B2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
& ( dvd_dvd @ int @ D2 @ A2 )
& ( dvd_dvd @ int @ D2 @ B2 )
& ! [E4: int] :
( ( ( dvd_dvd @ int @ E4 @ A2 )
& ( dvd_dvd @ int @ E4 @ B2 ) )
=> ( dvd_dvd @ int @ E4 @ D2 ) ) )
= ( D2
= ( gcd_gcd @ int @ A2 @ B2 ) ) ) ).
% gcd_unique_int
thf(fact_4229_gcd__non__0__int,axiom,
! [Y: int,X3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Y )
=> ( ( gcd_gcd @ int @ X3 @ Y )
= ( gcd_gcd @ int @ Y @ ( modulo_modulo @ int @ X3 @ Y ) ) ) ) ).
% gcd_non_0_int
thf(fact_4230_gcd__code__int,axiom,
( ( gcd_gcd @ int )
= ( ^ [K3: int,L2: int] :
( abs_abs @ int
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K3
@ ( gcd_gcd @ int @ L2 @ ( modulo_modulo @ int @ ( abs_abs @ int @ K3 ) @ ( abs_abs @ int @ L2 ) ) ) ) ) ) ) ).
% gcd_code_int
thf(fact_4231_strict__mono__on__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( preorder @ B ) )
=> ! [F2: A > B,A3: set @ A,X3: A,Y: A] :
( ( strict_mono_on @ A @ B @ F2 @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_4232_strict__mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [F2: A > B,A3: set @ A,R2: A,S: A] :
( ( strict_mono_on @ A @ B @ F2 @ A3 )
=> ( ( member @ A @ R2 @ A3 )
=> ( ( member @ A @ S @ A3 )
=> ( ( ord_less @ A @ R2 @ S )
=> ( ord_less @ B @ ( F2 @ R2 ) @ ( F2 @ S ) ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_4233_card__partition,axiom,
! [A: $tType,C5: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ ( set @ A ) @ C5 )
=> ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) )
=> ( ! [C3: set @ A] :
( ( member @ ( set @ A ) @ C3 @ C5 )
=> ( ( finite_card @ A @ C3 )
= K ) )
=> ( ! [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 @ K @ ( finite_card @ ( set @ A ) @ C5 ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_4234_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( 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 @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_4235_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( 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 @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_4236_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( 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 @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_4237_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_4238_gcd__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% gcd_nat.left_neutral
thf(fact_4239_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_4240_gcd__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( gcd_gcd @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% gcd_nat.right_neutral
thf(fact_4241_gcd__0__nat,axiom,
! [X3: nat] :
( ( gcd_gcd @ nat @ X3 @ ( zero_zero @ nat ) )
= X3 ) ).
% gcd_0_nat
thf(fact_4242_gcd__0__left__nat,axiom,
! [X3: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ X3 )
= X3 ) ).
% gcd_0_left_nat
thf(fact_4243_Int__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
= ( ( member @ A @ C2 @ A3 )
& ( member @ A @ C2 @ B3 ) ) ) ).
% Int_iff
thf(fact_4244_IntI,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_4245_gcd__1__nat,axiom,
! [M: nat] :
( ( gcd_gcd @ nat @ M @ ( one_one @ nat ) )
= ( one_one @ nat ) ) ).
% gcd_1_nat
thf(fact_4246_gcd__proj2__if__dvd__nat,axiom,
! [Y: nat,X3: nat] :
( ( dvd_dvd @ nat @ Y @ X3 )
=> ( ( gcd_gcd @ nat @ X3 @ Y )
= Y ) ) ).
% gcd_proj2_if_dvd_nat
thf(fact_4247_gcd__proj1__if__dvd__nat,axiom,
! [X3: nat,Y: nat] :
( ( dvd_dvd @ nat @ X3 @ Y )
=> ( ( gcd_gcd @ nat @ X3 @ Y )
= X3 ) ) ).
% gcd_proj1_if_dvd_nat
thf(fact_4248_gcd__nat_Obounded__iff,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) )
= ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ).
% gcd_nat.bounded_iff
thf(fact_4249_gcd__nat_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( dvd_dvd @ nat @ B2 @ A2 )
=> ( ( gcd_gcd @ nat @ A2 @ B2 )
= B2 ) ) ).
% gcd_nat.absorb2
thf(fact_4250_gcd__nat_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( gcd_gcd @ nat @ A2 @ B2 )
= A2 ) ) ).
% gcd_nat.absorb1
thf(fact_4251_gcd__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% gcd_Suc_0
thf(fact_4252_gcd__pos__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_gcd @ nat @ M @ N2 ) )
= ( ( M
!= ( zero_zero @ nat ) )
| ( N2
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_pos_nat
thf(fact_4253_boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X3 )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_left
thf(fact_4254_boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_right
thf(fact_4255_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_4256_Int__subset__iff,axiom,
! [A: $tType,C5: set @ A,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C5 @ A3 )
& ( ord_less_eq @ ( set @ A ) @ C5 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_4257_Int__insert__left__if0,axiom,
! [A: $tType,A2: A,C5: set @ A,B3: set @ A] :
( ~ ( member @ A @ A2 @ C5 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C5 )
= ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_4258_Int__insert__left__if1,axiom,
! [A: $tType,A2: A,C5: set @ A,B3: set @ A] :
( ( member @ A @ A2 @ C5 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C5 )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_4259_insert__inter__insert,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ ( insert @ A @ A2 @ B3 ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_4260_Int__insert__right__if0,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_4261_Int__insert__right__if1,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_4262_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_4263_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_4264_or__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
| ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% or_negative_int_iff
thf(fact_4265_Pow__Int__eq,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( pow2 @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
= ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B3 ) ) ) ).
% Pow_Int_eq
thf(fact_4266_gcd__int__int__eq,axiom,
! [M: nat,N2: nat] :
( ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N2 ) )
= ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ M @ N2 ) ) ) ).
% gcd_int_int_eq
thf(fact_4267_boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ ( uminus_uminus @ A @ X3 ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_right
thf(fact_4268_boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ X3 )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_left
thf(fact_4269_inf__compl__bot__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y @ ( uminus_uminus @ A @ X3 ) ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_right
thf(fact_4270_inf__compl__bot__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left2
thf(fact_4271_inf__compl__bot__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ ( inf_inf @ A @ X3 @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left1
thf(fact_4272_insert__disjoint_I1_J,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A2 @ B3 )
& ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_4273_insert__disjoint_I2_J,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ B3 ) )
= ( ~ ( member @ A @ A2 @ B3 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_4274_disjoint__insert_I1_J,axiom,
! [A: $tType,B3: set @ A,A2: A,A3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B3 @ ( insert @ A @ A2 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A2 @ B3 )
& ( ( inf_inf @ ( set @ A ) @ B3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_4275_disjoint__insert_I2_J,axiom,
! [A: $tType,A3: set @ A,B2: A,B3: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B3 ) ) )
= ( ~ ( member @ A @ B2 @ A3 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_4276_Diff__disjoint,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_4277_Compl__disjoint2,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_4278_Compl__disjoint,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_4279_Diff__Compl,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ).
% Diff_Compl
thf(fact_4280_or__nat__numerals_I4_J,axiom,
! [X3: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) ) ).
% or_nat_numerals(4)
thf(fact_4281_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_4282_gcd__nat__abs__right__eq,axiom,
! [N2: nat,K: int] :
( ( gcd_gcd @ nat @ N2 @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( nat2 @ ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ) ).
% gcd_nat_abs_right_eq
thf(fact_4283_gcd__nat__abs__left__eq,axiom,
! [K: int,N2: nat] :
( ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N2 )
= ( nat2 @ ( gcd_gcd @ int @ K @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% gcd_nat_abs_left_eq
thf(fact_4284_or__nat__numerals_I3_J,axiom,
! [X3: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) ) ).
% or_nat_numerals(3)
thf(fact_4285_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_4286_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A3: set @ B,P: B > $o,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X ) ) @ ( F2 @ X ) )
@ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_4287_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A3: set @ B,F2: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ ( zero_neq_one_of_bool @ A @ ( P @ X ) ) )
@ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_4288_sum__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A3: set @ B,P: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( zero_neq_one_of_bool @ A @ ( P @ X ) )
@ A3 )
= ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_4289_gcd__mult__distrib__nat,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times @ nat @ K @ ( gcd_gcd @ nat @ M @ N2 ) )
= ( gcd_gcd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ).
% gcd_mult_distrib_nat
thf(fact_4290_gcd__red__nat,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X: nat,Y2: nat] : ( gcd_gcd @ nat @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) ) ).
% gcd_red_nat
thf(fact_4291_gcd__unique__nat,axiom,
! [D2: nat,A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ D2 @ A2 )
& ( dvd_dvd @ nat @ D2 @ B2 )
& ! [E4: nat] :
( ( ( dvd_dvd @ nat @ E4 @ A2 )
& ( dvd_dvd @ nat @ E4 @ B2 ) )
=> ( dvd_dvd @ nat @ E4 @ D2 ) ) )
= ( D2
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ).
% gcd_unique_nat
thf(fact_4292_gcd__nat_Ostrict__coboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ( dvd_dvd @ nat @ B2 @ C2 )
& ( B2 != C2 ) )
=> ( ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 )
& ( ( gcd_gcd @ nat @ A2 @ B2 )
!= C2 ) ) ) ).
% gcd_nat.strict_coboundedI2
thf(fact_4293_gcd__nat_Ostrict__coboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) )
=> ( ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 )
& ( ( gcd_gcd @ nat @ A2 @ B2 )
!= C2 ) ) ) ).
% gcd_nat.strict_coboundedI1
thf(fact_4294_gcd__nat_Ostrict__order__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
= ( ( A2
= ( gcd_gcd @ nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ).
% gcd_nat.strict_order_iff
thf(fact_4295_gcd__nat_Ostrict__boundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) )
& ( A2
!= ( gcd_gcd @ nat @ B2 @ C2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ~ ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) ) ) ) ).
% gcd_nat.strict_boundedE
thf(fact_4296_gcd__nat_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( dvd_dvd @ nat @ B2 @ C2 )
=> ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 ) ) ).
% gcd_nat.coboundedI2
thf(fact_4297_gcd__nat_OcoboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( dvd_dvd @ nat @ A2 @ C2 )
=> ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 ) ) ).
% gcd_nat.coboundedI1
thf(fact_4298_gcd__nat_Oabsorb__iff2,axiom,
( ( dvd_dvd @ nat )
= ( ^ [B4: nat,A4: nat] :
( ( gcd_gcd @ nat @ A4 @ B4 )
= B4 ) ) ) ).
% gcd_nat.absorb_iff2
thf(fact_4299_gcd__nat_Oabsorb__iff1,axiom,
( ( dvd_dvd @ nat )
= ( ^ [A4: nat,B4: nat] :
( ( gcd_gcd @ nat @ A4 @ B4 )
= A4 ) ) ) ).
% gcd_nat.absorb_iff1
thf(fact_4300_gcd__nat_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 ) ).
% gcd_nat.cobounded2
thf(fact_4301_gcd__nat_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ A2 ) ).
% gcd_nat.cobounded1
thf(fact_4302_gcd__nat_Oorder__iff,axiom,
( ( dvd_dvd @ nat )
= ( ^ [A4: nat,B4: nat] :
( A4
= ( gcd_gcd @ nat @ A4 @ B4 ) ) ) ) ).
% gcd_nat.order_iff
thf(fact_4303_gcd__nat_OboundedI,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ A2 @ C2 )
=> ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) ) ) ) ).
% gcd_nat.boundedI
thf(fact_4304_gcd__nat_OboundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) )
=> ~ ( ( dvd_dvd @ nat @ A2 @ B2 )
=> ~ ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ).
% gcd_nat.boundedE
thf(fact_4305_gcd__nat_Oabsorb4,axiom,
! [B2: nat,A2: nat] :
( ( ( dvd_dvd @ nat @ B2 @ A2 )
& ( B2 != A2 ) )
=> ( ( gcd_gcd @ nat @ A2 @ B2 )
= B2 ) ) ).
% gcd_nat.absorb4
thf(fact_4306_gcd__nat_Oabsorb3,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( ( gcd_gcd @ nat @ A2 @ B2 )
= A2 ) ) ).
% gcd_nat.absorb3
thf(fact_4307_gcd__nat_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( gcd_gcd @ nat @ A2 @ B2 ) )
=> ( dvd_dvd @ nat @ A2 @ B2 ) ) ).
% gcd_nat.orderI
thf(fact_4308_gcd__nat_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( A2
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ).
% gcd_nat.orderE
thf(fact_4309_gcd__nat_Omono,axiom,
! [A2: nat,C2: nat,B2: nat,D2: nat] :
( ( dvd_dvd @ nat @ A2 @ C2 )
=> ( ( dvd_dvd @ nat @ B2 @ D2 )
=> ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ ( gcd_gcd @ nat @ C2 @ D2 ) ) ) ) ).
% gcd_nat.mono
thf(fact_4310_disjoint__iff__not__equal,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ B3 )
=> ( X != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_4311_Int__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_4312_Int__empty__left,axiom,
! [A: $tType,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_4313_disjoint__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ~ ( member @ A @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_4314_Int__emptyI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ~ ( member @ A @ X5 @ B3 ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_4315_Int__mono,axiom,
! [A: $tType,A3: set @ A,C5: set @ A,B3: set @ A,D6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ D6 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( inf_inf @ ( set @ A ) @ C5 @ D6 ) ) ) ) ).
% Int_mono
thf(fact_4316_Int__lower1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ A3 ) ).
% Int_lower1
thf(fact_4317_Int__lower2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_4318_Int__absorb1,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_4319_Int__absorb2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= A3 ) ) ).
% Int_absorb2
thf(fact_4320_Int__greatest,axiom,
! [A: $tType,C5: set @ A,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ C5 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ C5 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_4321_Int__Collect__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B3 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_4322_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_4323_Int__Collect,axiom,
! [A: $tType,X3: A,A3: set @ A,P: A > $o] :
( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X3 @ A3 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_4324_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A6 )
& ( member @ A @ X @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_4325_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_4326_bit_Odisj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se1065995026697491101ons_or @ A @ X3 @ ( zero_zero @ A ) )
= X3 ) ) ).
% bit.disj_zero_right
thf(fact_4327_Int__left__commute,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) )
= ( inf_inf @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% Int_left_commute
thf(fact_4328_Int__left__absorb,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ).
% Int_left_absorb
thf(fact_4329_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_4330_Int__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_4331_Int__assoc,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C5 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ).
% Int_assoc
thf(fact_4332_IntD2,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
=> ( member @ A @ C2 @ B3 ) ) ).
% IntD2
thf(fact_4333_IntD1,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
=> ( member @ A @ C2 @ A3 ) ) ).
% IntD1
thf(fact_4334_IntE,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ( ( member @ A @ C2 @ A3 )
=> ~ ( member @ A @ C2 @ B3 ) ) ) ).
% IntE
thf(fact_4335_Diff__Int__distrib2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ C5 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C5 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ).
% Diff_Int_distrib2
thf(fact_4336_Diff__Int__distrib,axiom,
! [A: $tType,C5: set @ A,A3: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ C5 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C5 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C5 @ B3 ) ) ) ).
% Diff_Int_distrib
thf(fact_4337_Diff__Diff__Int,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ).
% Diff_Diff_Int
thf(fact_4338_Diff__Int2,axiom,
! [A: $tType,A3: set @ A,C5: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C5 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C5 ) @ B3 ) ) ).
% Diff_Int2
thf(fact_4339_Int__Diff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C5 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ C5 ) ) ) ).
% Int_Diff
thf(fact_4340_Int__insert__left,axiom,
! [A: $tType,A2: A,C5: set @ A,B3: set @ A] :
( ( ( member @ A @ A2 @ C5 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C5 )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) )
& ( ~ ( member @ A @ A2 @ C5 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C5 )
= ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_4341_Int__insert__right,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: set @ A] :
( ( ( member @ A @ A2 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) )
& ( ~ ( member @ A @ A2 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_4342_Union__Int__subset,axiom,
! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A3 @ B3 ) ) @ ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ).
% Union_Int_subset
thf(fact_4343_Sup__inter__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B3: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ).
% Sup_inter_less_eq
thf(fact_4344_inf__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,A2: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ A2 ) @ ( inf_inf @ A @ X3 @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left2
thf(fact_4345_inf__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,A2: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ A2 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left1
thf(fact_4346_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_4347_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_4348_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B3: set @ A,A2: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B3 ) @ A2 )
= ( bot_bot @ A ) )
= ( ! [X: A] :
( ( member @ A @ X @ B3 )
=> ( ( inf_inf @ A @ X @ A2 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_4349_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_4350_gcd__diff1__nat,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 )
= ( gcd_gcd @ nat @ M @ N2 ) ) ) ).
% gcd_diff1_nat
thf(fact_4351_gcd__diff2__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ N2 @ M ) @ N2 )
= ( gcd_gcd @ nat @ M @ N2 ) ) ) ).
% gcd_diff2_nat
thf(fact_4352_gcd__nat_Oelims,axiom,
! [X3: nat,Xa2: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X3 @ Xa2 )
= Y )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y = X3 ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_4353_gcd__nat_Osimps,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X: nat,Y2: nat] :
( if @ nat
@ ( Y2
= ( zero_zero @ nat ) )
@ X
@ ( gcd_gcd @ nat @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_4354_gcd__non__0__nat,axiom,
! [Y: nat,X3: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ( ( gcd_gcd @ nat @ X3 @ Y )
= ( gcd_gcd @ nat @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) ) ).
% gcd_non_0_nat
thf(fact_4355_Int__Diff__disjoint,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_Diff_disjoint
thf(fact_4356_Diff__triv,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ B3 )
= A3 ) ) ).
% Diff_triv
thf(fact_4357_Union__disjoint,axiom,
! [A: $tType,C5: set @ ( set @ A ),A3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) @ A3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ C5 )
=> ( ( inf_inf @ ( set @ A ) @ X @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_disjoint
thf(fact_4358_ivl__disj__int__two_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(3)
thf(fact_4359_gcd__integer_Orep__eq,axiom,
! [X3: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( gcd_gcd @ code_integer @ X3 @ Xa2 ) )
= ( gcd_gcd @ int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% gcd_integer.rep_eq
thf(fact_4360_gcd__integer_Oabs__eq,axiom,
! [Xa2: int,X3: int] :
( ( gcd_gcd @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
= ( code_integer_of_int @ ( gcd_gcd @ int @ Xa2 @ X3 ) ) ) ).
% gcd_integer.abs_eq
thf(fact_4361_Diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ B6 ) ) ) ) ).
% Diff_eq
thf(fact_4362_inf__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ( inf_inf @ A @ X3 @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% inf_shunt
thf(fact_4363_finite__Inf__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( member @ A @ Y4 @ A3 )
=> ( member @ A @ ( inf_inf @ A @ X5 @ Y4 ) @ A3 ) ) )
=> ( member @ A @ ( complete_Inf_Inf @ A @ A3 ) @ A3 ) ) ) ) ) ).
% finite_Inf_in
thf(fact_4364_bezout__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ? [X5: nat,Y4: nat] :
( ( times_times @ nat @ A2 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).
% bezout_nat
thf(fact_4365_bezout__gcd__nat_H,axiom,
! [B2: nat,A2: nat] :
? [X5: nat,Y4: nat] :
( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( times_times @ nat @ A2 @ X5 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X5 ) @ ( times_times @ nat @ B2 @ Y4 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) )
| ( ( ord_less_eq @ nat @ ( times_times @ nat @ A2 @ Y4 ) @ ( times_times @ nat @ B2 @ X5 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X5 ) @ ( times_times @ nat @ A2 @ Y4 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_4366_ivl__disj__int__two_I7_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(7)
thf(fact_4367_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_4368_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_4369_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_4370_insert__partition,axiom,
! [A: $tType,X3: set @ A,F5: set @ ( set @ A )] :
( ~ ( member @ ( set @ A ) @ X3 @ F5 )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ ( insert @ ( set @ A ) @ X3 @ F5 ) )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ ( insert @ ( set @ A ) @ X3 @ F5 ) )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X5 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ ( complete_Sup_Sup @ ( set @ A ) @ F5 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_partition
thf(fact_4371_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,B3: set @ B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( if @ A @ ( member @ B @ X @ B3 ) @ ( G @ X ) @ ( zero_zero @ A ) )
@ A3 ) ) ) ) ).
% sum.inter_restrict
thf(fact_4372_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,B3: set @ B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( if @ A @ ( member @ B @ X @ B3 ) @ ( G @ X ) @ ( one_one @ A ) )
@ A3 ) ) ) ) ).
% prod.inter_restrict
thf(fact_4373_gcd__code__integer,axiom,
( ( gcd_gcd @ code_integer )
= ( ^ [K3: code_integer,L2: code_integer] :
( abs_abs @ code_integer
@ ( if @ code_integer
@ ( L2
= ( zero_zero @ code_integer ) )
@ K3
@ ( gcd_gcd @ code_integer @ L2 @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) ) ) ) ) ) ).
% gcd_code_integer
thf(fact_4374_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,X3: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ X3 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ X3 )
= ( 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 ) ) )
=> ( X3 = Y ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_4375_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S3: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H2 @ I4 )
= ( zero_zero @ A ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ S3 @ T4 ) )
=> ( ( G @ I4 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ S3 @ T4 ) )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_4376_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,K: A] :
( ( ( ord_less @ A @ X3 @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X3 @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_4377_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,B3: set @ B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_4378_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,B3: set @ B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_4379_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S3: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H2 @ I4 )
= ( one_one @ A ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ S3 @ T4 ) )
=> ( ( G @ I4 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ S3 @ T4 ) )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_4380_card__Diff__subset__Int,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_4381_gcd__int__def,axiom,
( ( gcd_gcd @ int )
= ( ^ [X: int,Y2: int] : ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ X ) ) @ ( nat2 @ ( abs_abs @ int @ Y2 ) ) ) ) ) ) ).
% gcd_int_def
thf(fact_4382_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,P: B > $o,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( H2 @ X ) @ ( G @ X ) )
@ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_4383_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,P: B > $o,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( H2 @ X ) @ ( G @ X ) )
@ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_4384_dvd__partition,axiom,
! [A: $tType,C5: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C5 )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ X5 ) ) )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C5 )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ C5 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X5 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_4385_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: set @ B,F2: B > A,B2: A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ B2 )
= ( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A4: B] : ( divide_divide @ A @ ( F2 @ A4 ) @ B2 )
@ ( inf_inf @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [A4: B] : ( dvd_dvd @ A @ B2 @ ( F2 @ A4 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F2
@ ( inf_inf @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [A4: B] :
~ ( dvd_dvd @ A @ B2 @ ( F2 @ A4 ) ) ) ) )
@ B2 ) ) ) ) ) ).
% sum_div_partition
thf(fact_4386_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,Zs: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys4 != Zs )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys4 ) @ ( set2 @ A @ Zs ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_4387_mask__Suc__exp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ).
% mask_Suc_exp
thf(fact_4388_bezw__aux,axiom,
! [X3: nat,Y: nat] :
( ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ X3 @ Y ) )
= ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ ( bezw @ X3 @ Y ) ) @ ( semiring_1_of_nat @ int @ X3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ X3 @ Y ) ) @ ( semiring_1_of_nat @ int @ Y ) ) ) ) ).
% bezw_aux
thf(fact_4389_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_4390_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_4391_mask__Suc__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ).
% mask_Suc_double
thf(fact_4392_OR__upper,axiom,
! [X3: int,N2: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less @ int @ X3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X3 @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% OR_upper
thf(fact_4393_or__Suc__0__eq,axiom,
! [N2: nat] :
( ( bit_se1065995026697491101ons_or @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% or_Suc_0_eq
thf(fact_4394_Suc__0__or__eq,axiom,
! [N2: nat] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% Suc_0_or_eq
thf(fact_4395_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M2: nat,N: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_4396_gcd__nat_Opelims,axiom,
! [X3: nat,Xa2: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) )
=> ~ ( ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y = X3 ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_4397_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X3 )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_4398_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_4399_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% inf.bounded_iff
thf(fact_4400_gcd__nat_Oidem,axiom,
! [A2: nat] :
( ( gcd_gcd @ nat @ A2 @ A2 )
= A2 ) ).
% gcd_nat.idem
thf(fact_4401_gcd__nat_Oleft__idem,axiom,
! [A2: nat,B2: nat] :
( ( gcd_gcd @ nat @ A2 @ ( gcd_gcd @ nat @ A2 @ B2 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ).
% gcd_nat.left_idem
thf(fact_4402_gcd__nat_Oright__idem,axiom,
! [A2: nat,B2: nat] :
( ( gcd_gcd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ).
% gcd_nat.right_idem
thf(fact_4403_gcd__idem__nat,axiom,
! [X3: nat] :
( ( gcd_gcd @ nat @ X3 @ X3 )
= X3 ) ).
% gcd_idem_nat
thf(fact_4404_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) )
= ( ( ord_less_eq @ A @ X3 @ Y )
& ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% le_inf_iff
thf(fact_4405_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A6 )
@ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_4406_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R )
@ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S3 ) )
= ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ) ) ).
% inf_Int_eq2
thf(fact_4407_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI2
thf(fact_4408_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI1
thf(fact_4409_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A4: A] :
( ( inf_inf @ A @ A4 @ B4 )
= B4 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_4410_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
( ( inf_inf @ A @ A4 @ B4 )
= A4 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_4411_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_4412_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_4413_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
( A4
= ( inf_inf @ A @ A4 @ B4 ) ) ) ) ) ).
% inf.order_iff
thf(fact_4414_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ) ).
% inf_greatest
thf(fact_4415_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ) ).
% inf.boundedI
thf(fact_4416_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% inf.boundedE
thf(fact_4417_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( inf_inf @ A @ X3 @ Y )
= Y ) ) ) ).
% inf_absorb2
thf(fact_4418_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( inf_inf @ A @ X3 @ Y )
= X3 ) ) ) ).
% inf_absorb1
thf(fact_4419_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_4420_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_4421_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y2: A] :
( ( inf_inf @ A @ X @ Y2 )
= X ) ) ) ) ).
% le_iff_inf
thf(fact_4422_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F2: A > A > A,X3: A,Y: A] :
( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ ( F2 @ X5 @ Y4 ) @ X5 )
=> ( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ ( F2 @ X5 @ Y4 ) @ Y4 )
=> ( ! [X5: A,Y4: A,Z3: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ( ord_less_eq @ A @ X5 @ Z3 )
=> ( ord_less_eq @ A @ X5 @ ( F2 @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf @ A @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ) ).
% inf_unique
thf(fact_4423_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_4424_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_4425_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X3: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ X3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% le_infI2
thf(fact_4426_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,X3: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% le_infI1
thf(fact_4427_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ ( inf_inf @ A @ C2 @ D2 ) ) ) ) ) ).
% inf_mono
thf(fact_4428_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X3 @ A2 )
=> ( ( ord_less_eq @ A @ X3 @ B2 )
=> ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_4429_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X3 @ A2 )
=> ~ ( ord_less_eq @ A @ X3 @ B2 ) ) ) ) ).
% le_infE
thf(fact_4430_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ Y ) ) ).
% inf_le2
thf(fact_4431_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ X3 ) ) ).
% inf_le1
thf(fact_4432_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ X3 ) ) ).
% inf_sup_ord(1)
thf(fact_4433_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ Y ) ) ).
% inf_sup_ord(2)
thf(fact_4434_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,X3: A,B2: A] :
( ( ord_less @ A @ A2 @ X3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% less_infI1
thf(fact_4435_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X3: A,A2: A] :
( ( ord_less @ A @ B2 @ X3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% less_infI2
thf(fact_4436_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_4437_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_4438_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% inf.strict_boundedE
thf(fact_4439_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B4: A] :
( ( A4
= ( inf_inf @ A @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ) ).
% inf.strict_order_iff
thf(fact_4440_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ A2 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.strict_coboundedI1
thf(fact_4441_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.strict_coboundedI2
thf(fact_4442_finite__inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A2: A,A3: set @ A] :
( ( inf_inf @ A @ A2 @ ( complete_Sup_Sup @ A @ A3 ) )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [B4: A] :
( ( Uu3
= ( inf_inf @ A @ A2 @ B4 ) )
& ( member @ A @ B4 @ A3 ) ) ) ) ) ) ).
% finite_inf_Sup
thf(fact_4443_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M2: nat,N: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N
@ ( if @ nat
@ ( N
= ( zero_zero @ nat ) )
@ M2
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_4444_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_4445_distinct__product__lists,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ! [X5: list @ A] :
( ( member @ ( list @ A ) @ X5 @ ( set2 @ ( list @ A ) @ Xss ) )
=> ( distinct @ A @ X5 ) )
=> ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_4446_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_4447_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_4448_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% max.bounded_iff
thf(fact_4449_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_4450_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_4451_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less @ A @ ( ord_max @ A @ X3 @ Y ) @ Z2 )
= ( ( ord_less @ A @ X3 @ Z2 )
& ( ord_less @ A @ Y @ Z2 ) ) ) ) ).
% max_less_iff_conj
thf(fact_4452_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_max @ A @ X3 @ ( bot_bot @ A ) )
= X3 ) ) ).
% max_bot2
thf(fact_4453_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X3 )
= X3 ) ) ).
% max_bot
thf(fact_4454_of__bool__or__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P
| Q ) )
= ( ord_max @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_or_iff
thf(fact_4455_max__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_max @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( suc @ ( ord_max @ nat @ M @ N2 ) ) ) ).
% max_Suc_Suc
thf(fact_4456_max__0R,axiom,
! [N2: nat] :
( ( ord_max @ nat @ N2 @ ( zero_zero @ nat ) )
= N2 ) ).
% max_0R
thf(fact_4457_max__0L,axiom,
! [N2: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N2 )
= N2 ) ).
% max_0L
thf(fact_4458_max__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% max_nat.right_neutral
thf(fact_4459_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_4460_max__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% max_nat.left_neutral
thf(fact_4461_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_4462_finite__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] : ( finite_finite2 @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ).
% finite_shuffles
thf(fact_4463_max__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(1)
thf(fact_4464_max__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X3 ) @ ( zero_zero @ A ) )
= ( numeral_numeral @ A @ X3 ) ) ) ).
% max_0_1(4)
thf(fact_4465_max__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X3 ) )
= ( numeral_numeral @ A @ X3 ) ) ) ).
% max_0_1(3)
thf(fact_4466_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_4467_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_4468_max__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(2)
thf(fact_4469_max__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(3)
thf(fact_4470_max__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(4)
thf(fact_4471_max__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( ord_max @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_max @ nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).
% max_Suc_numeral
thf(fact_4472_max__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ord_max @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
= ( suc @ ( ord_max @ nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).
% max_numeral_Suc
thf(fact_4473_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_4474_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ A2 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_4475_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A4: A] :
( ( A4
= ( ord_max @ A @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_4476_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% max.strict_boundedE
thf(fact_4477_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less @ A @ Z2 @ ( ord_max @ A @ X3 @ Y ) )
= ( ( ord_less @ A @ Z2 @ X3 )
| ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% less_max_iff_disj
thf(fact_4478_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,D2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D2 @ B2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D2 ) @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ) ).
% max.mono
thf(fact_4479_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_4480_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_4481_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% max.boundedE
thf(fact_4482_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% max.boundedI
thf(fact_4483_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A4: A] :
( A4
= ( ord_max @ A @ A4 @ B4 ) ) ) ) ) ).
% max.order_iff
thf(fact_4484_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_4485_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_4486_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less_eq @ A @ Z2 @ ( ord_max @ A @ X3 @ Y ) )
= ( ( ord_less_eq @ A @ Z2 @ X3 )
| ( ord_less_eq @ A @ Z2 @ Y ) ) ) ) ).
% le_max_iff_disj
thf(fact_4487_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A4: A] :
( ( ord_max @ A @ A4 @ B4 )
= A4 ) ) ) ) ).
% max.absorb_iff1
thf(fact_4488_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
( ( ord_max @ A @ A4 @ B4 )
= B4 ) ) ) ) ).
% max.absorb_iff2
thf(fact_4489_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI1
thf(fact_4490_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI2
thf(fact_4491_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A4: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B4 ) @ B4 @ A4 ) ) ) ) ).
% max_def_raw
thf(fact_4492_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A4: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B4 ) @ B4 @ A4 ) ) ) ) ).
% max_def
thf(fact_4493_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_max @ A @ X3 @ Y )
= X3 ) ) ) ).
% max_absorb1
thf(fact_4494_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_max @ A @ X3 @ Y )
= Y ) ) ) ).
% max_absorb2
thf(fact_4495_of__nat__max,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X3 @ Y ) )
= ( ord_max @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_max
thf(fact_4496_shuffles__commutes,axiom,
! [A: $tType] :
( ( shuffles @ A )
= ( ^ [Xs2: list @ A,Ys3: list @ A] : ( shuffles @ A @ Ys3 @ Xs2 ) ) ) ).
% shuffles_commutes
thf(fact_4497_of__int__max,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: int,Y: int] :
( ( ring_1_of_int @ A @ ( ord_max @ int @ X3 @ Y ) )
= ( ord_max @ A @ ( ring_1_of_int @ A @ X3 ) @ ( ring_1_of_int @ A @ Y ) ) ) ) ).
% of_int_max
thf(fact_4498_nat__add__max__right,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N2 @ Q4 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( plus_plus @ nat @ M @ Q4 ) ) ) ).
% nat_add_max_right
thf(fact_4499_nat__add__max__left,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N2 ) @ Q4 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q4 ) @ ( plus_plus @ nat @ N2 @ Q4 ) ) ) ).
% nat_add_max_left
thf(fact_4500_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( plus_plus @ A @ X3 @ ( ord_max @ A @ Y @ Z2 ) )
= ( ord_max @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( plus_plus @ A @ X3 @ Z2 ) ) ) ) ).
% max_add_distrib_right
thf(fact_4501_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X3 @ Y ) @ Z2 )
= ( ord_max @ A @ ( plus_plus @ A @ X3 @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% max_add_distrib_left
thf(fact_4502_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X3 @ Y ) @ Z2 )
= ( ord_max @ A @ ( minus_minus @ A @ X3 @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% max_diff_distrib_left
thf(fact_4503_nat__mult__max__left,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M @ N2 ) @ Q4 )
= ( ord_max @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N2 @ Q4 ) ) ) ).
% nat_mult_max_left
thf(fact_4504_nat__mult__max__right,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( times_times @ nat @ M @ ( ord_max @ nat @ N2 @ Q4 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M @ N2 ) @ ( times_times @ nat @ M @ Q4 ) ) ) ).
% nat_mult_max_right
thf(fact_4505_nat__minus__add__max,axiom,
! [N2: nat,M: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ N2 @ M ) @ M )
= ( ord_max @ nat @ N2 @ M ) ) ).
% nat_minus_add_max
thf(fact_4506_length__shuffles,axiom,
! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( size_size @ ( list @ A ) @ Zs2 )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ) ).
% length_shuffles
thf(fact_4507_max__Suc2,axiom,
! [M: nat,N2: nat] :
( ( ord_max @ nat @ M @ ( suc @ N2 ) )
= ( case_nat @ nat @ ( suc @ N2 )
@ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ M4 @ N2 ) )
@ M ) ) ).
% max_Suc2
thf(fact_4508_max__Suc1,axiom,
! [N2: nat,M: nat] :
( ( ord_max @ nat @ ( suc @ N2 ) @ M )
= ( case_nat @ nat @ ( suc @ N2 )
@ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ N2 @ M4 ) )
@ M ) ) ).
% max_Suc1
thf(fact_4509_Sup__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S3: set @ A,X3: A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( ( S3
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ X3 @ S3 ) )
= X3 ) )
& ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ X3 @ S3 ) )
= ( ord_max @ A @ X3 @ ( complete_Sup_Sup @ A @ S3 ) ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_4510_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: 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 ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) )
=> ( distinct @ A @ Zs2 ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_4511_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_4512_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F3: A > nat,R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X: A,Y2: A] :
( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y2 ) )
| ( ( ord_less_eq @ nat @ ( F3 @ X ) @ ( F3 @ Y2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R6 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_4513_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
@ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( product_apsnd @ code_integer @ code_integer @ code_integer ) @ ( times_times @ code_integer ) ) @ ( sgn_sgn @ code_integer ) @ L2
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( sgn_sgn @ code_integer @ K3 )
= ( sgn_sgn @ code_integer @ L2 ) )
@ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S8: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S8
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L2 ) @ S8 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_4514_totally__bounded__Union,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [M7: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ M7 )
=> ( ! [S5: set @ A] :
( ( member @ ( set @ A ) @ S5 @ M7 )
=> ( topolo6688025880775521714ounded @ A @ S5 ) )
=> ( topolo6688025880775521714ounded @ A @ ( complete_Sup_Sup @ ( set @ A ) @ M7 ) ) ) ) ) ).
% totally_bounded_Union
thf(fact_4515_nth__rotate1,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate1 @ A @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( suc @ N2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_4516_set__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rotate1 @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rotate1
thf(fact_4517_length__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate1
thf(fact_4518_distinct1__rotate,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rotate1 @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct1_rotate
thf(fact_4519_totally__bounded__empty,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( topolo6688025880775521714ounded @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% totally_bounded_empty
thf(fact_4520_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_4521_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F2: C > B,G: D > C,X3: product_prod @ A @ D] :
( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apsnd @ D @ C @ A @ G @ X3 ) )
= ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F2 @ G ) @ X3 ) ) ).
% apsnd_compose
thf(fact_4522_totally__bounded__subset,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [S3: set @ A,T4: set @ A] :
( ( topolo6688025880775521714ounded @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 )
=> ( topolo6688025880775521714ounded @ A @ T4 ) ) ) ) ).
% totally_bounded_subset
thf(fact_4523_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H2: B > A,G: C > B,A3: set @ C] :
( ( ( H2 @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X5: B,Y4: B] :
( ( H2 @ ( plus_plus @ B @ X5 @ Y4 ) )
= ( plus_plus @ A @ ( H2 @ X5 ) @ ( H2 @ Y4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H2 @ G ) @ A3 )
= ( H2 @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A3 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_4524_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F2: D > C > A,G: B > D,X3: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F2 @ G ) @ X3 )
= ( F2 @ ( G @ ( product_fst @ B @ C @ X3 ) ) @ ( product_snd @ B @ C @ X3 ) ) ) ).
% case_prod_comp
thf(fact_4525_mlex__leq,axiom,
! [A: $tType,F2: A > nat,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( mlex_prod @ A @ F2 @ R ) ) ) ) ).
% mlex_leq
thf(fact_4526_mlex__less,axiom,
! [A: $tType,F2: A > nat,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( mlex_prod @ A @ F2 @ R ) ) ) ).
% mlex_less
thf(fact_4527_mlex__iff,axiom,
! [A: $tType,X3: A,Y: A,F2: A > nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( mlex_prod @ A @ F2 @ R ) )
= ( ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
| ( ( ( F2 @ X3 )
= ( F2 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R ) ) ) ) ).
% mlex_iff
thf(fact_4528_set__remove1__eq,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( ( set2 @ A @ ( remove1 @ A @ X3 @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_remove1_eq
thf(fact_4529_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_4530_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F2: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 ) )
= ( product_fst @ A @ B ) ) ).
% fst_comp_apsnd
thf(fact_4531_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F2: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ F2 @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_apsnd
thf(fact_4532_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 )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% fst_diag_fst
thf(fact_4533_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 )
@ ^ [X: B] : ( product_Pair @ B @ B @ X @ X )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% snd_diag_snd
thf(fact_4534_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_4535_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 )
@ ^ [X: B] : ( product_Pair @ B @ B @ X @ X )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% fst_diag_snd
thf(fact_4536_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 )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% snd_diag_fst
thf(fact_4537_distinct__remove1,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( remove1 @ A @ X3 @ Xs ) ) ) ).
% distinct_remove1
thf(fact_4538_notin__set__remove1,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ~ ( member @ A @ X3 @ ( set2 @ A @ ( remove1 @ A @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_4539_remove1__idem,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X3 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_4540_remove1__commute,axiom,
! [A: $tType,X3: A,Y: A,Zs2: list @ A] :
( ( remove1 @ A @ X3 @ ( remove1 @ A @ Y @ Zs2 ) )
= ( remove1 @ A @ Y @ ( remove1 @ A @ X3 @ Zs2 ) ) ) ).
% remove1_commute
thf(fact_4541_set__remove1__subset,axiom,
! [A: $tType,X3: A,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( remove1 @ A @ X3 @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_remove1_subset
thf(fact_4542_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% sum.atLeast_Suc_atMost_Suc_shift
thf(fact_4543_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% sum.atLeast_Suc_lessThan_Suc_shift
thf(fact_4544_sum_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% sum.atLeastAtMost_shift_bounds
thf(fact_4545_sum_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% sum.atLeastLessThan_shift_bounds
thf(fact_4546_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% prod.atLeast_Suc_atMost_Suc_shift
thf(fact_4547_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% prod.atLeast_Suc_lessThan_Suc_shift
thf(fact_4548_prod_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% prod.atLeastAtMost_shift_bounds
thf(fact_4549_prod_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% prod.atLeastLessThan_shift_bounds
thf(fact_4550_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B3: set @ ( set @ B ),G: B > A] :
( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ B3 )
=> ( finite_finite2 @ B @ X5 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B3 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B3 )
=> ( ( A14 != A25 )
=> ! [X5: B] :
( ( member @ B @ X5 @ A14 )
=> ( ( member @ B @ X5 @ A25 )
=> ( ( G @ X5 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B3 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B3 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_4551_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ ( set @ B ),G: B > A] :
( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ C5 )
=> ( finite_finite2 @ B @ X5 ) )
=> ( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ C5 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C5 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X5 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( 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 ) @ G @ C5 ) ) ) ) ) ).
% prod.Union_disjoint
thf(fact_4552_sum_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc_shift
thf(fact_4553_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% sum.atLeast0_lessThan_Suc_shift
thf(fact_4554_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_4555_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_4556_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ).
% sum.atLeastLessThan_shift_0
thf(fact_4557_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ).
% prod.atLeastLessThan_shift_0
thf(fact_4558_length__remove1,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X3 @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X3 @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_remove1
thf(fact_4559_sum_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: set @ ( set @ B ),G: B > A] :
( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ B3 )
=> ( finite_finite2 @ B @ X5 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B3 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B3 )
=> ( ( A14 != A25 )
=> ! [X5: B] :
( ( member @ B @ X5 @ A14 )
=> ( ( member @ B @ X5 @ A25 )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B3 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ B3 ) ) ) ) ) ).
% sum.Union_comp
thf(fact_4560_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_4561_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_4562_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_4563_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_4564_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_4565_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_4566_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ ( set @ B ),G: B > A] :
( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ C5 )
=> ( finite_finite2 @ B @ X5 ) )
=> ( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ C5 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C5 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X5 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( 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 ) @ G @ C5 ) ) ) ) ) ).
% sum.Union_disjoint
thf(fact_4567_in__finite__psubset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A3 @ B3 ) @ ( finite_psubset @ A ) )
= ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
& ( finite_finite2 @ A @ B3 ) ) ) ).
% in_finite_psubset
thf(fact_4568_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 )
@ ^ [X: B,Y2: A] : ( product_Pair @ A @ B @ Y2 @ X ) ) ) ) ).
% snd_fst_flip
thf(fact_4569_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 )
@ ^ [X: A,Y2: B] : ( product_Pair @ B @ A @ Y2 @ X ) ) ) ) ).
% fst_snd_flip
thf(fact_4570_in__measure,axiom,
! [A: $tType,X3: A,Y: A,F2: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measure @ A @ F2 ) )
= ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ).
% in_measure
thf(fact_4571_fstI,axiom,
! [B: $tType,A: $tType,X3: product_prod @ A @ B,Y: A,Z2: B] :
( ( X3
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_fst @ A @ B @ X3 )
= Y ) ) ).
% fstI
thf(fact_4572_sndI,axiom,
! [A: $tType,B: $tType,X3: product_prod @ A @ B,Y: A,Z2: B] :
( ( X3
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_snd @ A @ B @ X3 )
= Z2 ) ) ).
% sndI
thf(fact_4573_finite__psubset__def,axiom,
! [A: $tType] :
( ( finite_psubset @ A )
= ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A6: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B6 )
& ( finite_finite2 @ A @ B6 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_4574_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N: nat,A4: A] :
( if @ A
@ ( N
= ( zero_zero @ nat ) )
@ A4
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_4575_root__powr__inverse,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( root @ N2 @ X3 )
= ( powr @ real @ X3 @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_4576_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A6: set @ C,F3: C > A,G2: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A4: C] :
( ( Uu3
= ( product_Pair @ A @ B @ ( F3 @ A4 ) @ ( G2 @ A4 ) ) )
& ( member @ C @ A4 @ A6 ) ) ) ) ) ).
% image2_def
thf(fact_4577_drop__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less @ int @ ( bit_se4197421643247451524op_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% drop_bit_negative_int_iff
thf(fact_4578_drop__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% drop_bit_of_0
thf(fact_4579_drop__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N2: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ A2 ) ) ) ).
% drop_bit_drop_bit
thf(fact_4580_real__root__Suc__0,axiom,
! [X3: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X3 )
= X3 ) ).
% real_root_Suc_0
thf(fact_4581_root__0,axiom,
! [X3: real] :
( ( root @ ( zero_zero @ nat ) @ X3 )
= ( zero_zero @ real ) ) ).
% root_0
thf(fact_4582_real__root__eq__iff,axiom,
! [N2: nat,X3: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( root @ N2 @ X3 )
= ( root @ N2 @ Y ) )
= ( X3 = Y ) ) ) ).
% real_root_eq_iff
thf(fact_4583_drop__bit__of__bool,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,B2: $o] :
( ( bit_se4197421643247451524op_bit @ A @ N2 @ ( zero_neq_one_of_bool @ A @ B2 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( N2
= ( zero_zero @ nat ) )
& B2 ) ) ) ) ).
% drop_bit_of_bool
thf(fact_4584_drop__bit__of__Suc__0,axiom,
! [N2: nat] :
( ( bit_se4197421643247451524op_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_neq_one_of_bool @ nat
@ ( N2
= ( zero_zero @ nat ) ) ) ) ).
% drop_bit_of_Suc_0
thf(fact_4585_drop__bit__Suc__minus__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_4586_drop__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_Suc_bit0
thf(fact_4587_drop__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_Suc_bit1
thf(fact_4588_real__root__eq__0__iff,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( root @ N2 @ X3 )
= ( zero_zero @ real ) )
= ( X3
= ( zero_zero @ real ) ) ) ) ).
% real_root_eq_0_iff
thf(fact_4589_real__root__less__iff,axiom,
! [N2: nat,X3: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y ) )
= ( ord_less @ real @ X3 @ Y ) ) ) ).
% real_root_less_iff
thf(fact_4590_real__root__le__iff,axiom,
! [N2: nat,X3: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y ) )
= ( ord_less_eq @ real @ X3 @ Y ) ) ) ).
% real_root_le_iff
thf(fact_4591_real__root__eq__1__iff,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( root @ N2 @ X3 )
= ( one_one @ real ) )
= ( X3
= ( one_one @ real ) ) ) ) ).
% real_root_eq_1_iff
thf(fact_4592_real__root__one,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( root @ N2 @ ( one_one @ real ) )
= ( one_one @ real ) ) ) ).
% real_root_one
thf(fact_4593_drop__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N2 @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A
@ ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% drop_bit_of_1
thf(fact_4594_real__root__lt__0__iff,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( root @ N2 @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ) ).
% real_root_lt_0_iff
thf(fact_4595_real__root__gt__0__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N2 @ Y ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_4596_real__root__le__0__iff,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( root @ N2 @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) ) ) ).
% real_root_le_0_iff
thf(fact_4597_real__root__ge__0__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N2 @ Y ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_4598_real__root__lt__1__iff,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( root @ N2 @ X3 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ) ).
% real_root_lt_1_iff
thf(fact_4599_real__root__gt__1__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N2 @ Y ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_4600_real__root__le__1__iff,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( root @ N2 @ X3 ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X3 @ ( one_one @ real ) ) ) ) ).
% real_root_le_1_iff
thf(fact_4601_real__root__ge__1__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N2 @ Y ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_4602_drop__bit__Suc__minus__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_4603_real__root__pow__pos2,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( power_power @ real @ ( root @ N2 @ X3 ) @ N2 )
= X3 ) ) ) ).
% real_root_pow_pos2
thf(fact_4604_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 )
= A2 )
= ( ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_4605_take__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N2: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) )
= ( bit_se4197421643247451524op_bit @ A @ N2 @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ A2 ) ) ) ) ).
% take_bit_drop_bit
thf(fact_4606_bit__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A2: A] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) )
= ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( plus_plus @ nat @ N2 ) ) ) ) ).
% bit_drop_bit_eq
thf(fact_4607_real__root__less__mono,axiom,
! [N2: nat,X3: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ X3 @ Y )
=> ( ord_less @ real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_4608_real__root__le__mono,axiom,
! [N2: nat,X3: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ X3 @ Y )
=> ( ord_less_eq @ real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_4609_real__root__power,axiom,
! [N2: nat,X3: real,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( root @ N2 @ ( power_power @ real @ X3 @ K ) )
= ( power_power @ real @ ( root @ N2 @ X3 ) @ K ) ) ) ).
% real_root_power
thf(fact_4610_real__root__abs,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( root @ N2 @ ( abs_abs @ real @ X3 ) )
= ( abs_abs @ real @ ( root @ N2 @ X3 ) ) ) ) ).
% real_root_abs
thf(fact_4611_sgn__root,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( sgn_sgn @ real @ ( root @ N2 @ X3 ) )
= ( sgn_sgn @ real @ X3 ) ) ) ).
% sgn_root
thf(fact_4612_bits__ident,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A2: A] :
( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) )
= A2 ) ) ).
% bits_ident
thf(fact_4613_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F2: B > A,X3: B,C2: C,G: B > C,A3: set @ B] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( C2
= ( G @ X3 ) )
=> ( ( member @ B @ X3 @ A3 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A3 @ F2 @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_4614_real__root__gt__zero,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N2 @ X3 ) ) ) ) ).
% real_root_gt_zero
thf(fact_4615_real__root__strict__decreasing,axiom,
! [N2: nat,N7: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ N2 @ N7 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ord_less @ real @ ( root @ N7 @ X3 ) @ ( root @ N2 @ X3 ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_4616_root__abs__power,axiom,
! [N2: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( abs_abs @ real @ ( root @ N2 @ ( power_power @ real @ Y @ N2 ) ) )
= ( abs_abs @ real @ Y ) ) ) ).
% root_abs_power
thf(fact_4617_real__root__pos__pos,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N2 @ X3 ) ) ) ) ).
% real_root_pos_pos
thf(fact_4618_real__root__strict__increasing,axiom,
! [N2: nat,N7: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ N2 @ N7 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( root @ N2 @ X3 ) @ ( root @ N7 @ X3 ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_4619_real__root__decreasing,axiom,
! [N2: nat,N7: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ N7 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( root @ N7 @ X3 ) @ ( root @ N2 @ X3 ) ) ) ) ) ).
% real_root_decreasing
thf(fact_4620_real__root__pow__pos,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( power_power @ real @ ( root @ N2 @ X3 ) @ N2 )
= X3 ) ) ) ).
% real_root_pow_pos
thf(fact_4621_real__root__pos__unique,axiom,
! [N2: nat,Y: real,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( power_power @ real @ Y @ N2 )
= X3 )
=> ( ( root @ N2 @ X3 )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_4622_real__root__power__cancel,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( root @ N2 @ ( power_power @ real @ X3 @ N2 ) )
= X3 ) ) ) ).
% real_root_power_cancel
thf(fact_4623_drop__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ A2 )
= ( bit_se4197421643247451524op_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% drop_bit_Suc
thf(fact_4624_real__root__increasing,axiom,
! [N2: nat,N7: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ N7 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( root @ N2 @ X3 ) @ ( root @ N7 @ X3 ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_4625_root__sgn__power,axiom,
! [N2: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( root @ N2 @ ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N2 ) ) )
= Y ) ) ).
% root_sgn_power
thf(fact_4626_sgn__power__root,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N2 @ X3 ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N2 @ X3 ) ) @ N2 ) )
= X3 ) ) ).
% sgn_power_root
thf(fact_4627_ln__root,axiom,
! [N2: nat,B2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ln_ln @ real @ ( root @ N2 @ B2 ) )
= ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).
% ln_root
thf(fact_4628_log__root,axiom,
! [N2: nat,A2: real,B2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( log @ B2 @ ( root @ N2 @ A2 ) )
= ( divide_divide @ real @ ( log @ B2 @ A2 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).
% log_root
thf(fact_4629_log__base__root,axiom,
! [N2: nat,B2: real,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( log @ ( root @ N2 @ B2 ) @ X3 )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ X3 ) ) ) ) ) ).
% log_base_root
thf(fact_4630_split__root,axiom,
! [P: real > $o,N2: nat,X3: real] :
( ( P @ ( root @ N2 @ X3 ) )
= ( ( ( N2
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ real ) ) )
& ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ! [Y2: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N2 ) )
= X3 )
=> ( P @ Y2 ) ) ) ) ) ).
% split_root
thf(fact_4631_in__lex__prod,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: 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 @ A7 @ B7 ) ) @ ( lex_prod @ A @ B @ R2 @ S ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A7 ) @ R2 )
| ( ( A2 = A7 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B7 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_4632_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X3: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X3 @ 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_4633_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X3: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X3 @ 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_4634_scale__right__distrib__NO__MATCH,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,Y: A,A2: real] :
( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X3 @ Y ) @ A2 )
=> ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ) ).
% scale_right_distrib_NO_MATCH
thf(fact_4635_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X3: B,Y: B,C2: A,A2: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ C2 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_4636_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X3: B,Y: B,A2: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ A2 )
=> ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_4637_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X3: B,Y: B,C2: A,A2: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ C2 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_4638_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X3: B,Y: B,A2: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ A2 )
=> ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_4639_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 )
@ ^ [A4: A,B4: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A9: A,B13: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A9 ) @ Ra )
| ( ( A4 = A9 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B4 @ B13 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_4640_scale__left__distrib__NO__MATCH,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,Y: A,C2: C,A2: real,B2: real] :
( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X3 @ Y ) @ C2 )
=> ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X3 )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) ) ) ) ).
% scale_left_distrib_NO_MATCH
thf(fact_4641_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F3: B > A,A4: A,Xs2: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( compow @ ( A > A ) @ N @ ( times_times @ A @ A4 ) @ ( F3 @ ( nth @ B @ Xs2 @ N ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_4642_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] :
? [I3: nat] :
( ( Uu3
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I3 ) @ ( nth @ B @ Ys @ I3 ) ) )
& ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% set_zip
thf(fact_4643_Fpow__Pow__finite,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A6: set @ A] : ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A6 ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_4644_Suc__funpow,axiom,
! [N2: nat] :
( ( compow @ ( nat > nat ) @ N2 @ suc )
= ( plus_plus @ nat @ N2 ) ) ).
% Suc_funpow
thf(fact_4645_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_4646_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_4647_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% min.bounded_iff
thf(fact_4648_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_4649_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_4650_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less @ A @ Z2 @ ( ord_min @ A @ X3 @ Y ) )
= ( ( ord_less @ A @ Z2 @ X3 )
& ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% min_less_iff_conj
thf(fact_4651_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_min @ A @ X3 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_4652_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X3 )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_4653_funpow__0,axiom,
! [A: $tType,F2: A > A,X3: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F2 @ X3 )
= X3 ) ).
% funpow_0
thf(fact_4654_max__min__same_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_max @ A @ X3 @ ( ord_min @ A @ X3 @ Y ) )
= X3 ) ) ).
% max_min_same(1)
thf(fact_4655_max__min__same_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X3 @ Y ) @ X3 )
= X3 ) ) ).
% max_min_same(2)
thf(fact_4656_max__min__same_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X3 @ Y ) @ Y )
= Y ) ) ).
% max_min_same(3)
thf(fact_4657_max__min__same_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X3: A] :
( ( ord_max @ A @ Y @ ( ord_min @ A @ X3 @ Y ) )
= Y ) ) ).
% max_min_same(4)
thf(fact_4658_min__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_min @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( suc @ ( ord_min @ nat @ M @ N2 ) ) ) ).
% min_Suc_Suc
thf(fact_4659_min__0R,axiom,
! [N2: nat] :
( ( ord_min @ nat @ N2 @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_4660_min__0L,axiom,
! [N2: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_4661_min__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% min_number_of(1)
thf(fact_4662_min__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X3 ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(4)
thf(fact_4663_min__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X3 ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(3)
thf(fact_4664_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_4665_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_4666_min__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ).
% min_number_of(2)
thf(fact_4667_min__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% min_number_of(3)
thf(fact_4668_min__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ).
% min_number_of(4)
thf(fact_4669_min__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( ord_min @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_min @ nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).
% min_Suc_numeral
thf(fact_4670_min__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ord_min @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
= ( suc @ ( ord_min @ nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).
% min_numeral_Suc
thf(fact_4671_zip__replicate,axiom,
! [A: $tType,B: $tType,I: nat,X3: A,J: nat,Y: B] :
( ( zip @ A @ B @ ( replicate @ A @ I @ X3 ) @ ( replicate @ B @ J @ Y ) )
= ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I @ J ) @ ( product_Pair @ A @ B @ X3 @ Y ) ) ) ).
% zip_replicate
thf(fact_4672_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_4673_comp__funpow,axiom,
! [B: $tType,A: $tType,N2: nat,F2: A > A] :
( ( compow @ ( ( B > A ) > B > A ) @ N2 @ ( comp @ A @ A @ B @ F2 ) )
= ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N2 @ F2 ) ) ) ).
% comp_funpow
thf(fact_4674_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.strict_coboundedI2
thf(fact_4675_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ A2 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.strict_coboundedI1
thf(fact_4676_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B4: A] :
( ( A4
= ( ord_min @ A @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_4677_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% min.strict_boundedE
thf(fact_4678_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 )
= ( ( ord_less @ A @ X3 @ Z2 )
| ( ord_less @ A @ Y @ Z2 ) ) ) ) ).
% min_less_iff_disj
thf(fact_4679_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ ( ord_min @ A @ C2 @ D2 ) ) ) ) ) ).
% min.mono
thf(fact_4680_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_4681_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_4682_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% min.boundedE
thf(fact_4683_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ) ).
% min.boundedI
thf(fact_4684_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
( A4
= ( ord_min @ A @ A4 @ B4 ) ) ) ) ) ).
% min.order_iff
thf(fact_4685_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_4686_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_4687_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
( ( ord_min @ A @ A4 @ B4 )
= A4 ) ) ) ) ).
% min.absorb_iff1
thf(fact_4688_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A4: A] :
( ( ord_min @ A @ A4 @ B4 )
= B4 ) ) ) ) ).
% min.absorb_iff2
thf(fact_4689_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI1
thf(fact_4690_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI2
thf(fact_4691_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X3 @ Z2 )
| ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% min_le_iff_disj
thf(fact_4692_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A4: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B4 ) @ A4 @ B4 ) ) ) ) ).
% min_def_raw
thf(fact_4693_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A4: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B4 ) @ A4 @ B4 ) ) ) ) ).
% min_def
thf(fact_4694_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_min @ A @ X3 @ Y )
= X3 ) ) ) ).
% min_absorb1
thf(fact_4695_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_min @ A @ X3 @ Y )
= Y ) ) ) ).
% min_absorb2
thf(fact_4696_of__nat__min,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X3 @ Y ) )
= ( ord_min @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_min
thf(fact_4697_funpow__swap1,axiom,
! [A: $tType,F2: A > A,N2: nat,X3: A] :
( ( F2 @ ( compow @ ( A > A ) @ N2 @ F2 @ X3 ) )
= ( compow @ ( A > A ) @ N2 @ F2 @ ( F2 @ X3 ) ) ) ).
% funpow_swap1
thf(fact_4698_of__int__min,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: int,Y: int] :
( ( ring_1_of_int @ A @ ( ord_min @ int @ X3 @ Y ) )
= ( ord_min @ A @ ( ring_1_of_int @ A @ X3 ) @ ( ring_1_of_int @ A @ Y ) ) ) ) ).
% of_int_min
thf(fact_4699_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( plus_plus @ A @ X3 @ ( ord_min @ A @ Y @ Z2 ) )
= ( ord_min @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( plus_plus @ A @ X3 @ Z2 ) ) ) ) ).
% min_add_distrib_right
thf(fact_4700_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 )
= ( ord_min @ A @ ( plus_plus @ A @ X3 @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% min_add_distrib_left
thf(fact_4701_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 )
= ( ord_min @ A @ ( minus_minus @ A @ X3 @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% min_diff_distrib_left
thf(fact_4702_min__diff,axiom,
! [M: nat,I: nat,N2: nat] :
( ( ord_min @ nat @ ( minus_minus @ nat @ M @ I ) @ ( minus_minus @ nat @ N2 @ I ) )
= ( minus_minus @ nat @ ( ord_min @ nat @ M @ N2 ) @ I ) ) ).
% min_diff
thf(fact_4703_funpow__mult,axiom,
! [A: $tType,N2: nat,M: nat,F2: A > A] :
( ( compow @ ( A > A ) @ N2 @ ( compow @ ( A > A ) @ M @ F2 ) )
= ( compow @ ( A > A ) @ ( times_times @ nat @ M @ N2 ) @ F2 ) ) ).
% funpow_mult
thf(fact_4704_nat__mult__min__left,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M @ N2 ) @ Q4 )
= ( ord_min @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N2 @ Q4 ) ) ) ).
% nat_mult_min_left
thf(fact_4705_nat__mult__min__right,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( times_times @ nat @ M @ ( ord_min @ nat @ N2 @ Q4 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M @ N2 ) @ ( times_times @ nat @ M @ Q4 ) ) ) ).
% nat_mult_min_right
thf(fact_4706_inf__nat__def,axiom,
( ( inf_inf @ nat )
= ( ord_min @ nat ) ) ).
% inf_nat_def
thf(fact_4707_funpow__Suc__right,axiom,
! [A: $tType,N2: nat,F2: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N2 ) @ F2 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ F2 ) ) ).
% funpow_Suc_right
thf(fact_4708_funpow_Osimps_I2_J,axiom,
! [A: $tType,N2: nat,F2: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N2 ) @ F2 )
= ( comp @ A @ A @ A @ F2 @ ( compow @ ( A > A ) @ N2 @ F2 ) ) ) ).
% funpow.simps(2)
thf(fact_4709_funpow__add,axiom,
! [A: $tType,M: nat,N2: nat,F2: A > A] :
( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M @ N2 ) @ F2 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M @ F2 ) @ ( compow @ ( A > A ) @ N2 @ F2 ) ) ) ).
% funpow_add
thf(fact_4710_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X3: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X3 @ Y ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_min_eq_max
thf(fact_4711_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X3: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X3 @ Y ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_max_eq_min
thf(fact_4712_empty__in__Fpow,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) ) ).
% empty_in_Fpow
thf(fact_4713_Fpow__not__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_Fpow @ A @ A3 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Fpow_not_empty
thf(fact_4714_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X3 @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X3 @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_4715_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X3 @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X3 @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_4716_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X3 @ Y ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X3 @ Y ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_4717_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X3 @ Y ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X3 @ Y ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_4718_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N: nat] : ( compow @ ( A > A ) @ N @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_4719_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,A2: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A2 )
= ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A2 ) ) ) ).
% numeral_add_unfold_funpow
thf(fact_4720_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P4: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X3 @ Y ) @ P4 )
= ( ord_min @ A @ ( divide_divide @ A @ X3 @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X3 @ Y ) @ P4 )
= ( ord_max @ A @ ( divide_divide @ A @ X3 @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_4721_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P4: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X3 @ Y ) @ P4 )
= ( ord_max @ A @ ( divide_divide @ A @ X3 @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X3 @ Y ) @ P4 )
= ( ord_min @ A @ ( divide_divide @ A @ X3 @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_4722_min__Suc2,axiom,
! [M: nat,N2: nat] :
( ( ord_min @ nat @ M @ ( suc @ N2 ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ M4 @ N2 ) )
@ M ) ) ).
% min_Suc2
thf(fact_4723_min__Suc1,axiom,
! [N2: nat,M: nat] :
( ( ord_min @ nat @ ( suc @ N2 ) @ M )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ N2 @ M4 ) )
@ M ) ) ).
% min_Suc1
thf(fact_4724_Inf__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S3: set @ A,X3: A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( ( S3
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ X3 @ S3 ) )
= X3 ) )
& ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ X3 @ S3 ) )
= ( ord_min @ A @ X3 @ ( complete_Inf_Inf @ A @ S3 ) ) ) ) ) ) ) ).
% Inf_insert_finite
thf(fact_4725_Fpow__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) @ ( finite_Fpow @ A @ B3 ) ) ) ).
% Fpow_mono
thf(fact_4726_Fpow__def,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A6: set @ A] :
( collect @ ( set @ A )
@ ^ [X6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ A6 )
& ( finite_finite2 @ A @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_4727_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_4728_relpowp__bot,axiom,
! [A: $tType,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( compow @ ( A > A > $o ) @ N2 @ ( bot_bot @ ( A > A > $o ) ) )
= ( bot_bot @ ( A > A > $o ) ) ) ) ).
% relpowp_bot
thf(fact_4729_relpowp__fun__conv,axiom,
! [A: $tType] :
( ( compow @ ( A > A > $o ) )
= ( ^ [N: nat,P3: A > A > $o,X: A,Y2: A] :
? [F3: nat > A] :
( ( ( F3 @ ( zero_zero @ nat ) )
= X )
& ( ( F3 @ N )
= Y2 )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( P3 @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) ) ) ) ) ).
% relpowp_fun_conv
thf(fact_4730_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P3: A > $o,R6: A > ( set @ ( product_prod @ B @ B ) )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [X9: A,Y6: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X: A,Y2: B] :
( ( X9 = X )
& ( P3 @ X )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y2 ) @ ( R6 @ X ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_4731_same__fstI,axiom,
! [B: $tType,A: $tType,P: A > $o,X3: A,Y8: B,Y: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P @ X3 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y ) @ ( R @ X3 ) )
=> ( 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 @ X3 @ Y8 ) @ ( product_Pair @ A @ B @ X3 @ Y ) ) @ ( same_fst @ A @ B @ P @ R ) ) ) ) ).
% same_fstI
thf(fact_4732_inf__int__def,axiom,
( ( inf_inf @ int )
= ( ord_min @ int ) ) ).
% inf_int_def
thf(fact_4733_relpowp__Suc__E,axiom,
! [A: $tType,N2: nat,P: A > A > $o,X3: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X3 @ Z2 )
=> ~ ! [Y4: A] :
( ( compow @ ( A > A > $o ) @ N2 @ P @ X3 @ Y4 )
=> ~ ( P @ Y4 @ Z2 ) ) ) ).
% relpowp_Suc_E
thf(fact_4734_relpowp__Suc__I,axiom,
! [A: $tType,N2: nat,P: A > A > $o,X3: A,Y: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ N2 @ P @ X3 @ Y )
=> ( ( P @ Y @ Z2 )
=> ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X3 @ Z2 ) ) ) ).
% relpowp_Suc_I
thf(fact_4735_relpowp__Suc__D2,axiom,
! [A: $tType,N2: nat,P: A > A > $o,X3: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X3 @ Z2 )
=> ? [Y4: A] :
( ( P @ X3 @ Y4 )
& ( compow @ ( A > A > $o ) @ N2 @ P @ Y4 @ Z2 ) ) ) ).
% relpowp_Suc_D2
thf(fact_4736_relpowp__Suc__E2,axiom,
! [A: $tType,N2: nat,P: A > A > $o,X3: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X3 @ Z2 )
=> ~ ! [Y4: A] :
( ( P @ X3 @ Y4 )
=> ~ ( compow @ ( A > A > $o ) @ N2 @ P @ Y4 @ Z2 ) ) ) ).
% relpowp_Suc_E2
thf(fact_4737_relpowp__Suc__I2,axiom,
! [A: $tType,P: A > A > $o,X3: A,Y: A,N2: nat,Z2: A] :
( ( P @ X3 @ Y )
=> ( ( compow @ ( A > A > $o ) @ N2 @ P @ Y @ Z2 )
=> ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X3 @ Z2 ) ) ) ).
% relpowp_Suc_I2
thf(fact_4738_relpowp__0__I,axiom,
! [A: $tType,P: A > A > $o,X3: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X3 @ X3 ) ).
% relpowp_0_I
thf(fact_4739_relpowp__0__E,axiom,
! [A: $tType,P: A > A > $o,X3: A,Y: A] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X3 @ Y )
=> ( X3 = Y ) ) ).
% relpowp_0_E
thf(fact_4740_relpowp_Osimps_I1_J,axiom,
! [A: $tType,R: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R )
= ( ^ [Y3: A,Z: A] : Y3 = Z ) ) ).
% relpowp.simps(1)
thf(fact_4741_relpowp__E2,axiom,
! [A: $tType,N2: nat,P: A > A > $o,X3: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ N2 @ P @ X3 @ Z2 )
=> ( ( ( N2
= ( zero_zero @ nat ) )
=> ( X3 != Z2 ) )
=> ~ ! [Y4: A,M3: nat] :
( ( N2
= ( suc @ M3 ) )
=> ( ( P @ X3 @ Y4 )
=> ~ ( compow @ ( A > A > $o ) @ M3 @ P @ Y4 @ Z2 ) ) ) ) ) ).
% relpowp_E2
thf(fact_4742_relpowp__E,axiom,
! [A: $tType,N2: nat,P: A > A > $o,X3: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ N2 @ P @ X3 @ Z2 )
=> ( ( ( N2
= ( zero_zero @ nat ) )
=> ( X3 != Z2 ) )
=> ~ ! [Y4: A,M3: nat] :
( ( N2
= ( suc @ M3 ) )
=> ( ( compow @ ( A > A > $o ) @ M3 @ P @ X3 @ Y4 )
=> ~ ( P @ Y4 @ Z2 ) ) ) ) ) ).
% relpowp_E
thf(fact_4743_Nat_Ofunpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% Nat.funpow_code_def
thf(fact_4744_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X: nat,Y2: nat] : ( ord_less_eq @ nat @ Y2 @ X )
@ ^ [X: nat,Y2: nat] : ( ord_less @ nat @ Y2 @ X ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_4745_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F2: C > B,G: D > A,X3: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apfst @ D @ A @ C @ G @ X3 ) )
= ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X3 ) ) @ ( F2 @ ( product_snd @ D @ C @ X3 ) ) ) ) ).
% apsnd_apfst
thf(fact_4746_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F2: C > A,X3: C,Y: B] :
( ( product_apfst @ C @ A @ B @ F2 @ ( product_Pair @ C @ B @ X3 @ Y ) )
= ( product_Pair @ A @ B @ ( F2 @ X3 ) @ Y ) ) ).
% apfst_conv
thf(fact_4747_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > A,X3: product_prod @ C @ B,G: C > A] :
( ( ( product_apfst @ C @ A @ B @ F2 @ X3 )
= ( product_apfst @ C @ A @ B @ G @ X3 ) )
= ( ( F2 @ ( product_fst @ C @ B @ X3 ) )
= ( G @ ( product_fst @ C @ B @ X3 ) ) ) ) ).
% apfst_eq_conv
thf(fact_4748_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > A,X3: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F2 @ X3 ) )
= ( F2 @ ( product_fst @ C @ B @ X3 ) ) ) ).
% fst_apfst
thf(fact_4749_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F2: C > B,X3: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F2 @ X3 ) )
= ( product_snd @ C @ A @ X3 ) ) ).
% snd_apfst
thf(fact_4750_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 ) )
= ( product_snd @ A @ B ) ) ).
% snd_comp_apfst
thf(fact_4751_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F2 @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_apfst
thf(fact_4752_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F2: C > A,G: D > B,X3: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F2 @ ( product_apsnd @ D @ B @ C @ G @ X3 ) )
= ( product_Pair @ A @ B @ ( F2 @ ( product_fst @ C @ D @ X3 ) ) @ ( G @ ( product_snd @ C @ D @ X3 ) ) ) ) ).
% apfst_apsnd
thf(fact_4753_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: C > A,G: D > C,X3: product_prod @ D @ B] :
( ( product_apfst @ C @ A @ B @ F2 @ ( product_apfst @ D @ C @ B @ G @ X3 ) )
= ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F2 @ G ) @ X3 ) ) ).
% apfst_compose
thf(fact_4754_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F2: C > B,G: D > A,P4: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apfst @ D @ A @ C @ G @ P4 ) )
= ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F2 @ P4 ) ) ) ).
% apsnd_apfst_commute
thf(fact_4755_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat )
@ ^ [M2: nat,N: nat] :
( ( dvd_dvd @ nat @ M2 @ N )
& ( M2 != N ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_4756_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q4: product_prod @ A @ B,F2: C > A,P4: product_prod @ C @ B] :
( ( Q4
= ( product_apfst @ C @ A @ B @ F2 @ P4 ) )
=> ~ ! [X5: C,Y4: B] :
( ( P4
= ( product_Pair @ C @ B @ X5 @ Y4 ) )
=> ( Q4
!= ( product_Pair @ A @ B @ ( F2 @ X5 ) @ Y4 ) ) ) ) ).
% apfst_convE
thf(fact_4757_set__removeAll,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( set2 @ A @ ( removeAll @ A @ X3 @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_removeAll
thf(fact_4758_times__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y2 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V5 ) @ ( times_times @ nat @ Y2 @ U2 ) ) ) )
@ Xa2
@ X3 ) ) ) ).
% times_int.abs_eq
thf(fact_4759_removeAll__id,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( removeAll @ A @ X3 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_4760_distinct__removeAll,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( removeAll @ A @ X3 @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_4761_int_Oabs__induct,axiom,
! [P: int > $o,X3: int] :
( ! [Y4: product_prod @ nat @ nat] : ( P @ ( abs_Integ @ Y4 ) )
=> ( P @ X3 ) ) ).
% int.abs_induct
thf(fact_4762_eq__Abs__Integ,axiom,
! [Z2: int] :
~ ! [X5: nat,Y4: nat] :
( Z2
!= ( abs_Integ @ ( product_Pair @ nat @ nat @ X5 @ Y4 ) ) ) ).
% eq_Abs_Integ
thf(fact_4763_nat_Oabs__eq,axiom,
! [X3: product_prod @ nat @ nat] :
( ( nat2 @ ( abs_Integ @ X3 ) )
= ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ X3 ) ) ).
% nat.abs_eq
thf(fact_4764_length__removeAll__less__eq,axiom,
! [A: $tType,X3: A,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_4765_distinct__remove1__removeAll,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( ( remove1 @ A @ X3 @ Xs )
= ( removeAll @ A @ X3 @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_4766_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_4767_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_4768_length__removeAll__less,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_4769_uminus__int_Oabs__eq,axiom,
! [X3: product_prod @ nat @ nat] :
( ( uminus_uminus @ int @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X: nat,Y2: nat] : ( product_Pair @ nat @ nat @ Y2 @ X )
@ X3 ) ) ) ).
% uminus_int.abs_eq
thf(fact_4770_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_4771_of__int_Oabs__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: product_prod @ nat @ nat] :
( ( ring_1_of_int @ A @ ( abs_Integ @ X3 ) )
= ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ X3 ) ) ) ).
% of_int.abs_eq
thf(fact_4772_less__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y2 ) ) )
@ Xa2
@ X3 ) ) ).
% less_int.abs_eq
thf(fact_4773_less__eq__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y2 ) ) )
@ Xa2
@ X3 ) ) ).
% less_eq_int.abs_eq
thf(fact_4774_plus__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y2 @ V5 ) ) )
@ Xa2
@ X3 ) ) ) ).
% plus_int.abs_eq
thf(fact_4775_minus__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ Y2 @ U2 ) ) )
@ Xa2
@ X3 ) ) ) ).
% minus_int.abs_eq
thf(fact_4776_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_4777_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_4778_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_4779_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_4780_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_4781_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_4782_list__update__nonempty,axiom,
! [A: $tType,Xs: list @ A,K: nat,X3: A] :
( ( ( list_update @ A @ Xs @ K @ X3 )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% list_update_nonempty
thf(fact_4783_concat__replicate__trivial,axiom,
! [A: $tType,I: nat] :
( ( concat @ A @ ( replicate @ ( list @ A ) @ I @ ( nil @ A ) ) )
= ( nil @ A ) ) ).
% concat_replicate_trivial
thf(fact_4784_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_4785_enumerate__simps_I1_J,axiom,
! [A: $tType,N2: nat] :
( ( enumerate @ A @ N2 @ ( nil @ A ) )
= ( nil @ ( product_prod @ nat @ A ) ) ) ).
% enumerate_simps(1)
thf(fact_4786_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_4787_set__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( set2 @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_4788_set__empty2,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_4789_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_4790_empty__replicate,axiom,
! [A: $tType,N2: nat,X3: A] :
( ( ( nil @ A )
= ( replicate @ A @ N2 @ X3 ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_4791_replicate__empty,axiom,
! [A: $tType,N2: nat,X3: A] :
( ( ( replicate @ A @ N2 @ X3 )
= ( nil @ A ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_4792_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F2: B > A,A2: A] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ ( nil @ B ) )
= ( zero_zero @ A ) ) ) ).
% horner_sum_simps(1)
thf(fact_4793_concat__eq__Nil__conv,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( ( concat @ A @ Xss )
= ( nil @ A ) )
= ( ! [X: list @ A] :
( ( member @ ( list @ A ) @ X @ ( set2 @ ( list @ A ) @ Xss ) )
=> ( X
= ( nil @ A ) ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_4794_Nil__eq__concat__conv,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( ( nil @ A )
= ( concat @ A @ Xss ) )
= ( ! [X: list @ A] :
( ( member @ ( list @ A ) @ X @ ( set2 @ ( list @ A ) @ Xss ) )
=> ( X
= ( nil @ A ) ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_4795_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_4796_removeAll_Osimps_I1_J,axiom,
! [A: $tType,X3: A] :
( ( removeAll @ A @ X3 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% removeAll.simps(1)
thf(fact_4797_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_4798_distinct_Osimps_I1_J,axiom,
! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% distinct.simps(1)
thf(fact_4799_concat_Osimps_I1_J,axiom,
! [A: $tType] :
( ( concat @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ).
% concat.simps(1)
thf(fact_4800_list__update_Osimps_I1_J,axiom,
! [A: $tType,I: nat,V2: A] :
( ( list_update @ A @ ( nil @ A ) @ I @ V2 )
= ( nil @ A ) ) ).
% list_update.simps(1)
thf(fact_4801_list__update__code_I1_J,axiom,
! [A: $tType,I: nat,Y: A] :
( ( list_update @ A @ ( nil @ A ) @ I @ Y )
= ( nil @ A ) ) ).
% list_update_code(1)
thf(fact_4802_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_4803_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_4804_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_4805_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_4806_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_4807_remove1_Osimps_I1_J,axiom,
! [A: $tType,X3: A] :
( ( remove1 @ A @ X3 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remove1.simps(1)
thf(fact_4808_gen__length__code_I1_J,axiom,
! [A: $tType,N2: nat] :
( ( gen_length @ A @ N2 @ ( nil @ A ) )
= N2 ) ).
% gen_length_code(1)
thf(fact_4809_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_4810_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_4811_replicate__0,axiom,
! [A: $tType,X3: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X3 )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_4812_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_4813_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_4814_list_Osize__gen_I1_J,axiom,
! [A: $tType,X3: A > nat] :
( ( size_list @ A @ X3 @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size_gen(1)
thf(fact_4815_find_Osimps_I1_J,axiom,
! [A: $tType,Uu: A > $o] :
( ( find @ A @ Uu @ ( nil @ A ) )
= ( none @ A ) ) ).
% find.simps(1)
thf(fact_4816_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_4817_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_4818_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y2: nat,Z6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) )
@ ( rep_Integ @ X )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_4819_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y2: nat,Z6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) )
@ ( rep_Integ @ X )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_4820_prod__encode__def,axiom,
( nat_prod_encode
= ( product_case_prod @ nat @ nat @ nat
@ ^ [M2: nat,N: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M2 @ N ) ) @ M2 ) ) ) ).
% prod_encode_def
thf(fact_4821_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_4822_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_4823_nat_Orep__eq,axiom,
( nat2
= ( ^ [X: int] : ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ ( rep_Integ @ X ) ) ) ) ).
% nat.rep_eq
thf(fact_4824_of__int_Orep__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [X: int] :
( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ ( rep_Integ @ X ) ) ) ) ) ).
% of_int.rep_eq
thf(fact_4825_prod__encode__prod__decode__aux,axiom,
! [K: nat,M: nat] :
( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
= ( plus_plus @ nat @ ( nat_triangle @ K ) @ M ) ) ).
% prod_encode_prod_decode_aux
thf(fact_4826_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_4827_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 )
@ ^ [X: nat,Y2: nat] : ( product_Pair @ nat @ nat @ Y2 @ X ) ) ) ) ).
% uminus_int_def
thf(fact_4828_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P4: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P4 ) )
= ( ? [A4: B] :
( P4
= ( product_Pair @ B @ A @ A4 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_4829_prod_Osize__neq,axiom,
! [A: $tType,B: $tType,X3: product_prod @ A @ B] :
( ( size_size @ ( product_prod @ A @ B ) @ X3 )
!= ( zero_zero @ nat ) ) ).
% prod.size_neq
thf(fact_4830_sum_Osize__neq,axiom,
! [A: $tType,B: $tType,X3: sum_sum @ A @ B] :
( ( size_size @ ( sum_sum @ A @ B ) @ X3 )
!= ( zero_zero @ nat ) ) ).
% sum.size_neq
thf(fact_4831_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A2: A,P4: product_prod @ A @ B] :
( ( A2
= ( product_fst @ A @ B @ P4 ) )
= ( ? [B4: B] :
( P4
= ( product_Pair @ A @ B @ A2 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_4832_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 ) )
@ ^ [X: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y2 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V5 ) @ ( times_times @ nat @ Y2 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_4833_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 ) )
@ ^ [X: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ Y2 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_4834_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 ) )
@ ^ [X: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y2 @ V5 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_4835_Gcd__remove0__nat,axiom,
! [M7: set @ nat] :
( ( finite_finite2 @ nat @ M7 )
=> ( ( gcd_Gcd @ nat @ M7 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M7 @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_4836_add__neg__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N2 ) )
= ( neg_numeral_sub @ A @ N2 @ one2 ) ) ) ).
% add_neg_numeral_special(4)
thf(fact_4837_add__neg__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ M @ one2 ) ) ) ).
% add_neg_numeral_special(3)
thf(fact_4838_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_4839_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_4840_Gcd__insert,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A2: A,A3: set @ A] :
( ( gcd_Gcd @ A @ ( insert @ A @ A2 @ A3 ) )
= ( gcd_gcd @ A @ A2 @ ( gcd_Gcd @ A @ A3 ) ) ) ) ).
% Gcd_insert
thf(fact_4841_semiring__norm_I167_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ W2 @ V2 ) @ Y ) ) ) ).
% semiring_norm(167)
thf(fact_4842_semiring__norm_I166_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ V2 @ W2 ) @ Y ) ) ) ).
% semiring_norm(166)
thf(fact_4843_add__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N2: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( neg_numeral_sub @ A @ M @ N2 ) ) ) ).
% add_neg_numeral_simps(1)
thf(fact_4844_add__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) )
= ( neg_numeral_sub @ A @ N2 @ M ) ) ) ).
% add_neg_numeral_simps(2)
thf(fact_4845_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_4846_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A] :
( ( ( gcd_Gcd @ A @ A3 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_4847_add__neg__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) )
= ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% add_neg_numeral_special(1)
thf(fact_4848_add__neg__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% add_neg_numeral_special(2)
thf(fact_4849_Gcd__dvd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( dvd_dvd @ A @ ( gcd_Gcd @ A @ A3 ) @ A2 ) ) ) ).
% Gcd_dvd
thf(fact_4850_dvd__GcdD,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [X3: A,A3: set @ A,Y: A] :
( ( dvd_dvd @ A @ X3 @ ( gcd_Gcd @ A @ A3 ) )
=> ( ( member @ A @ Y @ A3 )
=> ( dvd_dvd @ A @ X3 @ Y ) ) ) ) ).
% dvd_GcdD
thf(fact_4851_dvd__Gcd__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [X3: A,A3: set @ A] :
( ( dvd_dvd @ A @ X3 @ ( gcd_Gcd @ A @ A3 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( dvd_dvd @ A @ X3 @ X ) ) ) ) ) ).
% dvd_Gcd_iff
thf(fact_4852_Gcd__greatest,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A,A2: A] :
( ! [B5: A] :
( ( member @ A @ B5 @ A3 )
=> ( dvd_dvd @ A @ A2 @ B5 ) )
=> ( dvd_dvd @ A @ A2 @ ( gcd_Gcd @ A @ A3 ) ) ) ) ).
% Gcd_greatest
thf(fact_4853_Gcd__dvd__nat,axiom,
! [A2: nat,A3: set @ nat] :
( ( member @ nat @ A2 @ A3 )
=> ( dvd_dvd @ nat @ ( gcd_Gcd @ nat @ A3 ) @ A2 ) ) ).
% Gcd_dvd_nat
thf(fact_4854_Gcd__greatest__nat,axiom,
! [A3: set @ nat,A2: nat] :
( ! [B5: nat] :
( ( member @ nat @ B5 @ A3 )
=> ( dvd_dvd @ nat @ A2 @ B5 ) )
=> ( dvd_dvd @ nat @ A2 @ ( gcd_Gcd @ nat @ A3 ) ) ) ).
% Gcd_greatest_nat
thf(fact_4855_Gcd__1,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A] :
( ( member @ A @ ( one_one @ A ) @ A3 )
=> ( ( gcd_Gcd @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% Gcd_1
thf(fact_4856_Gcd__nat__eq__one,axiom,
! [N7: set @ nat] :
( ( member @ nat @ ( one_one @ nat ) @ N7 )
=> ( ( gcd_Gcd @ nat @ N7 )
= ( one_one @ nat ) ) ) ).
% Gcd_nat_eq_one
thf(fact_4857_Gcd__eq__1__I,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A2: A,A3: set @ A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( member @ A @ A2 @ A3 )
=> ( ( gcd_Gcd @ A @ A3 )
= ( one_one @ A ) ) ) ) ) ).
% Gcd_eq_1_I
thf(fact_4858_Gcd__in,axiom,
! [A3: set @ nat] :
( ! [A5: nat,B5: nat] :
( ( member @ nat @ A5 @ A3 )
=> ( ( member @ nat @ B5 @ A3 )
=> ( member @ nat @ ( gcd_gcd @ nat @ A5 @ B5 ) @ A3 ) ) )
=> ( ( A3
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( gcd_Gcd @ nat @ A3 ) @ A3 ) ) ) ).
% Gcd_in
thf(fact_4859_sub__non__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num,M: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N2 @ M ) )
= ( ord_less_eq @ num @ M @ N2 ) ) ) ).
% sub_non_negative
thf(fact_4860_sub__non__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num,M: num] :
( ( ord_less_eq @ A @ ( neg_numeral_sub @ A @ N2 @ M ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ num @ N2 @ M ) ) ) ).
% sub_non_positive
thf(fact_4861_sub__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num,M: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N2 @ M ) )
= ( ord_less @ num @ M @ N2 ) ) ) ).
% sub_positive
thf(fact_4862_sub__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num,M: num] :
( ( ord_less @ A @ ( neg_numeral_sub @ A @ N2 @ M ) @ ( zero_zero @ A ) )
= ( ord_less @ num @ N2 @ M ) ) ) ).
% sub_negative
thf(fact_4863_sub__BitM__One__eq,axiom,
! [N2: num] :
( ( neg_numeral_sub @ int @ ( bitM @ N2 ) @ one2 )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( neg_numeral_sub @ int @ N2 @ one2 ) ) ) ).
% sub_BitM_One_eq
thf(fact_4864_semiring__char__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiri4206861660011772517g_char @ A )
= ( ^ [Uu4: itself @ A] :
( gcd_Gcd @ nat
@ ( collect @ nat
@ ^ [N: nat] :
( ( semiring_1_of_nat @ A @ N )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% semiring_char_def
thf(fact_4865_num__of__nat_Osimps_I2_J,axiom,
! [N2: nat] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( num_of_nat @ ( suc @ N2 ) )
= ( inc @ ( num_of_nat @ N2 ) ) ) )
& ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( num_of_nat @ ( suc @ N2 ) )
= one2 ) ) ) ).
% num_of_nat.simps(2)
thf(fact_4866_pred__nat__def,axiom,
( pred_nat
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [M2: nat,N: nat] :
( N
= ( suc @ M2 ) ) ) ) ) ).
% pred_nat_def
thf(fact_4867_abs__Gcd__eq,axiom,
! [K5: set @ int] :
( ( abs_abs @ int @ ( gcd_Gcd @ int @ K5 ) )
= ( gcd_Gcd @ int @ K5 ) ) ).
% abs_Gcd_eq
thf(fact_4868_Gcd__dvd__int,axiom,
! [A2: int,A3: set @ int] :
( ( member @ int @ A2 @ A3 )
=> ( dvd_dvd @ int @ ( gcd_Gcd @ int @ A3 ) @ A2 ) ) ).
% Gcd_dvd_int
thf(fact_4869_Gcd__greatest__int,axiom,
! [A3: set @ int,A2: int] :
( ! [B5: int] :
( ( member @ int @ B5 @ A3 )
=> ( dvd_dvd @ int @ A2 @ B5 ) )
=> ( dvd_dvd @ int @ A2 @ ( gcd_Gcd @ int @ A3 ) ) ) ).
% Gcd_greatest_int
thf(fact_4870_Gcd__int__greater__eq__0,axiom,
! [K5: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Gcd @ int @ K5 ) ) ).
% Gcd_int_greater_eq_0
thf(fact_4871_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ ( zero_zero @ nat ) )
= one2 ) ).
% num_of_nat.simps(1)
thf(fact_4872_numeral__num__of__nat,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( numeral_numeral @ nat @ ( num_of_nat @ N2 ) )
= N2 ) ) ).
% numeral_num_of_nat
thf(fact_4873_num__of__nat__One,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ ( one_one @ nat ) )
=> ( ( num_of_nat @ N2 )
= one2 ) ) ).
% num_of_nat_One
thf(fact_4874_numeral__num__of__nat__unfold,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N2 ) )
= ( one_one @ A ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N2 ) )
= ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).
% numeral_num_of_nat_unfold
thf(fact_4875_num__of__nat__double,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( num_of_nat @ ( plus_plus @ nat @ N2 @ N2 ) )
= ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ).
% num_of_nat_double
thf(fact_4876_num__of__nat__plus__distrib,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( num_of_nat @ ( plus_plus @ nat @ M @ N2 ) )
= ( plus_plus @ num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_4877_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,P4: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I6 )
& ( ( P4 @ X )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P4 @ I6 ) ) )
& ( ~ ( member @ B @ I @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) )
= ( times_times @ A @ ( P4 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P4 @ I6 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_4878_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( remove1 @ A @ X3 @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_4879_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X3
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa2
= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( 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 ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
@ ( 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 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_4880_ball__empty,axiom,
! [A: $tType,P: A > $o,X2: A] :
( ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X2 ) ) ).
% ball_empty
thf(fact_4881_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_4882_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ A3 )
= ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_4883_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( set2 @ A @ ( linord4507533701916653071of_set @ A @ A3 ) )
= A3 ) ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_4884_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P4: B > A] :
( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty'
thf(fact_4885_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( size_size @ ( list @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) )
= ( finite_card @ A @ A3 ) ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_4886_prod_Oeq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,P4: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ I6 )
= ( groups7121269368397514597t_prod @ B @ A @ P4 @ I6 ) ) ) ) ).
% prod.eq_sum
thf(fact_4887_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( linord4507533701916653071of_set @ A @ A3 )
= ( nil @ A ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_4888_finite__set__of__finite__funs,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,D2: B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( finite_finite2 @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F3: A > B] :
! [X: A] :
( ( ( member @ A @ X @ A3 )
=> ( member @ B @ ( F3 @ X ) @ B3 ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ( F3 @ X )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_4889_option_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > B,X22: A] :
( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X22 ) )
= ( F22 @ X22 ) ) ).
% option.simps(5)
thf(fact_4890_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_4891_Ball__Collect,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A6: set @ A,P3: A > $o] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( collect @ A @ P3 ) ) ) ) ).
% Ball_Collect
thf(fact_4892_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 )
@ ^ [X: A] : ( H2 @ ( F22 @ X ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_4893_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A6: set @ A,P3: A > $o] :
! [X: A] :
( ( member @ A @ X @ A6 )
=> ( P3 @ X ) ) ) ) ).
% Ball_def
thf(fact_4894_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( ( linord4507533701916653071of_set @ A @ A3 )
= ( linord4507533701916653071of_set @ A @ B3 ) )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( A3 = B3 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_4895_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] : ( distinct @ A @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_4896_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_4897_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_4898_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I3: B] : ( times_times @ A @ ( G @ I3 ) @ ( H2 @ I3 ) )
@ I6 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% prod.distrib_triv'
thf(fact_4899_case__optionE,axiom,
! [A: $tType,P: $o,Q: A > $o,X3: option @ A] :
( ( case_option @ $o @ A @ P @ Q @ X3 )
=> ( ( ( X3
= ( none @ A ) )
=> ~ P )
=> ~ ! [Y4: A] :
( ( X3
= ( some @ A @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_4900_Sup__eq__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A )
= ( ^ [A6: set @ A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [B4: A] :
! [X: A] :
( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ X @ B4 ) ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_4901_Inf__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A )
= ( ^ [A6: set @ A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [B4: A] :
! [X: A] :
( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ B4 @ X ) ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_4902_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_4903_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T4: set @ B,G: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T4 )
= ( groups1962203154675924110t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_4904_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T4: set @ B,H2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H2 @ I4 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 )
= ( groups1962203154675924110t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_4905_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T4 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_4906_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ T4 ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_4907_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I6 )
& ( ( G @ X )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I6 )
& ( ( H2 @ X )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I3: B] : ( times_times @ A @ ( G @ I3 ) @ ( H2 @ I3 ) )
@ I6 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_4908_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
@ ^ [X: B] :
( ( member @ B @ X @ I8 )
& ( ( P6 @ X )
!= ( one_one @ A ) ) ) ) )
@ ( groups7121269368397514597t_prod @ B @ A @ P6
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I8 )
& ( ( P6 @ X )
!= ( one_one @ A ) ) ) ) )
@ ( one_one @ A ) ) ) ) ) ).
% prod.G_def
thf(fact_4909_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P @ F1 ) )
| ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
& ~ ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_4910_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P @ F1 ) )
& ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
=> ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_4911_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 )
= ( ( Deg = Deg4 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X @ ( 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 ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
@ ( 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 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_4912_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X3
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Y
= ( Xa2
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Deg2 = Xa2 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X @ ( 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 ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
@ ( 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 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_4913_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X3 @ Xa2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X3
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa2
!= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
@ ( 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 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_4914_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X3
= ( 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,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Y
= ( ( Deg2 = Xa2 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X @ ( 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 ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
@ ( 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 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_4915_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X3
= ( 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,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
@ ( 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 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_4916_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X3
= ( 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,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( 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 ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
@ ( 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 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_4917_Union__maximal__sets,axiom,
! [A: $tType,F8: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F8 )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [T8: set @ A] :
( ( member @ ( set @ A ) @ T8 @ F8 )
& ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ F8 )
=> ~ ( ord_less @ ( set @ A ) @ T8 @ X ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ F8 ) ) ) ).
% Union_maximal_sets
thf(fact_4918_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 ) )
=> ( ! [X5: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X5 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y2: list @ A,Z6: list @ A] :
( ( size_size @ ( list @ A ) @ Y2 )
= ( size_size @ ( list @ A ) @ Z6 ) )
@ X5 ) )
=> ( Xs = Ys ) ) ) ) ).
% concat_injective
thf(fact_4919_concat__eq__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ! [X5: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X5 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y2: list @ A,Z6: list @ A] :
( ( size_size @ ( list @ A ) @ Y2 )
= ( size_size @ ( list @ A ) @ Z6 ) )
@ X5 ) )
=> ( ( ( 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_4920_list__eq__iff__zip__eq,axiom,
! [A: $tType] :
( ( ^ [Y3: list @ A,Z: list @ A] : Y3 = Z )
= ( ^ [Xs2: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [X: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs2 @ Ys3 ) ) )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Z: A] : Y3 = Z
@ X ) ) ) ) ) ).
% list_eq_iff_zip_eq
thf(fact_4921_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X: A] :
( ( P3 @ X )
& ! [Y2: A] :
( ( P3 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_4922_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq: A > A > $o,P3: A > $o] :
( the @ A
@ ^ [X: A] :
( ( P3 @ X )
& ! [Y2: A] :
( ( P3 @ Y2 )
=> ( Less_eq @ X @ Y2 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_4923_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 ) )
& ! [X: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( product_case_prod @ A @ B @ $o
@ ^ [Y2: A,Z6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y2 @ Z6 ) @ R2 )
@ X ) ) ) ) ).
% listrel_iff_zip
thf(fact_4924_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_4925_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_4926_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_4927_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_4928_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_4929_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_4930_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_4931_GreatestI__ex__nat,axiom,
! [P: nat > $o,B2: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_4932_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_4933_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A,Q: A > $o] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X5 ) )
=> ( Q @ X5 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_4934_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( order_Greatest @ A @ P )
= X3 ) ) ) ) ).
% Greatest_equality
thf(fact_4935_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 ) )
& ! [N: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N ) @ ( nth @ B @ Ys @ N ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_4936_take__bit__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ A @ N2 ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N2 ) ) ) ) ).
% take_bit_numeral_numeral
thf(fact_4937_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X3 @ A3 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X3
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_4938_sorted__list__of__set__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A )
= ( linord144544945434240204of_set @ A @ A
@ ^ [X: A] : X ) ) ) ).
% sorted_list_of_set_def
thf(fact_4939_take__bit__num__simps_I1_J,axiom,
! [M: num] :
( ( bit_take_bit_num @ ( zero_zero @ nat ) @ M )
= ( none @ num ) ) ).
% take_bit_num_simps(1)
thf(fact_4940_remove1__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [X3: B,F2: B > A,Xs: list @ B] :
( ( remove1 @ B @ X3 @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) )
= Xs ) ) ).
% remove1_insort_key
thf(fact_4941_length__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X3: B,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ).
% length_insort
thf(fact_4942_take__bit__num__simps_I2_J,axiom,
! [N2: nat] :
( ( bit_take_bit_num @ ( suc @ N2 ) @ one2 )
= ( some @ num @ one2 ) ) ).
% take_bit_num_simps(2)
thf(fact_4943_take__bit__num__simps_I3_J,axiom,
! [N2: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ N2 @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_4944_take__bit__num__simps_I4_J,axiom,
! [N2: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_4945_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X3 @ A3 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X3
@ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_4946_take__bit__num__simps_I6_J,axiom,
! [R2: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R2 ) @ ( bit0 @ M ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% take_bit_num_simps(6)
thf(fact_4947_insort__not__Nil,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,A2: B,Xs: list @ B] :
( ( linorder_insort_key @ B @ A @ F2 @ A2 @ Xs )
!= ( nil @ B ) ) ) ).
% insort_not_Nil
thf(fact_4948_insort__key__left__comm,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X3: B,Y: B,Xs: list @ B] :
( ( ( F2 @ X3 )
!= ( F2 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F2 @ Y @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) )
= ( linorder_insort_key @ B @ A @ F2 @ X3 @ ( linorder_insort_key @ B @ A @ F2 @ Y @ Xs ) ) ) ) ) ).
% insort_key_left_comm
thf(fact_4949_insort__left__comm,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A,Xs: list @ A] :
( ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X3
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ Y
@ Xs ) )
= ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ Y
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X3
@ Xs ) ) ) ) ).
% insort_left_comm
thf(fact_4950_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X3: A] :
( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ Y )
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X3 ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X3 )
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ Y ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
thf(fact_4951_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
! [N2: nat,M: num] :
( ( bit_take_bit_num @ N2 @ ( bit0 @ M ) )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N: nat] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ N @ M ) )
@ N2 ) ) ).
% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_4952_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
! [N2: nat] :
( ( bit_take_bit_num @ N2 @ one2 )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N: nat] : ( some @ num @ one2 )
@ N2 ) ) ).
% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_4953_set__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X3: B,Xs: list @ B] :
( ( set2 @ B @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) )
= ( insert @ B @ X3 @ ( set2 @ B @ Xs ) ) ) ) ).
% set_insort_key
thf(fact_4954_distinct__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X3: B,Xs: list @ B] :
( ( distinct @ B @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) )
= ( ~ ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
& ( distinct @ B @ Xs ) ) ) ) ).
% distinct_insort
thf(fact_4955_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
! [N2: nat,M: num] :
( ( bit_take_bit_num @ N2 @ ( bit1 @ M ) )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N: nat] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N @ M ) ) )
@ N2 ) ) ).
% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_4956_take__bit__num__eq__None__imp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N2: num] :
( ( ( bit_take_bit_num @ M @ N2 )
= ( none @ num ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N2 ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_num_eq_None_imp
thf(fact_4957_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ A3 )
= ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X3
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_4958_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N: nat,M2: num] :
( if @ ( option @ num )
@ ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( numeral_numeral @ nat @ M2 ) )
= ( zero_zero @ nat ) )
@ ( none @ num )
@ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ ( numeral_numeral @ nat @ M2 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_4959_Bit__Operations_Otake__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N: nat,M2: num] :
( product_case_prod @ nat @ num @ ( option @ num )
@ ^ [A4: nat,X: num] :
( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [O: nat] :
( case_num @ ( option @ num ) @ ( some @ num @ one2 )
@ ^ [P6: num] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ O @ P6 ) )
@ ^ [P6: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
@ X )
@ A4 )
@ ( product_Pair @ nat @ num @ N @ M2 ) ) ) ) ).
% Bit_Operations.take_bit_num_code
thf(fact_4960_set__nths,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs @ I6 ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [I3: nat] :
( ( Uu3
= ( nth @ A @ Xs @ I3 ) )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I3 @ I6 ) ) ) ) ).
% set_nths
thf(fact_4961_Rats__eq__int__div__nat,axiom,
( ( field_char_0_Rats @ real )
= ( collect @ real
@ ^ [Uu3: real] :
? [I3: int,N: nat] :
( ( Uu3
= ( divide_divide @ real @ ( ring_1_of_int @ real @ I3 ) @ ( semiring_1_of_nat @ real @ N ) ) )
& ( N
!= ( zero_zero @ nat ) ) ) ) ) ).
% Rats_eq_int_div_nat
thf(fact_4962_nths__nil,axiom,
! [A: $tType,A3: set @ nat] :
( ( nths @ A @ ( nil @ A ) @ A3 )
= ( nil @ A ) ) ).
% nths_nil
thf(fact_4963_nths__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( nths @ A @ Xs @ ( bot_bot @ ( set @ nat ) ) )
= ( nil @ A ) ) ).
% nths_empty
thf(fact_4964_Rats__infinite,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ~ ( finite_finite2 @ A @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_infinite
thf(fact_4965_distinct__nthsI,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( nths @ A @ Xs @ I6 ) ) ) ).
% distinct_nthsI
thf(fact_4966_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_4967_Rats__dense__in__real,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ X3 @ Y )
=> ? [X5: real] :
( ( member @ real @ X5 @ ( field_char_0_Rats @ real ) )
& ( ord_less @ real @ X3 @ X5 )
& ( ord_less @ real @ X5 @ Y ) ) ) ).
% Rats_dense_in_real
thf(fact_4968_Rats__no__bot__less,axiom,
! [X3: real] :
? [X5: real] :
( ( member @ real @ X5 @ ( field_char_0_Rats @ real ) )
& ( ord_less @ real @ X5 @ X3 ) ) ).
% Rats_no_bot_less
thf(fact_4969_Rats__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_0
thf(fact_4970_notin__set__nthsI,axiom,
! [A: $tType,X3: A,Xs: list @ A,I6: set @ nat] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ~ ( member @ A @ X3 @ ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) ) ) ).
% notin_set_nthsI
thf(fact_4971_in__set__nthsD,axiom,
! [A: $tType,X3: A,Xs: list @ A,I6: set @ nat] :
( ( member @ A @ X3 @ ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_4972_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_4973_nths__all,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ nat @ I4 @ I6 ) )
=> ( ( nths @ A @ Xs @ I6 )
= Xs ) ) ).
% nths_all
thf(fact_4974_length__nths,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I6 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I3 @ I6 ) ) ) ) ) ).
% length_nths
thf(fact_4975_Nats__altdef1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [Uu3: A] :
? [N: int] :
( ( Uu3
= ( ring_1_of_int @ A @ N ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ N ) ) ) ) ) ).
% Nats_altdef1
thf(fact_4976_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_4977_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X3: nat] :
( ( ( ord_less @ nat @ C2 @ Y )
=> ( ( image @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X3 @ C2 ) @ ( minus_minus @ nat @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ nat @ C2 @ Y )
=> ( ( ( ord_less @ nat @ X3 @ Y )
=> ( ( image @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X3 @ Y )
=> ( ( image @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_4978_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A,X3: B,A3: set @ B] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member @ B @ X3 @ A3 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ).
% image_eqI
thf(fact_4979_image__ident,axiom,
! [A: $tType,Y7: set @ A] :
( ( image @ A @ A
@ ^ [X: A] : X
@ Y7 )
= Y7 ) ).
% image_ident
thf(fact_4980_image__is__empty,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( ( image @ B @ A @ F2 @ A3 )
= ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_4981_empty__is__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F2 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_4982_image__empty,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_4983_finite__imageI,axiom,
! [B: $tType,A: $tType,F5: set @ A,H2: A > B] :
( ( finite_finite2 @ A @ F5 )
=> ( finite_finite2 @ B @ ( image @ A @ B @ H2 @ F5 ) ) ) ).
% finite_imageI
thf(fact_4984_image__insert,axiom,
! [A: $tType,B: $tType,F2: B > A,A2: B,B3: set @ B] :
( ( image @ B @ A @ F2 @ ( insert @ B @ A2 @ B3 ) )
= ( insert @ A @ ( F2 @ A2 ) @ ( image @ B @ A @ F2 @ B3 ) ) ) ).
% image_insert
thf(fact_4985_insert__image,axiom,
! [B: $tType,A: $tType,X3: A,A3: set @ A,F2: A > B] :
( ( member @ A @ X3 @ A3 )
=> ( ( insert @ B @ ( F2 @ X3 ) @ ( image @ A @ B @ F2 @ A3 ) )
= ( image @ A @ B @ F2 @ A3 ) ) ) ).
% insert_image
thf(fact_4986_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S3 )
= S3 ) ) ).
% image_add_0
thf(fact_4987_image__add__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost
thf(fact_4988_image__add__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan
thf(fact_4989_image__add__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [C2: A,A2: A] :
( ( image @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_ord_atMost @ A @ A2 ) )
= ( set_ord_atMost @ A @ ( plus_plus @ A @ C2 @ A2 ) ) ) ) ).
% image_add_atMost
thf(fact_4990_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] :
( ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
= ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_4991_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) )
= ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_4992_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: B > A,A3: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image @ B @ A @ B3 @ A3 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( ( B3 @ X )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_4993_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: B > A,A3: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ B3 @ A3 ) )
= ( bot_bot @ A ) )
= ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( ( B3 @ X )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_4994_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X: B] : ( bot_bot @ A )
@ A3 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_4995_ccSUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X: B] : ( bot_bot @ A )
@ A3 ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_bot
thf(fact_4996_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I3: B] : F2
@ A3 ) )
= F2 ) ) ) ).
% SUP_const
thf(fact_4997_ccSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: set @ B,F2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I3: B] : F2
@ A3 ) )
= F2 ) ) ) ).
% ccSUP_const
thf(fact_4998_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% cSUP_const
thf(fact_4999_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A
@ ^ [N: A] : ( plus_plus @ A @ N @ K )
@ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_5000_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I3: B] : F2
@ A3 ) )
= F2 ) ) ) ).
% INF_const
thf(fact_5001_ccINF__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: set @ B,F2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I3: B] : F2
@ A3 ) )
= F2 ) ) ) ).
% ccINF_const
thf(fact_5002_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% cINF_const
thf(fact_5003_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A
@ ^ [N: A] : ( plus_plus @ A @ N @ K )
@ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_5004_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A3: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) )
= ( bot_bot @ A ) )
= ( ! [X: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X )
=> ? [Y2: B] :
( ( member @ B @ Y2 @ A3 )
& ( ord_less @ A @ ( F2 @ Y2 ) @ X ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_5005_ccSUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F2: B > A] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_empty
thf(fact_5006_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A @ ( times_times @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D2 @ A2 ) @ ( times_times @ A @ D2 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_5007_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A
@ ^ [C4: A] : ( divide_divide @ A @ C4 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A2 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_5008_less__rat,axiom,
! [B2: int,D2: int,A2: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( ord_less @ int @ ( times_times @ int @ ( times_times @ int @ A2 @ D2 ) @ ( times_times @ int @ B2 @ D2 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ) ).
% less_rat
thf(fact_5009_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F2: B > A,P: B > $o] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X: B] :
( ( Uu3
= ( F2 @ X ) )
& ( P @ X ) ) )
= ( image @ B @ A @ F2 @ ( collect @ B @ P ) ) ) ).
% setcompr_eq_image
thf(fact_5010_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X: B] :
( ( Uu3
= ( F2 @ X ) )
& ( member @ B @ X @ A3 ) ) )
= ( image @ B @ A @ F2 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_5011_of__nat__in__Nats,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_1_Nats @ A ) ) ) ).
% of_nat_in_Nats
thf(fact_5012_Nats__induct,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X3: A,P: A > $o] :
( ( member @ A @ X3 @ ( semiring_1_Nats @ A ) )
=> ( ! [N3: nat] : ( P @ ( semiring_1_of_nat @ A @ N3 ) )
=> ( P @ X3 ) ) ) ) ).
% Nats_induct
thf(fact_5013_Nats__cases,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X3: A] :
( ( member @ A @ X3 @ ( semiring_1_Nats @ A ) )
=> ~ ! [N3: nat] :
( X3
!= ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% Nats_cases
thf(fact_5014_image__Fpow__mono,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B3 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F2 ) @ ( finite_Fpow @ B @ A3 ) ) @ ( finite_Fpow @ A @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_5015_imageI,axiom,
! [B: $tType,A: $tType,X3: A,A3: set @ A,F2: A > B] :
( ( member @ A @ X3 @ A3 )
=> ( member @ B @ ( F2 @ X3 ) @ ( image @ A @ B @ F2 @ A3 ) ) ) ).
% imageI
thf(fact_5016_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F2: B > A,A3: set @ B] :
( ( member @ A @ Z2 @ ( image @ B @ A @ F2 @ A3 ) )
= ( ? [X: B] :
( ( member @ B @ X @ A3 )
& ( Z2
= ( F2 @ X ) ) ) ) ) ).
% image_iff
thf(fact_5017_bex__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( image @ B @ A @ F2 @ A3 ) )
& ( P @ X2 ) )
=> ? [X5: B] :
( ( member @ B @ X5 @ A3 )
& ( P @ ( F2 @ X5 ) ) ) ) ).
% bex_imageD
thf(fact_5018_image__cong,axiom,
! [B: $tType,A: $tType,M7: set @ A,N7: set @ A,F2: A > B,G: A > B] :
( ( M7 = N7 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ N7 )
=> ( ( F2 @ X5 )
= ( G @ X5 ) ) )
=> ( ( image @ A @ B @ F2 @ M7 )
= ( image @ A @ B @ G @ N7 ) ) ) ) ).
% image_cong
thf(fact_5019_ball__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F2 @ A3 ) )
=> ( P @ X5 ) )
=> ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( P @ ( F2 @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_5020_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X3: A,A3: set @ A,B2: B,F2: A > B] :
( ( member @ A @ X3 @ A3 )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_5021_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A,A3: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A3 ) )
=> ~ ! [X5: B] :
( ( B2
= ( F2 @ X5 ) )
=> ~ ( member @ B @ X5 @ A3 ) ) ) ).
% imageE
thf(fact_5022_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,G: C > B,A3: set @ C] :
( ( image @ B @ A @ F2 @ ( image @ C @ B @ G @ A3 ) )
= ( image @ C @ A
@ ^ [X: C] : ( F2 @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_5023_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( image @ B @ A @ F2 @ A3 ) )
& ( P @ X ) ) )
= ( image @ B @ A @ F2
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A3 )
& ( P @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_5024_all__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] :
( ( ! [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F2 @ A3 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A3 )
=> ( P @ ( image @ B @ A @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_5025_subset__image__iff,axiom,
! [A: $tType,B: $tType,B3: set @ A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F2 @ A3 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A3 )
& ( B3
= ( image @ B @ A @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_5026_image__subset__iff,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B3 )
= ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( member @ A @ ( F2 @ X ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_5027_subset__imageE,axiom,
! [A: $tType,B: $tType,B3: set @ A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F2 @ A3 ) )
=> ~ ! [C8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C8 @ A3 )
=> ( B3
!= ( image @ B @ A @ F2 @ C8 ) ) ) ) ).
% subset_imageE
thf(fact_5028_image__subsetI,axiom,
! [A: $tType,B: $tType,A3: set @ A,F2: A > B,B3: set @ B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( member @ B @ ( F2 @ X5 ) @ B3 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_5029_image__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,F2: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B3 ) ) ) ).
% image_mono
thf(fact_5030_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B3: set @ A] :
( ( ( image @ B @ A @ F2 @ A3 )
= B3 )
=> ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F2 ) @ ( pow2 @ B @ A3 ) )
= ( pow2 @ A @ B3 ) ) ) ).
% image_Pow_surj
thf(fact_5031_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: A > B,B3: set @ B] :
( ! [X5: A] :
( ( P @ X5 )
=> ( member @ B @ ( F2 @ X5 ) @ B3 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( collect @ A @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_5032_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B3 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F2 ) @ ( pow2 @ B @ A3 ) ) @ ( pow2 @ A @ B3 ) ) ) ).
% image_Pow_mono
thf(fact_5033_Nats__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_0
thf(fact_5034_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_5035_Nats__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_1
thf(fact_5036_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_5037_Nats__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [W2: num] : ( member @ A @ ( numeral_numeral @ A @ W2 ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_numeral
thf(fact_5038_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X: B] : ( insert @ A @ ( F2 @ X ) @ ( bot_bot @ ( set @ A ) ) )
@ A3 ) )
= ( image @ B @ A @ F2 @ A3 ) ) ).
% UNION_singleton_eq_range
thf(fact_5039_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F2 @ A3 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A3 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A4: A] :
( ( member @ A @ A4 @ A3 )
& ( ( F2 @ A4 )
= ( F2 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_5040_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B3: set @ C,F2: B > A,G: C > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ? [X2: C] :
( ( member @ C @ X2 @ B3 )
& ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B3 )
=> ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ord_less_eq @ A @ ( G @ J2 ) @ ( F2 @ X2 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) )
= ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B3 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_5041_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B3: set @ C,G: C > A,F2: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ? [X2: C] :
( ( member @ C @ X2 @ B3 )
& ( ord_less_eq @ A @ ( G @ X2 ) @ ( F2 @ I4 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B3 )
=> ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) )
= ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B3 ) ) ) ) ) ) ).
% INF_eq
thf(fact_5042_zero__notin__Suc__image,axiom,
! [A3: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A3 ) ) ).
% zero_notin_Suc_image
thf(fact_5043_finite__surj,axiom,
! [A: $tType,B: $tType,A3: set @ A,B3: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ ( image @ A @ B @ F2 @ A3 ) )
=> ( finite_finite2 @ B @ B3 ) ) ) ).
% finite_surj
thf(fact_5044_finite__subset__image,axiom,
! [A: $tType,B: $tType,B3: set @ A,F2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F2 @ A3 ) )
=> ? [C8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C8 @ A3 )
& ( finite_finite2 @ B @ C8 )
& ( B3
= ( image @ B @ A @ F2 @ C8 ) ) ) ) ) ).
% finite_subset_image
thf(fact_5045_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] :
( ( ? [B6: set @ A] :
( ( finite_finite2 @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F2 @ A3 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set @ B] :
( ( finite_finite2 @ B @ B6 )
& ( ord_less_eq @ ( set @ B ) @ B6 @ A3 )
& ( P @ ( image @ B @ A @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_5046_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] :
( ( ! [B6: set @ A] :
( ( ( finite_finite2 @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F2 @ A3 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set @ B] :
( ( ( finite_finite2 @ B @ B6 )
& ( ord_less_eq @ ( set @ B ) @ B6 @ A3 ) )
=> ( P @ ( image @ B @ A @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_5047_translation__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S: set @ A,T2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_Int
thf(fact_5048_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F2: B > A,X3: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ( F2 @ I4 )
= X3 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ I6 ) )
= X3 ) ) ) ) ).
% SUP_eq_const
thf(fact_5049_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S: set @ A,T2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_diff
thf(fact_5050_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F2: B > A,X3: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ( F2 @ I4 )
= X3 ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) )
= X3 ) ) ) ) ).
% INF_eq_const
thf(fact_5051_finite__image__absD,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ ( image @ A @ A @ ( abs_abs @ A ) @ S3 ) )
=> ( finite_finite2 @ A @ S3 ) ) ) ).
% finite_image_absD
thf(fact_5052_image__Int__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B3: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ ( image @ B @ A @ F2 @ B3 ) ) ) ).
% image_Int_subset
thf(fact_5053_image__diff__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B3: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ ( image @ B @ A @ F2 @ B3 ) ) @ ( image @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_5054_Rat__induct__pos,axiom,
! [P: rat > $o,Q4: rat] :
( ! [A5: int,B5: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( P @ ( fract @ A5 @ B5 ) ) )
=> ( P @ Q4 ) ) ).
% Rat_induct_pos
thf(fact_5055_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,T2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_Compl
thf(fact_5056_quotient__of__eq,axiom,
! [A2: int,B2: int,P4: int,Q4: int] :
( ( ( quotient_of @ ( fract @ A2 @ B2 ) )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( ( fract @ P4 @ Q4 )
= ( fract @ A2 @ B2 ) ) ) ).
% quotient_of_eq
thf(fact_5057_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: B > A,X3: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ X3 ) )
=> ( ! [Y4: A] :
( ! [I5: B] :
( ( member @ B @ I5 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I5 ) @ Y4 ) )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) )
= X3 ) ) ) ) ).
% SUP_eqI
thf(fact_5058_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B3: set @ C,F2: B > A,G: C > A] :
( ! [N3: B] :
( ( member @ B @ N3 @ A3 )
=> ? [X2: C] :
( ( member @ C @ X2 @ B3 )
& ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ X2 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B3 ) ) ) ) ) ).
% SUP_mono
thf(fact_5059_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: B > A,U: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U ) ) ) ).
% SUP_least
thf(fact_5060_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G: B > A,A3: set @ B] :
( ! [X5: B] : ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ).
% SUP_mono'
thf(fact_5061_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A3: set @ B,F2: B > A] :
( ( member @ B @ I @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% SUP_upper
thf(fact_5062_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A3: set @ B,U: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U )
= ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X ) @ U ) ) ) ) ) ).
% SUP_le_iff
thf(fact_5063_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A3: set @ B,U: A,F2: B > A] :
( ( member @ B @ I @ A3 )
=> ( ( ord_less_eq @ A @ U @ ( F2 @ I ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_5064_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: A,F2: B > A,A3: set @ B] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A3 )
& ( ord_less @ A @ A2 @ ( F2 @ X ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_5065_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A3: set @ B,Y: A,I: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ Y )
=> ( ( member @ B @ I @ A3 )
=> ( ord_less @ A @ ( F2 @ I ) @ Y ) ) ) ) ).
% SUP_lessD
thf(fact_5066_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,X3: A,F2: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ( ord_less_eq @ A @ X3 @ ( F2 @ I4 ) ) )
=> ( ! [Y4: A] :
( ! [I5: B] :
( ( member @ B @ I5 @ A3 )
=> ( ord_less_eq @ A @ Y4 @ ( F2 @ I5 ) ) )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) )
= X3 ) ) ) ) ).
% INF_eqI
thf(fact_5067_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: set @ B,A3: set @ C,F2: C > A,G: B > A] :
( ! [M3: B] :
( ( member @ B @ M3 @ B3 )
=> ? [X2: C] :
( ( member @ C @ X2 @ A3 )
& ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G @ M3 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ).
% INF_mono
thf(fact_5068_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A3: set @ B,F2: B > A] :
( ( member @ B @ I @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( F2 @ I ) ) ) ) ).
% INF_lower
thf(fact_5069_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G: B > A,A3: set @ B] :
( ! [X5: B] : ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ).
% INF_mono'
thf(fact_5070_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A3: set @ B,F2: B > A,U: A] :
( ( member @ B @ I @ A3 )
=> ( ( ord_less_eq @ A @ ( F2 @ I ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U ) ) ) ) ).
% INF_lower2
thf(fact_5071_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( ord_less_eq @ A @ U @ ( F2 @ X ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_5072_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,U: A,F2: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ( ord_less_eq @ A @ U @ ( F2 @ I4 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% INF_greatest
thf(fact_5073_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A3: set @ B,A2: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ A2 )
= ( ? [X: B] :
( ( member @ B @ X @ A3 )
& ( ord_less @ A @ ( F2 @ X ) @ A2 ) ) ) ) ) ).
% INF_less_iff
thf(fact_5074_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A,F2: B > A,A3: set @ B,I: B] :
( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
=> ( ( member @ B @ I @ A3 )
=> ( ord_less @ A @ Y @ ( F2 @ I ) ) ) ) ) ).
% less_INF_D
thf(fact_5075_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] :
? [N: nat,F3: nat > A] :
( A6
= ( image @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_5076_nat__seg__image__imp__finite,axiom,
! [A: $tType,A3: set @ A,F2: nat > A,N2: nat] :
( ( A3
= ( image @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) ) )
=> ( finite_finite2 @ A @ A3 ) ) ).
% nat_seg_image_imp_finite
thf(fact_5077_image__constant,axiom,
! [A: $tType,B: $tType,X3: A,A3: set @ A,C2: B] :
( ( member @ A @ X3 @ A3 )
=> ( ( image @ A @ B
@ ^ [X: A] : C2
@ A3 )
= ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_5078_image__constant__conv,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X: B] : C2
@ A3 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X: B] : C2
@ A3 )
= ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_5079_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,H2: B > A,G: B > C] :
( ( finite_finite2 @ B @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y2: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ S3 )
& ( ( G @ X )
= Y2 ) ) ) )
@ ( image @ B @ C @ G @ S3 ) ) ) ) ) ).
% sum.image_gen
thf(fact_5080_normalize__eq,axiom,
! [A2: int,B2: int,P4: int,Q4: int] :
( ( ( normalize @ ( product_Pair @ int @ int @ A2 @ B2 ) )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( ( fract @ P4 @ Q4 )
= ( fract @ A2 @ B2 ) ) ) ).
% normalize_eq
thf(fact_5081_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,H2: B > A,G: B > C] :
( ( finite_finite2 @ B @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y2: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ S3 )
& ( ( G @ X )
= Y2 ) ) ) )
@ ( image @ B @ C @ G @ S3 ) ) ) ) ) ).
% prod.image_gen
thf(fact_5082_Gcd__mono,axiom,
! [A: $tType,B: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( dvd_dvd @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( dvd_dvd @ A @ ( gcd_Gcd @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( gcd_Gcd @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ).
% Gcd_mono
thf(fact_5083_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B,X3: A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ A3 )
=> ( ( F2 @ Y4 )
= ( F2 @ X3 ) ) )
=> ( ( the_elem @ B @ ( image @ A @ B @ F2 @ A3 ) )
= ( F2 @ X3 ) ) ) ) ).
% the_elem_image_unique
thf(fact_5084_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X3: A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
= ( ! [Y2: A] :
( ( ord_less @ A @ Y2 @ X3 )
=> ? [X: B] :
( ( member @ B @ X @ A3 )
& ( ord_less @ A @ Y2 @ ( F2 @ X ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_5085_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A3: set @ B,X3: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ X3 )
= ( ! [Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ? [X: B] :
( ( member @ B @ X @ A3 )
& ( ord_less @ A @ ( F2 @ X ) @ Y2 ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_5086_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,C2: A,F2: B > A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ord_less_eq @ A @ C2 @ ( F2 @ I4 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ I6 ) )
= C2 )
= ( ! [X: B] :
( ( member @ B @ X @ I6 )
=> ( ( F2 @ X )
= C2 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_5087_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,M7: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ M7 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ M7 ) ) ) ) ).
% cSUP_least
thf(fact_5088_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F2: B > A,C2: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) )
= C2 )
= ( ! [X: B] :
( ( member @ B @ X @ I6 )
=> ( ( F2 @ X )
= C2 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_5089_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,M: A,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ M @ ( F2 @ X5 ) ) )
=> ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_5090_card__image__le,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image @ A @ B @ F2 @ A3 ) ) @ ( finite_card @ A @ A3 ) ) ) ).
% card_image_le
thf(fact_5091_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_5092_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B3: set @ B,F2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_5093_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: set @ B,A3: set @ B,F2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ B3 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_5094_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% SUP_empty
thf(fact_5095_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y2: B] : C2
@ A3 ) )
= ( bot_bot @ A ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y2: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% SUP_constant
thf(fact_5096_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T4: set @ C,G: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S3 ) @ T4 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y2: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ S3 )
& ( ( G @ X )
= Y2 ) ) ) )
@ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% sum.group
thf(fact_5097_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F2: B > A,X3: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I3: B] : ( inf_inf @ A @ ( F2 @ I3 ) @ X3 )
@ I6 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) ) @ X3 ) ) ) ) ).
% INF_inf_const2
thf(fact_5098_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,X3: A,F2: B > A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I3: B] : ( inf_inf @ A @ X3 @ ( F2 @ I3 ) )
@ I6 ) )
= ( inf_inf @ A @ X3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_5099_Nats__subset__Rats,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ord_less_eq @ ( set @ A ) @ ( semiring_1_Nats @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Nats_subset_Rats
thf(fact_5100_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T4: set @ C,G: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S3 ) @ T4 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y2: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ S3 )
& ( ( G @ X )
= Y2 ) ) ) )
@ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% prod.group
thf(fact_5101_quotient__of__Fract,axiom,
! [A2: int,B2: int] :
( ( quotient_of @ ( fract @ A2 @ B2 ) )
= ( normalize @ ( product_Pair @ int @ int @ A2 @ B2 ) ) ) ).
% quotient_of_Fract
thf(fact_5102_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,H2: B > C,G: C > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X5: B,Y4: B] :
( ( member @ B @ X5 @ A3 )
=> ( ( member @ B @ Y4 @ A3 )
=> ( ( X5 != Y4 )
=> ( ( ( H2 @ X5 )
= ( H2 @ Y4 ) )
=> ( ( G @ ( H2 @ X5 ) )
= ( zero_zero @ A ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image @ B @ C @ H2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A3 ) ) ) ) ) ).
% sum.reindex_nontrivial
thf(fact_5103_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_5104_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,H2: B > C,G: C > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X5: B,Y4: B] :
( ( member @ B @ X5 @ A3 )
=> ( ( member @ B @ Y4 @ A3 )
=> ( ( X5 != Y4 )
=> ( ( ( H2 @ X5 )
= ( H2 @ Y4 ) )
=> ( ( G @ ( H2 @ X5 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image @ B @ C @ H2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A3 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_5105_surj__card__le,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ ( image @ A @ B @ F2 @ A3 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B3 ) @ ( finite_card @ A @ A3 ) ) ) ) ).
% surj_card_le
thf(fact_5106_scaleR__image__atLeastAtMost,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,X3: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( image @ A @ A @ ( real_V8093663219630862766scaleR @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ Y ) ) ) ) ) ).
% scaleR_image_atLeastAtMost
thf(fact_5107_image__Suc__lessThan,axiom,
! [N2: nat] :
( ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N2 ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ).
% image_Suc_lessThan
thf(fact_5108_image__Suc__atMost,axiom,
! [N2: nat] :
( ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N2 ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N2 ) ) ) ).
% image_Suc_atMost
thf(fact_5109_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N2: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_5110_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_5111_lessThan__Suc__eq__insert__0,axiom,
! [N2: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ N2 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_5112_atMost__Suc__eq__insert__0,axiom,
! [N2: nat] :
( ( set_ord_atMost @ nat @ ( suc @ N2 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_5113_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_5114_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_5115_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_5116_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_5117_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I6: set @ C,G: A > B,F2: C > A] :
( ( finite_finite2 @ C @ I6 )
=> ( ! [I4: C] :
( ( member @ C @ I4 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F2 @ I4 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image @ C @ A @ F2 @ I6 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F2 ) @ I6 ) ) ) ) ) ).
% sum_image_le
thf(fact_5118_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_5119_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_5120_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_5121_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_5122_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X3: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X3 ) @ ( times_times @ A @ C2 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y ) @ ( times_times @ A @ C2 @ X3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ X3 @ Y )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_5123_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A,C2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A
@ ^ [X: A] : ( times_times @ A @ X @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X3 @ C2 ) @ ( times_times @ A @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A
@ ^ [X: A] : ( times_times @ A @ X @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C2 ) @ ( times_times @ A @ X3 @ C2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X3 @ Y )
=> ( ( image @ A @ A
@ ^ [X: A] : ( times_times @ A @ X @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_5124_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_5125_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_5126_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_5127_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_5128_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S3: set @ A,R: set @ B,G: A > B,F2: B > C] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ R )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ G @ S3 ) @ R )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X: A] : ( F2 @ ( G @ X ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y2: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ S3 )
& ( ( G @ X )
= Y2 ) ) ) ) )
@ ( F2 @ Y2 ) )
@ R ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_5129_INF__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: A,B3: A] :
( ( inf_inf @ A @ A3
@ ( complete_Inf_Inf @ A
@ ( image @ nat @ A
@ ^ [X: nat] : B3
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( inf_inf @ A @ A3 @ B3 ) ) ) ).
% INF_nat_binary
thf(fact_5130_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F2: A > C,G: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu3: C] : ( bot_bot @ ( set @ B ) )
@ F2 )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image @ D @ B @ G )
@ ^ [Uu3: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_5131_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
@ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R5 ) ) ) ) ) ) ).
% listrel_def
thf(fact_5132_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set @ ( product_prod @ A @ B ),F2: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A3 )
=> ( member @ C @ ( F2 @ A2 @ B2 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F2 ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_5133_Gcd__abs__eq,axiom,
! [K5: set @ int] :
( ( gcd_Gcd @ int @ ( image @ int @ int @ ( abs_abs @ int ) @ K5 ) )
= ( gcd_Gcd @ int @ K5 ) ) ).
% Gcd_abs_eq
thf(fact_5134_finite__UN,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( finite_finite2 @ B @ ( B3 @ X ) ) ) ) ) ) ).
% finite_UN
thf(fact_5135_UN__constant,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y2: B] : C2
@ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y2: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_5136_finite__UN__I,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A3 )
=> ( finite_finite2 @ B @ ( B3 @ A5 ) ) )
=> ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ) ).
% finite_UN_I
thf(fact_5137_finite__INT,axiom,
! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B )] :
( ? [X2: A] :
( ( member @ A @ X2 @ I6 )
& ( finite_finite2 @ B @ ( A3 @ X2 ) ) )
=> ( finite_finite2 @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) ) ).
% finite_INT
thf(fact_5138_Gcd__int__eq,axiom,
! [N7: set @ nat] :
( ( gcd_Gcd @ int @ ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ N7 ) )
= ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ N7 ) ) ) ).
% Gcd_int_eq
thf(fact_5139_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set @ B,A2: A,B3: B > ( set @ A )] :
( ( ( C5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X: B] : ( insert @ A @ A2 @ ( B3 @ X ) )
@ C5 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X: B] : ( insert @ A @ A2 @ ( B3 @ X ) )
@ C5 ) )
= ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ C5 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_5140_UN__singleton,axiom,
! [A: $tType,A3: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ A @ ( set @ A )
@ ^ [X: A] : ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
@ A3 ) )
= A3 ) ).
% UN_singleton
thf(fact_5141_set__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( set2 @ A @ ( concat @ A @ Xs ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xs ) ) ) ) ).
% set_concat
thf(fact_5142_Gcd__nat__abs__eq,axiom,
! [K5: set @ int] :
( ( gcd_Gcd @ nat
@ ( image @ int @ nat
@ ^ [K3: int] : ( nat2 @ ( abs_abs @ int @ K3 ) )
@ K5 ) )
= ( nat2 @ ( gcd_Gcd @ int @ K5 ) ) ) ).
% Gcd_nat_abs_eq
thf(fact_5143_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image @ C @ ( A > B > $o )
@ ^ [I3: C,X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( R2 @ I3 ) )
@ S3 ) )
= ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_5144_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image @ C @ ( A > B > $o )
@ ^ [I3: C,X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( R2 @ I3 ) )
@ S3 ) )
= ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_5145_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S3: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I3: set @ ( product_prod @ A @ B ),X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ I3 )
@ S3 ) )
= ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ).
% INF_Int_eq2
thf(fact_5146_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S3: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I3: set @ ( product_prod @ A @ B ),X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ I3 )
@ S3 ) )
= ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_5147_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S7: set @ ( A > B > $o ),X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S7 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_5148_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S7: set @ ( A > B > $o ),X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S7 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_5149_UN__image__subset,axiom,
! [C: $tType,A: $tType,B: $tType,F2: B > ( set @ A ),G: C > ( set @ B ),X3: C,X8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ ( G @ X3 ) ) ) @ X8 )
= ( ord_less_eq @ ( set @ B ) @ ( G @ X3 )
@ ( collect @ B
@ ^ [X: B] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ X ) @ X8 ) ) ) ) ).
% UN_image_subset
thf(fact_5150_listrelp_ONil,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] : ( listrelp @ A @ B @ R2 @ ( nil @ A ) @ ( nil @ B ) ) ).
% listrelp.Nil
thf(fact_5151_None__notin__image__Some,axiom,
! [A: $tType,A3: set @ A] :
~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ) ).
% None_notin_image_Some
thf(fact_5152_INF__filter__not__bot,axiom,
! [I7: $tType,A: $tType,B3: set @ I7,F5: I7 > ( filter @ A )] :
( ! [X10: set @ I7] :
( ( ord_less_eq @ ( set @ I7 ) @ X10 @ B3 )
=> ( ( finite_finite2 @ I7 @ X10 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ I7 @ ( filter @ A ) @ F5 @ X10 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ I7 @ ( filter @ A ) @ F5 @ B3 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_5153_finite__int__iff__bounded,axiom,
( ( finite_finite2 @ int )
= ( ^ [S7: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S7 ) @ ( set_ord_lessThan @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded
thf(fact_5154_finite__int__iff__bounded__le,axiom,
( ( finite_finite2 @ int )
= ( ^ [S7: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S7 ) @ ( set_ord_atMost @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded_le
thf(fact_5155_UN__empty2,axiom,
! [B: $tType,A: $tType,A3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X: B] : ( bot_bot @ ( set @ A ) )
@ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_5156_UN__empty,axiom,
! [B: $tType,A: $tType,B3: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_5157_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( ( B3 @ X )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_5158_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( ( B3 @ X )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_5159_UN__subset__iff,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),I6: set @ B,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) @ B3 )
= ( ! [X: B] :
( ( member @ B @ X @ I6 )
=> ( ord_less_eq @ ( set @ A ) @ ( A3 @ X ) @ B3 ) ) ) ) ).
% UN_subset_iff
thf(fact_5160_UN__upper,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,B3: A > ( set @ B )] :
( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( B3 @ A2 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ).
% UN_upper
thf(fact_5161_UN__least,axiom,
! [A: $tType,B: $tType,A3: set @ A,B3: A > ( set @ B ),C5: set @ B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( B3 @ X5 ) @ C5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) @ C5 ) ) ).
% UN_least
thf(fact_5162_UN__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,F2: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_5163_INT__subset__iff,axiom,
! [A: $tType,B: $tType,B3: set @ A,A3: B > ( set @ A ),I6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ I6 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ ( A3 @ X ) ) ) ) ) ).
% INT_subset_iff
thf(fact_5164_INT__anti__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,F2: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ B3 ) ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ A3 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_5165_INT__greatest,axiom,
! [B: $tType,A: $tType,A3: set @ A,C5: set @ B,B3: A > ( set @ B )] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ C5 @ ( B3 @ X5 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ C5 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ).
% INT_greatest
thf(fact_5166_INT__lower,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,B3: A > ( set @ B )] :
( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) @ ( B3 @ A2 ) ) ) ).
% INT_lower
thf(fact_5167_in__image__insert__iff,axiom,
! [A: $tType,B3: set @ ( set @ A ),X3: A,A3: set @ A] :
( ! [C8: set @ A] :
( ( member @ ( set @ A ) @ C8 @ B3 )
=> ~ ( member @ A @ X3 @ C8 ) )
=> ( ( member @ ( set @ A ) @ A3 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X3 ) @ B3 ) )
= ( ( member @ A @ X3 @ A3 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ) ) ).
% in_image_insert_iff
thf(fact_5168_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set @ B,A2: A,B3: B > ( set @ A )] :
( ( ( C5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ C5 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X: B] : ( insert @ A @ A2 @ ( B3 @ X ) )
@ C5 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_5169_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set @ D,A3: set @ C,B3: D > ( set @ C )] :
( ( ( C5
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D @ ( set @ C ) @ B3 @ C5 ) ) )
= A3 ) )
& ( ( C5
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D @ ( set @ C ) @ B3 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image @ D @ ( set @ C )
@ ^ [X: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B3 @ X ) )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_5170_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C5: set @ A,A3: A > ( set @ B ),B3: set @ B] :
( ( ( C5
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ C5 ) ) @ B3 )
= B3 ) )
& ( ( C5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ C5 ) ) @ B3 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X ) @ B3 )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_5171_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B12: set @ ( set @ A ),A3: set @ A] :
( ( ( B12
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) @ A3 )
= A3 ) )
& ( ( B12
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) @ A3 )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ ( set @ A ) @ ( set @ A )
@ ^ [B6: set @ A] : ( inf_inf @ ( set @ A ) @ B6 @ A3 )
@ B12 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_5172_Int__Inter__eq_I1_J,axiom,
! [A: $tType,B12: set @ ( set @ A ),A3: set @ A] :
( ( ( B12
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) )
= A3 ) )
& ( ( B12
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 ) @ B12 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_5173_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X8: set @ A,A3: set @ ( product_prod @ A @ B ),Y7: set @ B,P: A > B > $o,Q: A > B > $o] :
( ( X8
= ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) )
=> ( ( Y7
= ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ Y7 )
=> ( ( P @ X5 @ Xa3 )
=> ( Q @ X5 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_5174_INT__extend__simps_I4_J,axiom,
! [G3: $tType,H4: $tType,C5: set @ H4,A3: set @ G3,B3: H4 > ( set @ G3 )] :
( ( ( C5
= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( minus_minus @ ( set @ G3 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image @ H4 @ ( set @ G3 ) @ B3 @ C5 ) ) )
= A3 ) )
& ( ( C5
!= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( minus_minus @ ( set @ G3 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image @ H4 @ ( set @ G3 ) @ B3 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ G3 )
@ ( image @ H4 @ ( set @ G3 )
@ ^ [X: H4] : ( minus_minus @ ( set @ G3 ) @ A3 @ ( B3 @ X ) )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_5175_infinite__int__iff__infinite__nat__abs,axiom,
! [S3: set @ int] :
( ( ~ ( finite_finite2 @ int @ S3 ) )
= ( ~ ( finite_finite2 @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ S3 ) ) ) ) ).
% infinite_int_iff_infinite_nat_abs
thf(fact_5176_UN__le__add__shift__strict,axiom,
! [A: $tType,M7: nat > ( set @ A ),K: nat,N2: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [I3: nat] : ( M7 @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ N2 @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_5177_UN__le__add__shift,axiom,
! [A: $tType,M7: nat > ( set @ A ),K: nat,N2: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [I3: nat] : ( M7 @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or1337092689740270186AtMost @ nat @ K @ ( plus_plus @ nat @ N2 @ K ) ) ) ) ) ).
% UN_le_add_shift
thf(fact_5178_subset__subseqs,axiom,
! [A: $tType,X8: set @ A,Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X8 @ ( set2 @ A @ Xs ) )
=> ( member @ ( set @ A ) @ X8 @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_5179_subseqs__powset,axiom,
! [A: $tType,Xs: list @ A] :
( ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
= ( pow2 @ A @ ( set2 @ A @ Xs ) ) ) ).
% subseqs_powset
thf(fact_5180_conj__subset__def,axiom,
! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A3
@ ( collect @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_5181_finite__Inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A3: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A3 ) )
@ ( complete_Sup_Sup @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F3: ( set @ A ) > A] :
( ( Uu3
= ( image @ ( set @ A ) @ A @ F3 @ A3 ) )
& ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A3 )
=> ( member @ A @ ( F3 @ X ) @ X ) ) ) ) ) ) ) ) ).
% finite_Inf_Sup
thf(fact_5182_Inf__Sup__le,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A3 ) )
@ ( complete_Sup_Sup @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F3: ( set @ A ) > A] :
( ( Uu3
= ( image @ ( set @ A ) @ A @ F3 @ A3 ) )
& ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A3 )
=> ( member @ A @ ( F3 @ X ) @ X ) ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_5183_Sup__Inf__le,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ ( set @ A )] :
( ord_less_eq @ A
@ ( complete_Sup_Sup @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F3: ( set @ A ) > A] :
( ( Uu3
= ( image @ ( set @ A ) @ A @ F3 @ A3 ) )
& ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A3 )
=> ( member @ A @ ( F3 @ X ) @ X ) ) ) ) ) )
@ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A3 ) ) ) ) ).
% Sup_Inf_le
thf(fact_5184_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,A3: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ( finite_finite2 @ C @ ( A3 @ X5 ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I6 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A3 @ X5 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ A3 @ I6 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( groups7311177749621191930dd_sum @ C @ A @ G @ ( A3 @ X ) )
@ I6 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_5185_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,A3: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ( finite_finite2 @ C @ ( A3 @ X5 ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I6 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A3 @ X5 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ A3 @ I6 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( groups7121269368397514597t_prod @ C @ A @ G @ ( A3 @ X ) )
@ I6 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_5186_card__UN__le,axiom,
! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B )] :
( ( finite_finite2 @ A @ I6 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] : ( finite_card @ B @ ( A3 @ I3 ) )
@ I6 ) ) ) ).
% card_UN_le
thf(fact_5187_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( listrelp @ A @ B
@ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R2 ) )
= ( ^ [X: list @ A,Y2: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X @ Y2 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_5188_Gcd__int__def,axiom,
( ( gcd_Gcd @ int )
= ( ^ [K6: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K6 ) ) ) ) ) ).
% Gcd_int_def
thf(fact_5189_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B )] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ I6 )
=> ( finite_finite2 @ B @ ( A3 @ X5 ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ I6 )
=> ! [Xa3: A] :
( ( member @ A @ Xa3 @ I6 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ ( A3 @ X5 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] : ( finite_card @ B @ ( A3 @ I3 ) )
@ I6 ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_5190_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F2: nat > ( set @ A ),S3: set @ A] :
( ! [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ I4 ) @ S3 )
=> ( ( finite_finite2 @ A @ S3 )
=> ( ? [N8: nat] :
( ! [N3: nat] :
( ( ord_less_eq @ nat @ N3 @ N8 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N8 )
=> ( ( ord_less @ nat @ M3 @ N3 )
=> ( ord_less @ ( set @ A ) @ ( F2 @ M3 ) @ ( F2 @ N3 ) ) ) ) )
& ! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ( F2 @ N8 )
= ( F2 @ N3 ) ) ) )
=> ( ( F2 @ ( finite_card @ A @ S3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ F2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_5191_length__remdups__concat,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ ( concat @ A @ Xss ) ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xss ) ) ) ) ) ).
% length_remdups_concat
thf(fact_5192_INT__simps_I4_J,axiom,
! [G3: $tType,H4: $tType,C5: set @ H4,A3: set @ G3,B3: H4 > ( set @ G3 )] :
( ( ( C5
= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G3 )
@ ( image @ H4 @ ( set @ G3 )
@ ^ [X: H4] : ( minus_minus @ ( set @ G3 ) @ A3 @ ( B3 @ X ) )
@ C5 ) )
= ( top_top @ ( set @ G3 ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G3 )
@ ( image @ H4 @ ( set @ G3 )
@ ^ [X: H4] : ( minus_minus @ ( set @ G3 ) @ A3 @ ( B3 @ X ) )
@ C5 ) )
= ( minus_minus @ ( set @ G3 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image @ H4 @ ( set @ G3 ) @ B3 @ C5 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_5193_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_5194_UNIV__I,axiom,
! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_5195_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_5196_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_5197_Int__UNIV,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( top_top @ ( set @ A ) ) )
= ( ( A3
= ( top_top @ ( set @ A ) ) )
& ( B3
= ( top_top @ ( set @ A ) ) ) ) ) ).
% Int_UNIV
thf(fact_5198_range__mult,axiom,
! [A2: real] :
( ( ( A2
= ( zero_zero @ real ) )
=> ( ( image @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) )
= ( insert @ real @ ( zero_zero @ real ) @ ( bot_bot @ ( set @ real ) ) ) ) )
& ( ( A2
!= ( zero_zero @ real ) )
=> ( ( image @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) )
= ( top_top @ ( set @ real ) ) ) ) ) ).
% range_mult
thf(fact_5199_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X3 )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_5200_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_max @ A @ X3 @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_5201_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X3 )
= X3 ) ) ).
% min_top
thf(fact_5202_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_min @ A @ X3 @ ( top_top @ A ) )
= X3 ) ) ).
% min_top2
thf(fact_5203_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_5204_remdups__eq__nil__right__iff,axiom,
! [A: $tType,X3: list @ A] :
( ( ( nil @ A )
= ( remdups @ A @ X3 ) )
= ( X3
= ( nil @ A ) ) ) ).
% remdups_eq_nil_right_iff
thf(fact_5205_remdups__eq__nil__iff,axiom,
! [A: $tType,X3: list @ A] :
( ( ( remdups @ A @ X3 )
= ( nil @ A ) )
= ( X3
= ( nil @ A ) ) ) ).
% remdups_eq_nil_iff
thf(fact_5206_set__remdups,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( remdups @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_remdups
thf(fact_5207_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_5208_remdups__id__iff__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( remdups @ A @ Xs )
= Xs )
= ( distinct @ A @ Xs ) ) ).
% remdups_id_iff_distinct
thf(fact_5209_distinct__remdups,axiom,
! [A: $tType,Xs: list @ A] : ( distinct @ A @ ( remdups @ A @ Xs ) ) ).
% distinct_remdups
thf(fact_5210_Collect__const,axiom,
! [A: $tType,P: $o] :
( ( P
=> ( ( collect @ A
@ ^ [S8: A] : P )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P
=> ( ( collect @ A
@ ^ [S8: A] : P )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_5211_finite__Collect__not,axiom,
! [A: $tType,P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
~ ( P @ X ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_5212_range__abs__Nats,axiom,
( ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) )
= ( semiring_1_Nats @ int ) ) ).
% range_abs_Nats
thf(fact_5213_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: $o] :
( ( P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A4: A,B4: B] : P ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A4: A,B4: B] : P ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_5214_range__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% range_add
thf(fact_5215_surj__plus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_plus
thf(fact_5216_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( ( complete_Sup_Sup @ A @ A3 )
= ( top_top @ A ) )
= ( ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ A3 )
& ( ord_less @ A @ X @ Y2 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_5217_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.compl_one
thf(fact_5218_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.compl_zero
thf(fact_5219_ball__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( top_top @ ( set @ A ) ) )
=> ( P @ X ) ) )
= ( ! [X6: A] : ( P @ X6 ) ) ) ).
% ball_UNIV
thf(fact_5220_Inf__UNIV,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Inf_UNIV
thf(fact_5221_ccInf__empty,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% ccInf_empty
thf(fact_5222_Inf__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Inf_empty
thf(fact_5223_Diff__UNIV,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_5224_surj__fn,axiom,
! [A: $tType,F2: A > A,N2: nat] :
( ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_fn
thf(fact_5225_finite__compl,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_compl
thf(fact_5226_Gcd__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Gcd_UNIV
thf(fact_5227_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_5228_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A3: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) )
= ( top_top @ A ) )
= ( ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ? [Y2: B] :
( ( member @ B @ Y2 @ A3 )
& ( ord_less @ A @ X @ ( F2 @ Y2 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_5229_range__constant,axiom,
! [B: $tType,A: $tType,X3: A] :
( ( image @ B @ A
@ ^ [Uu3: B] : X3
@ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_5230_ccINF__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F2: B > A] :
( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% ccINF_empty
thf(fact_5231_INT__constant,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y2: B] : C2
@ A3 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y2: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_5232_Inf__atMostLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( top_top @ A ) @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X3 ) )
= ( bot_bot @ A ) ) ) ) ).
% Inf_atMostLessThan
thf(fact_5233_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set @ A,A3: A > ( set @ B ),B3: set @ B] :
( ( ( C5
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X ) @ B3 )
@ C5 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X ) @ B3 )
@ C5 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ C5 ) ) @ B3 ) ) ) ) ).
% INT_simps(1)
thf(fact_5234_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set @ D,A3: set @ C,B3: D > ( set @ C )] :
( ( ( C5
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image @ D @ ( set @ C )
@ ^ [X: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B3 @ X ) )
@ C5 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image @ D @ ( set @ C )
@ ^ [X: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B3 @ X ) )
@ C5 ) )
= ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D @ ( set @ C ) @ B3 @ C5 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_5235_INT__simps_I3_J,axiom,
! [E2: $tType,F: $tType,C5: set @ E2,A3: E2 > ( set @ F ),B3: set @ F] :
( ( ( C5
= ( bot_bot @ ( set @ E2 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E2 @ ( set @ F )
@ ^ [X: E2] : ( minus_minus @ ( set @ F ) @ ( A3 @ X ) @ B3 )
@ C5 ) )
= ( top_top @ ( set @ F ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ E2 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E2 @ ( set @ F )
@ ^ [X: E2] : ( minus_minus @ ( set @ F ) @ ( A3 @ X ) @ B3 )
@ C5 ) )
= ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E2 @ ( set @ F ) @ A3 @ C5 ) ) @ B3 ) ) ) ) ).
% INT_simps(3)
thf(fact_5236_UNIV__option__conv,axiom,
! [A: $tType] :
( ( top_top @ ( set @ ( option @ A ) ) )
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% UNIV_option_conv
thf(fact_5237_rangeI,axiom,
! [A: $tType,B: $tType,F2: B > A,X3: B] : ( member @ A @ ( F2 @ X3 ) @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_5238_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A,X3: B] :
( ( B2
= ( F2 @ X3 ) )
=> ( member @ A @ B2 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_5239_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A] :
( ( member @ A @ B2 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X5: B] :
( B2
!= ( F2 @ X5 ) ) ) ).
% rangeE
thf(fact_5240_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F2: C > A,G: B > C] :
( ( image @ B @ A
@ ^ [X: B] : ( F2 @ ( G @ X ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C @ A @ F2 @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_5241_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I6: set @ A,F5: A > ( filter @ B )] :
( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I6 )
=> ? [X2: A] :
( ( member @ A @ X2 @ I6 )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X2 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ I4 ) @ ( F5 @ J2 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F5 @ I6 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X: A] :
( ( member @ A @ X @ I6 )
& ( ( F5 @ X )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_5242_remdups_Osimps_I1_J,axiom,
! [A: $tType] :
( ( remdups @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remdups.simps(1)
thf(fact_5243_Int__UNIV__right,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= A3 ) ).
% Int_UNIV_right
thf(fact_5244_Int__UNIV__left,axiom,
! [A: $tType,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 )
= B3 ) ).
% Int_UNIV_left
thf(fact_5245_remdups__remdups,axiom,
! [A: $tType,Xs: list @ A] :
( ( remdups @ A @ ( remdups @ A @ Xs ) )
= ( remdups @ A @ Xs ) ) ).
% remdups_remdups
thf(fact_5246_UNIV__eq__I,axiom,
! [A: $tType,A3: set @ A] :
( ! [X5: A] : ( member @ A @ X5 @ A3 )
=> ( ( top_top @ ( set @ A ) )
= A3 ) ) ).
% UNIV_eq_I
thf(fact_5247_UNIV__witness,axiom,
! [A: $tType] :
? [X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_5248_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% top.extremum_strict
thf(fact_5249_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_5250_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X: A] : $true ) ) ).
% UNIV_def
thf(fact_5251_subset__UNIV,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_5252_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_5253_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_5254_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_5255_distinct__remdups__id,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( remdups @ A @ Xs )
= Xs ) ) ).
% distinct_remdups_id
thf(fact_5256_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_5257_atLeastAtMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( bounded_lattice @ A )
=> ! [X3: A,Y: A] :
( ( ( set_or1337092689740270186AtMost @ A @ X3 @ Y )
= ( top_top @ ( set @ A ) ) )
= ( ( X3
= ( bot_bot @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_5258_insert__UNIV,axiom,
! [A: $tType,X3: A] :
( ( insert @ A @ X3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_5259_Inf__filter__not__bot,axiom,
! [A: $tType,B3: set @ ( filter @ A )] :
( ! [X10: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X10 @ B3 )
=> ( ( finite_finite2 @ ( filter @ A ) @ X10 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ X10 )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ B3 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% Inf_filter_not_bot
thf(fact_5260_infinite__UNIV__int,axiom,
~ ( finite_finite2 @ int @ ( top_top @ ( set @ int ) ) ) ).
% infinite_UNIV_int
thf(fact_5261_nat__not__finite,axiom,
~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ).
% nat_not_finite
thf(fact_5262_infinite__UNIV__nat,axiom,
~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ).
% infinite_UNIV_nat
thf(fact_5263_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_5264_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_5265_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_5266_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_5267_finite__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_UNIV
thf(fact_5268_ex__new__if__finite,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ A3 )
=> ? [A5: A] :
~ ( member @ A @ A5 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_5269_infinite__UNIV__char__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% infinite_UNIV_char_0
thf(fact_5270_infinite__UNIV__listI,axiom,
! [A: $tType] :
~ ( finite_finite2 @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% infinite_UNIV_listI
thf(fact_5271_range__subsetD,axiom,
! [B: $tType,A: $tType,F2: B > A,B3: set @ A,I: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) @ B3 )
=> ( member @ A @ ( F2 @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_5272_perfect__space__class_OUNIV__not__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X3: A] :
( ( top_top @ ( set @ A ) )
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% perfect_space_class.UNIV_not_singleton
thf(fact_5273_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_5274_finite__range__Some,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_range_Some
thf(fact_5275_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_5276_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( ( finite_card @ A @ A3 )
= ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( A3
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_5277_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_5278_notin__range__Some,axiom,
! [A: $tType,X3: option @ A] :
( ( ~ ( member @ ( option @ A ) @ X3 @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
= ( X3
= ( none @ A ) ) ) ).
% notin_range_Some
thf(fact_5279_Inter__UNIV,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Inter_UNIV
thf(fact_5280_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_5281_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_5282_int__in__range__abs,axiom,
! [N2: nat] : ( member @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) ) ).
% int_in_range_abs
thf(fact_5283_Compl__eq__Diff__UNIV,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_5284_Inter__empty,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Inter_empty
thf(fact_5285_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F2: B > A] :
( ( collect @ A
@ ^ [U2: A] :
? [X: B] :
( U2
= ( F2 @ X ) ) )
= ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% full_SetCompr_eq
thf(fact_5286_finite__range__imageI,axiom,
! [C: $tType,A: $tType,B: $tType,G: B > A,F2: A > C] :
( ( finite_finite2 @ A @ ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ C
@ ( image @ B @ C
@ ^ [X: B] : ( F2 @ ( G @ X ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_imageI
thf(fact_5287_remove1__remdups,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( ( remove1 @ A @ X3 @ ( remdups @ A @ Xs ) )
= ( remdups @ A @ ( remove1 @ A @ X3 @ Xs ) ) ) ) ).
% remove1_remdups
thf(fact_5288_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_5289_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F2: B > A,A2: A,X3: B] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( F2 @ X3 )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_5290_INF__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A] :
( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% INF_empty
thf(fact_5291_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [Y2: B] : C2
@ A3 ) )
= ( top_top @ A ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [Y2: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% INF_constant
thf(fact_5292_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_5293_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_5294_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) ) @ ( image @ B @ A @ F2 @ ( uminus_uminus @ ( set @ B ) @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_5295_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_5296_Nats__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% Nats_def
thf(fact_5297_Collect__ex__eq,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ( collect @ A
@ ^ [X: A] :
? [X6: B] : ( P @ X @ X6 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y2: B] :
( collect @ A
@ ^ [X: A] : ( P @ X @ Y2 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% Collect_ex_eq
thf(fact_5298_INT__empty,axiom,
! [B: $tType,A: $tType,B3: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_5299_UN__UN__finite__eq,axiom,
! [A: $tType,A3: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [N: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_5300_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_5301_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_5302_UN__finite__subset,axiom,
! [A: $tType,A3: nat > ( set @ A ),C5: set @ A] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) @ C5 ) ) ).
% UN_finite_subset
thf(fact_5303_UNIV__nat__eq,axiom,
( ( top_top @ ( set @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UNIV_nat_eq
thf(fact_5304_UN__finite2__eq,axiom,
! [A: $tType,A3: nat > ( set @ A ),B3: nat > ( set @ A ),K: nat] :
( ! [N3: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_5305_INT__extend__simps_I3_J,axiom,
! [F: $tType,E2: $tType,C5: set @ E2,A3: E2 > ( set @ F ),B3: set @ F] :
( ( ( C5
= ( bot_bot @ ( set @ E2 ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E2 @ ( set @ F ) @ A3 @ C5 ) ) @ B3 )
= ( minus_minus @ ( set @ F ) @ ( top_top @ ( set @ F ) ) @ B3 ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ E2 ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E2 @ ( set @ F ) @ A3 @ C5 ) ) @ B3 )
= ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E2 @ ( set @ F )
@ ^ [X: E2] : ( minus_minus @ ( set @ F ) @ ( A3 @ X ) @ B3 )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_5306_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F2: B > A] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_5307_UN__finite2__subset,axiom,
! [A: $tType,A3: nat > ( set @ A ),B3: nat > ( set @ A ),K: nat] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_5308_range__mod,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( image @ nat @ nat
@ ^ [M2: nat] : ( modulo_modulo @ nat @ M2 @ N2 )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% range_mod
thf(fact_5309_cclfp__def,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( order_532582986084564980_cclfp @ A )
= ( ^ [F3: A > A] :
( complete_Sup_Sup @ A
@ ( image @ nat @ A
@ ^ [I3: nat] : ( compow @ ( A > A ) @ I3 @ F3 @ ( bot_bot @ A ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% cclfp_def
thf(fact_5310_root__def,axiom,
( root
= ( ^ [N: nat,X: real] :
( if @ real
@ ( N
= ( zero_zero @ nat ) )
@ ( zero_zero @ real )
@ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) )
@ ^ [Y2: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N ) )
@ X ) ) ) ) ).
% root_def
thf(fact_5311_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A6: set @ A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A6 ) ) ) ) ).
% Id_on_def
thf(fact_5312_Id__onI,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A3 ) ) ) ).
% Id_onI
thf(fact_5313_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_5314_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_5315_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_5316_less__filter__def,axiom,
! [A: $tType] :
( ( ord_less @ ( filter @ A ) )
= ( ^ [F9: filter @ A,F10: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F9 @ F10 )
& ~ ( ord_less_eq @ ( filter @ A ) @ F10 @ F9 ) ) ) ) ).
% less_filter_def
thf(fact_5317_Id__on__iff,axiom,
! [A: $tType,X3: A,Y: A,A3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( id_on @ A @ A3 ) )
= ( ( X3 = Y )
& ( member @ A @ X3 @ A3 ) ) ) ).
% Id_on_iff
thf(fact_5318_Id__on__eqI,axiom,
! [A: $tType,A2: A,B2: A,A3: set @ A] :
( ( A2 = B2 )
=> ( ( member @ A @ A2 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id_on @ A @ A3 ) ) ) ) ).
% Id_on_eqI
thf(fact_5319_Id__onE,axiom,
! [A: $tType,C2: product_prod @ A @ A,A3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A3 ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( C2
!= ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ) ).
% Id_onE
thf(fact_5320_these__insert__Some,axiom,
! [A: $tType,X3: A,A3: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( some @ A @ X3 ) @ A3 ) )
= ( insert @ A @ X3 @ ( these @ A @ A3 ) ) ) ).
% these_insert_Some
thf(fact_5321_DERIV__even__real__root,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N2 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N2 ) ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_5322_DERIV__real__root__generic,axiom,
! [N2: nat,X3: real,D6: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( X3
!= ( zero_zero @ real ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( D6
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( D6
= ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( D6
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ( has_field_derivative @ real @ ( root @ N2 ) @ D6 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_5323_top1I,axiom,
! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ).
% top1I
thf(fact_5324_top2I,axiom,
! [A: $tType,B: $tType,X3: A,Y: B] : ( top_top @ ( A > B > $o ) @ X3 @ Y ) ).
% top2I
thf(fact_5325_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_5326_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_5327_these__image__Some__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( these @ A @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) )
= A3 ) ).
% these_image_Some_eq
thf(fact_5328_these__insert__None,axiom,
! [A: $tType,A3: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( none @ A ) @ A3 ) )
= ( these @ A @ A3 ) ) ).
% these_insert_None
thf(fact_5329_DERIV__at__within__shift__lemma,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y: A,Z2: A,X3: A,S3: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z2 @ X3 ) @ ( image @ A @ A @ ( plus_plus @ A @ Z2 ) @ S3 ) ) )
=> ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F2 @ ( plus_plus @ A @ Z2 ) ) @ Y @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ).
% DERIV_at_within_shift_lemma
thf(fact_5330_DERIV__at__within__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y: A,Z2: A,X3: A,S3: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z2 @ X3 ) @ ( image @ A @ A @ ( plus_plus @ A @ Z2 ) @ S3 ) ) )
= ( has_field_derivative @ A
@ ^ [X: A] : ( F2 @ ( plus_plus @ A @ Z2 @ X ) )
@ Y
@ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ).
% DERIV_at_within_shift
thf(fact_5331_DERIV__neg__dec__right,axiom,
! [F2: real > real,L: real,X3: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_5332_DERIV__pos__inc__right,axiom,
! [F2: real > real,L: real,X3: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D3 )
=> ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_5333_DERIV__pos__inc__left,axiom,
! [F2: real > real,L: real,X3: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_5334_DERIV__neg__dec__left,axiom,
! [F2: real > real,L: real,X3: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D3 )
=> ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_5335_DERIV__neg__imp__decreasing,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
=> ( ( ord_less_eq @ real @ X5 @ B2 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_5336_DERIV__pos__imp__increasing,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
=> ( ( ord_less_eq @ real @ X5 @ B2 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) ) )
=> ( ord_less @ real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_5337_DERIV__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F11: A,X3: A,S: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_field_derivative @ A @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ) ).
% DERIV_subset
thf(fact_5338_has__field__derivative__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y: A,X3: A,S: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ) ).
% has_field_derivative_subset
thf(fact_5339_DERIV__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y: A,X3: A,Z2: A] :
( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ X3 @ Z2 ) @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A
@ ^ [X: A] : ( F2 @ ( plus_plus @ A @ X @ Z2 ) )
@ Y
@ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_shift
thf(fact_5340_at__discrete,axiom,
! [A: $tType] :
( ( topolo8865339358273720382pology @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [X: A,S7: set @ A] : ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% at_discrete
thf(fact_5341_DERIV__inverse_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X3: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( F2 @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( inverse_inverse @ A @ ( F2 @ X ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) @ D6 ) @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% DERIV_inverse'
thf(fact_5342_has__real__derivative__pos__inc__right,axiom,
! [F2: real > real,L: real,X3: real,S3: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( plus_plus @ real @ X3 @ H5 ) @ S3 )
=> ( ( ord_less @ real @ H5 @ D3 )
=> ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_5343_has__real__derivative__neg__dec__right,axiom,
! [F2: real > real,L: real,X3: real,S3: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S3 ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( plus_plus @ real @ X3 @ H5 ) @ S3 )
=> ( ( ord_less @ real @ H5 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_5344_has__real__derivative__neg__dec__left,axiom,
! [F2: real > real,L: real,X3: real,S3: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S3 ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( minus_minus @ real @ X3 @ H5 ) @ S3 )
=> ( ( ord_less @ real @ H5 @ D3 )
=> ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_5345_has__real__derivative__pos__inc__left,axiom,
! [F2: real > real,L: real,X3: real,S3: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( minus_minus @ real @ X3 @ H5 ) @ S3 )
=> ( ( ord_less @ real @ H5 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_5346_DERIV__const,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [K: A,F5: filter @ A] :
( has_field_derivative @ A
@ ^ [X: A] : K
@ ( zero_zero @ A )
@ F5 ) ) ).
% DERIV_const
thf(fact_5347_field__differentiable__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F11: A,F5: filter @ A,G: A > A,G7: A] :
( ( has_field_derivative @ A @ F2 @ F11 @ F5 )
=> ( ( has_field_derivative @ A @ G @ G7 @ F5 )
=> ( has_field_derivative @ A
@ ^ [Z6: A] : ( plus_plus @ A @ ( F2 @ Z6 ) @ ( G @ Z6 ) )
@ ( plus_plus @ A @ F11 @ G7 )
@ F5 ) ) ) ) ).
% field_differentiable_add
thf(fact_5348_DERIV__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X3: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( plus_plus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( plus_plus @ A @ D6 @ E5 )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% DERIV_add
thf(fact_5349_DERIV__mult_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X3: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X3 ) @ E5 ) @ ( times_times @ A @ D6 @ ( G @ X3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% DERIV_mult'
thf(fact_5350_DERIV__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Da: A,X3: A,S: set @ A,G: A > A,Db: A] :
( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( plus_plus @ A @ ( times_times @ A @ Da @ ( G @ X3 ) ) @ ( times_times @ A @ Db @ ( F2 @ X3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% DERIV_mult
thf(fact_5351_DERIV__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X3: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( G @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D6 @ ( G @ X3 ) ) @ ( times_times @ A @ ( F2 @ X3 ) @ E5 ) ) @ ( times_times @ A @ ( G @ X3 ) @ ( G @ X3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ).
% DERIV_divide
thf(fact_5352_in__these__eq,axiom,
! [A: $tType,X3: A,A3: set @ ( option @ A )] :
( ( member @ A @ X3 @ ( these @ A @ A3 ) )
= ( member @ ( option @ A ) @ ( some @ A @ X3 ) @ A3 ) ) ).
% in_these_eq
thf(fact_5353_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_5354_at__le,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,T2: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ord_less_eq @ ( filter @ A ) @ ( topolo174197925503356063within @ A @ X3 @ S ) @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ).
% at_le
thf(fact_5355_trivial__limit__at__left__real,axiom,
! [A: $tType] :
( ( ( dense_order @ A )
& ( no_bot @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X3: A] :
( ( topolo174197925503356063within @ A @ X3 @ ( set_ord_lessThan @ A @ X3 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_left_real
thf(fact_5356_MVT2,axiom,
! [A2: real,B2: real,F2: real > real,F11: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
=> ( ( ord_less_eq @ real @ X5 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( F11 @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( F11 @ Z3 ) ) ) ) ) ) ).
% MVT2
thf(fact_5357_DERIV__local__const,axiom,
! [F2: real > real,L: real,X3: real,D2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y4 ) ) @ D2 )
=> ( ( F2 @ X3 )
= ( F2 @ Y4 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_const
thf(fact_5358_DERIV__ln,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( inverse_inverse @ real @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln
thf(fact_5359_DERIV__cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K: A,Xa2: A] :
( has_field_derivative @ A
@ ^ [X: A] : ( cos @ A @ ( plus_plus @ A @ X @ K ) )
@ ( uminus_uminus @ A @ ( sin @ A @ ( plus_plus @ A @ Xa2 @ K ) ) )
@ ( topolo174197925503356063within @ A @ Xa2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% DERIV_cos_add
thf(fact_5360_DERIV__power__Suc,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X3: A,S: set @ A,N2: nat] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( power_power @ A @ ( F2 @ X ) @ ( suc @ N2 ) )
@ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( times_times @ A @ D6 @ ( power_power @ A @ ( F2 @ X3 ) @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ).
% DERIV_power_Suc
thf(fact_5361_DERIV__inverse,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X3: A,S: set @ A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ).
% DERIV_inverse
thf(fact_5362_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X3: A,S: set @ A,N2: nat] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( power_power @ A @ ( F2 @ X ) @ N2 )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( times_times @ A @ D6 @ ( power_power @ A @ ( F2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ).
% DERIV_power
thf(fact_5363_DERIV__local__max,axiom,
! [F2: real > real,L: real,X3: real,D2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y4 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_max
thf(fact_5364_DERIV__local__min,axiom,
! [F2: real > real,L: real,X3: real,D2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y4 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_min
thf(fact_5365_DERIV__ln__divide,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln_divide
thf(fact_5366_DERIV__pow,axiom,
! [N2: nat,X3: real,S: set @ real] :
( has_field_derivative @ real
@ ^ [X: real] : ( power_power @ real @ X @ N2 )
@ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ X3 @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ S ) ) ).
% DERIV_pow
thf(fact_5367_at__within__Icc__at,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,X3: A,B2: A] :
( ( ord_less @ A @ A2 @ X3 )
=> ( ( ord_less @ A @ X3 @ B2 )
=> ( ( topolo174197925503356063within @ A @ X3 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% at_within_Icc_at
thf(fact_5368_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_5369_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_5370_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D2: A,X3: A,S: set @ A,G: A > A,E3: A] :
( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( has_field_derivative @ A @ G @ E3 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( G @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y2: A] : ( divide_divide @ A @ ( F2 @ Y2 ) @ ( G @ Y2 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D2 @ ( G @ X3 ) ) @ ( times_times @ A @ E3 @ ( F2 @ X3 ) ) ) @ ( power_power @ A @ ( G @ X3 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ).
% DERIV_quotient
thf(fact_5371_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D2: A,X3: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( F2 @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( inverse_inverse @ A @ ( F2 @ X ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ D2 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F2 @ X3 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_5372_termdiffs__sums__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K5: real,C2: nat > A,F2: A > A,F11: A,Z2: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 )
=> ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ Z3 @ N ) )
@ ( F2 @ Z3 ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F11 @ ( topolo174197925503356063within @ A @ Z2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K5 )
=> ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ Z2 @ N ) )
@ F11 ) ) ) ) ) ).
% termdiffs_sums_strong
thf(fact_5373_has__real__derivative__powr,axiom,
! [Z2: real,R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Z2 )
=> ( has_field_derivative @ real
@ ^ [Z6: real] : ( powr @ real @ Z6 @ R2 )
@ ( times_times @ real @ R2 @ ( powr @ real @ Z2 @ ( minus_minus @ real @ R2 @ ( one_one @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% has_real_derivative_powr
thf(fact_5374_termdiffs__strong_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K5: real,C2: nat > A,Z2: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 )
=> ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ Z3 @ N ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K5 )
=> ( has_field_derivative @ A
@ ^ [Z6: A] :
( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ Z6 @ N ) ) )
@ ( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) )
@ ( topolo174197925503356063within @ A @ Z2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong'
thf(fact_5375_termdiffs__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K5: A,X3: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ K5 @ N ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] :
( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ X @ N ) ) )
@ ( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong
thf(fact_5376_termdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K5: A,X3: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ K5 @ N ) ) )
=> ( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ K5 @ N ) ) )
=> ( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N ) @ ( power_power @ A @ K5 @ N ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] :
( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ X @ N ) ) )
@ ( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% termdiffs
thf(fact_5377_DERIV__log,axiom,
! [X3: real,B2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( has_field_derivative @ real @ ( log @ B2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( times_times @ real @ ( ln_ln @ real @ B2 ) @ X3 ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_log
thf(fact_5378_DERIV__fun__powr,axiom,
! [G: real > real,M: real,X3: real,R2: real] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) )
=> ( has_field_derivative @ real
@ ^ [X: real] : ( powr @ real @ ( G @ X ) @ R2 )
@ ( times_times @ real @ ( times_times @ real @ R2 @ ( powr @ real @ ( G @ X3 ) @ ( minus_minus @ real @ R2 @ ( semiring_1_of_nat @ real @ ( one_one @ nat ) ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_fun_powr
thf(fact_5379_DERIV__powr,axiom,
! [G: real > real,M: real,X3: real,F2: real > real,R2: real] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) )
=> ( ( has_field_derivative @ real @ F2 @ R2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X: real] : ( powr @ real @ ( G @ X ) @ ( F2 @ X ) )
@ ( times_times @ real @ ( powr @ real @ ( G @ X3 ) @ ( F2 @ X3 ) ) @ ( plus_plus @ real @ ( times_times @ real @ R2 @ ( ln_ln @ real @ ( G @ X3 ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ M @ ( F2 @ X3 ) ) @ ( G @ X3 ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_powr
thf(fact_5380_DERIV__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( tan @ A ) @ ( inverse_inverse @ A @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_tan
thf(fact_5381_arcosh__real__has__field__derivative,axiom,
! [X3: real,A3: set @ real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( has_field_derivative @ real @ ( arcosh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ A3 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_5382_DERIV__real__sqrt,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( has_field_derivative @ real @ sqrt @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_real_sqrt
thf(fact_5383_DERIV__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( sin @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( cot @ A ) @ ( uminus_uminus @ A @ ( inverse_inverse @ A @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_cot
thf(fact_5384_Option_Othese__def,axiom,
! [A: $tType] :
( ( these @ A )
= ( ^ [A6: set @ ( option @ A )] :
( image @ ( option @ A ) @ A @ ( the2 @ A )
@ ( collect @ ( option @ A )
@ ^ [X: option @ A] :
( ( member @ ( option @ A ) @ X @ A6 )
& ( X
!= ( none @ A ) ) ) ) ) ) ) ).
% Option.these_def
thf(fact_5385_has__field__derivative__tanh,axiom,
! [A10: $tType] :
( ( ( real_Vector_banach @ A10 )
& ( real_V3459762299906320749_field @ A10 ) )
=> ! [G: A10 > A10,X3: A10,Db: A10,S: set @ A10] :
( ( ( cosh @ A10 @ ( G @ X3 ) )
!= ( zero_zero @ A10 ) )
=> ( ( has_field_derivative @ A10 @ G @ Db @ ( topolo174197925503356063within @ A10 @ X3 @ S ) )
=> ( has_field_derivative @ A10
@ ^ [X: A10] : ( tanh @ A10 @ ( G @ X ) )
@ ( times_times @ A10 @ ( minus_minus @ A10 @ ( one_one @ A10 ) @ ( power_power @ A10 @ ( tanh @ A10 @ ( G @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A10 @ X3 @ S ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_5386_DERIV__real__sqrt__generic,axiom,
! [X3: real,D6: real] :
( ( X3
!= ( zero_zero @ real ) )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( D6
= ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( D6
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( has_field_derivative @ real @ sqrt @ D6 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_5387_artanh__real__has__field__derivative,axiom,
! [X3: real,A3: set @ real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ ( artanh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ A3 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_5388_DERIV__real__root,axiom,
! [N2: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( has_field_derivative @ real @ ( root @ N2 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_real_root
thf(fact_5389_DERIV__arcsin,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arcsin @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arcsin
thf(fact_5390_these__not__empty__eq,axiom,
! [A: $tType,B3: set @ ( option @ A )] :
( ( ( these @ A @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B3
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B3
!= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_5391_these__empty__eq,axiom,
! [A: $tType,B3: set @ ( option @ A )] :
( ( ( these @ A @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B3
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B3
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_5392_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F2: real > real,X3: real,N2: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
& ! [M3: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) )
& ( ( F2 @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X3 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_5393_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F2: real > real,X3: real,N2: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) )
& ( ( F2 @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X3 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_5394_Some__image__these__eq,axiom,
! [A: $tType,A3: set @ ( option @ A )] :
( ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A3 ) )
= ( collect @ ( option @ A )
@ ^ [X: option @ A] :
( ( member @ ( option @ A ) @ X @ A3 )
& ( X
!= ( none @ A ) ) ) ) ) ).
% Some_image_these_eq
thf(fact_5395_DERIV__odd__real__root,axiom,
! [N2: nat,X3: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( X3
!= ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N2 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_5396_Maclaurin__minus,axiom,
! [H2: real,N2: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ H2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ H2 @ T7 )
& ( ord_less_eq @ real @ T7 @ ( zero_zero @ real ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ H2 @ T7 )
& ( ord_less @ real @ T7 @ ( zero_zero @ real ) )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_5397_Maclaurin2,axiom,
! [H2: real,Diff: nat > real > real,F2: real > real,N2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ H2 @ N2 ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_5398_Maclaurin,axiom,
! [H2: real,N2: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ H2 )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_5399_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F2: real > real,N2: nat,X3: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( X3
!= ( zero_zero @ real ) )
=> ( ! [M3: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( 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 @ X3 ) )
& ( ( F2 @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X3 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_5400_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F2: real > real,N2: nat,X3: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) )
& ( ( F2 @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X3 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_5401_Taylor,axiom,
! [N2: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ A2 @ T7 )
& ( ord_less_eq @ real @ T7 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B2 )
=> ( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ( ( X3 != C2 )
=> ? [T7: real] :
( ( ( ord_less @ real @ X3 @ C2 )
=> ( ( ord_less @ real @ X3 @ T7 )
& ( ord_less @ real @ T7 @ C2 ) ) )
& ( ~ ( ord_less @ real @ X3 @ C2 )
=> ( ( ord_less @ real @ C2 @ T7 )
& ( ord_less @ real @ T7 @ X3 ) ) )
& ( ( F2 @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ C2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X3 @ C2 ) @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X3 @ C2 ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_5402_Taylor__up,axiom,
! [N2: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ A2 @ T7 )
& ( ord_less_eq @ real @ T7 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ C2 )
=> ( ( ord_less @ real @ C2 @ B2 )
=> ? [T7: real] :
( ( ord_less @ real @ C2 @ T7 )
& ( ord_less @ real @ T7 @ B2 )
& ( ( F2 @ B2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ C2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ N2 ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_5403_Taylor__down,axiom,
! [N2: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ A2 @ T7 )
& ( ord_less_eq @ real @ T7 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less @ real @ A2 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B2 )
=> ? [T7: real] :
( ( ord_less @ real @ A2 @ T7 )
& ( ord_less @ real @ T7 @ C2 )
& ( ( F2 @ A2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ C2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ M2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ N2 ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_5404_Maclaurin__lemma2,axiom,
! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B3: real] :
( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( N2
= ( suc @ K ) )
=> ! [M6: nat,T9: real] :
( ( ( ord_less @ nat @ M6 @ N2 )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T9 )
& ( ord_less_eq @ real @ T9 @ H2 ) )
=> ( has_field_derivative @ real
@ ^ [U2: real] :
( minus_minus @ real @ ( Diff @ M6 @ U2 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M6 @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ U2 @ P6 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ M6 ) ) )
@ ( times_times @ real @ B3 @ ( divide_divide @ real @ ( power_power @ real @ U2 @ ( minus_minus @ nat @ N2 @ M6 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N2 @ M6 ) ) ) ) ) )
@ ( minus_minus @ real @ ( Diff @ ( suc @ M6 ) @ T9 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M6 ) @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ T9 @ P6 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ M6 ) ) ) )
@ ( times_times @ real @ B3 @ ( divide_divide @ real @ ( power_power @ real @ T9 @ ( minus_minus @ nat @ N2 @ ( suc @ M6 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N2 @ ( suc @ M6 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ T9 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_5405_DERIV__arctan__series,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real
@ ^ [X9: real] :
( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X9 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) )
@ ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( power_power @ real @ X3 @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_arctan_series
thf(fact_5406_DERIV__arccos,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arccos @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arccos
thf(fact_5407_DERIV__power__series_H,axiom,
! [R: real,F2: nat > real,X0: real] :
( ! [X5: real] :
( ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( summable @ real
@ ^ [N: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N ) @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) @ ( power_power @ real @ X5 @ N ) ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( has_field_derivative @ real
@ ^ [X: real] :
( suminf @ real
@ ^ [N: nat] : ( times_times @ real @ ( F2 @ N ) @ ( power_power @ real @ X @ ( suc @ N ) ) ) )
@ ( suminf @ real
@ ^ [N: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N ) @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) @ ( power_power @ real @ X0 @ N ) ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_5408_has__derivative__arcsin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X3: A,G7: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X3 ) )
=> ( ( ord_less @ real @ ( G @ X3 ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G7 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( arcsin @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G7 @ X ) @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ).
% has_derivative_arcsin
thf(fact_5409_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_5410_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_5411_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_5412_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K: A] :
( ( ord_less_eq @ A @ L @ K )
=> ( ( set_or5935395276787703475ssThan @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanLessThan_empty
thf(fact_5413_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_5414_cSup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ X3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X3 ) )
= X3 ) ) ) ).
% cSup_greaterThanLessThan
thf(fact_5415_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X3 @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanLessThan
thf(fact_5416_cInf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X3 ) )
= Y ) ) ) ).
% cInf_greaterThanLessThan
thf(fact_5417_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X3 @ Y ) )
= X3 ) ) ) ).
% Inf_greaterThanLessThan
thf(fact_5418_has__derivative__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F11: A > B,X3: A,S: set @ A,T2: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ) ).
% has_derivative_subset
thf(fact_5419_has__derivative__scaleR,axiom,
! [C: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ D )
& ( real_V822414075346904944vector @ C ) )
=> ! [F2: D > real,F11: D > real,X3: D,S: set @ D,G: D > C,G7: D > C] :
( ( has_derivative @ D @ real @ F2 @ F11 @ ( topolo174197925503356063within @ D @ X3 @ S ) )
=> ( ( has_derivative @ D @ C @ G @ G7 @ ( topolo174197925503356063within @ D @ X3 @ S ) )
=> ( has_derivative @ D @ C
@ ^ [X: D] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [H: D] : ( plus_plus @ C @ ( real_V8093663219630862766scaleR @ C @ ( F2 @ X3 ) @ ( G7 @ H ) ) @ ( real_V8093663219630862766scaleR @ C @ ( F11 @ H ) @ ( G @ X3 ) ) )
@ ( topolo174197925503356063within @ D @ X3 @ S ) ) ) ) ) ).
% has_derivative_scaleR
thf(fact_5420_has__derivative__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F11: A > B,F5: filter @ A,G: A > B,G7: A > B] :
( ( has_derivative @ A @ B @ F2 @ F11 @ F5 )
=> ( ( has_derivative @ A @ B @ G @ G7 @ F5 )
=> ( has_derivative @ A @ B
@ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [X: A] : ( plus_plus @ B @ ( F11 @ X ) @ ( G7 @ X ) )
@ F5 ) ) ) ) ).
% has_derivative_add
thf(fact_5421_has__derivative__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [C2: B,F5: filter @ A] :
( has_derivative @ A @ B
@ ^ [X: A] : C2
@ ^ [X: A] : ( zero_zero @ B )
@ F5 ) ) ).
% has_derivative_const
thf(fact_5422_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_5423_has__derivative__zero__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F5: A > B,X3: A] :
( ( has_derivative @ A @ B
@ ^ [X: A] : ( zero_zero @ B )
@ F5
@ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
=> ( F5
= ( ^ [H: A] : ( zero_zero @ B ) ) ) ) ) ).
% has_derivative_zero_unique
thf(fact_5424_has__derivative__mult,axiom,
! [A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ D )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F2: D > A,F11: D > A,X3: D,S: set @ D,G: D > A,G7: D > A] :
( ( has_derivative @ D @ A @ F2 @ F11 @ ( topolo174197925503356063within @ D @ X3 @ S ) )
=> ( ( has_derivative @ D @ A @ G @ G7 @ ( topolo174197925503356063within @ D @ X3 @ S ) )
=> ( has_derivative @ D @ A
@ ^ [X: D] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [H: D] : ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X3 ) @ ( G7 @ H ) ) @ ( times_times @ A @ ( F11 @ H ) @ ( G @ X3 ) ) )
@ ( topolo174197925503356063within @ D @ X3 @ S ) ) ) ) ) ).
% has_derivative_mult
thf(fact_5425_has__derivative__in__compose2,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [T2: set @ A,G: A > B,G7: A > A > B,F2: C > A,S: set @ C,X3: C,F11: C > A] :
( ! [X5: A] :
( ( member @ A @ X5 @ T2 )
=> ( has_derivative @ A @ B @ G @ ( G7 @ X5 ) @ ( topolo174197925503356063within @ A @ X5 @ T2 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ S ) @ T2 )
=> ( ( member @ C @ X3 @ S )
=> ( ( has_derivative @ C @ A @ F2 @ F11 @ ( topolo174197925503356063within @ C @ X3 @ S ) )
=> ( has_derivative @ C @ B
@ ^ [X: C] : ( G @ ( F2 @ X ) )
@ ^ [Y2: C] : ( G7 @ ( F2 @ X3 ) @ ( F11 @ Y2 ) )
@ ( topolo174197925503356063within @ C @ X3 @ S ) ) ) ) ) ) ) ).
% has_derivative_in_compose2
thf(fact_5426_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_5427_ivl__disj__int__two_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(4)
thf(fact_5428_ivl__disj__int__two_I5_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(5)
thf(fact_5429_ivl__disj__int__two_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(1)
thf(fact_5430_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_5431_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_5432_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_5433_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_5434_has__derivative__divide_H,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,F11: C > A,X3: C,S3: set @ C,G: C > A,G7: C > A] :
( ( has_derivative @ C @ A @ F2 @ F11 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) )
=> ( ( has_derivative @ C @ A @ G @ G7 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) )
=> ( ( ( G @ X3 )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X: C] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [H: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F11 @ H ) @ ( G @ X3 ) ) @ ( times_times @ A @ ( F2 @ X3 ) @ ( G7 @ H ) ) ) @ ( times_times @ A @ ( G @ X3 ) @ ( G @ X3 ) ) )
@ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_5435_arccos__less__arccos,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arccos @ Y ) @ ( arccos @ X3 ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_5436_arccos__less__mono,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arccos @ X3 ) @ ( arccos @ Y ) )
= ( ord_less @ real @ Y @ X3 ) ) ) ) ).
% arccos_less_mono
thf(fact_5437_has__derivative__inverse,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,X3: C,F11: C > A,S3: set @ C] :
( ( ( F2 @ X3 )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F2 @ F11 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) )
=> ( has_derivative @ C @ A
@ ^ [X: C] : ( inverse_inverse @ A @ ( F2 @ X ) )
@ ^ [H: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) @ ( F11 @ H ) ) @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) ) )
@ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) ) ) ) ).
% has_derivative_inverse
thf(fact_5438_has__derivative__inverse_H,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X3: A,S3: set @ A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A @ ( inverse_inverse @ A )
@ ^ [H: A] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ X3 ) @ H ) @ ( inverse_inverse @ A @ X3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ).
% has_derivative_inverse'
thf(fact_5439_DERIV__isconst3,axiom,
! [A2: real,B2: real,X3: real,Y: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( member @ real @ Y @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ! [X5: real] :
( ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( F2 @ X3 )
= ( F2 @ Y ) ) ) ) ) ) ).
% DERIV_isconst3
thf(fact_5440_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_5441_has__derivative__ln,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X3: A,G7: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) )
=> ( ( has_derivative @ A @ real @ G @ G7 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( ln_ln @ real @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G7 @ X ) @ ( inverse_inverse @ real @ ( G @ X3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% has_derivative_ln
thf(fact_5442_sin__arccos__nonzero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X3 ) )
!= ( zero_zero @ real ) ) ) ) ).
% sin_arccos_nonzero
thf(fact_5443_has__derivative__arccos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X3: A,G7: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X3 ) )
=> ( ( ord_less @ real @ ( G @ X3 ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G7 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( arccos @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G7 @ X ) @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ).
% has_derivative_arccos
thf(fact_5444_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,F11: C > A,X3: C,S3: set @ C,G: C > A,G7: C > A] :
( ( has_derivative @ C @ A @ F2 @ F11 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) )
=> ( ( has_derivative @ C @ A @ G @ G7 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) )
=> ( ( ( G @ X3 )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X: C] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [H: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F2 @ X3 ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G @ X3 ) ) @ ( G7 @ H ) ) @ ( inverse_inverse @ A @ ( G @ X3 ) ) ) ) @ ( divide_divide @ A @ ( F11 @ H ) @ ( G @ X3 ) ) )
@ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_5445_has__derivative__prod,axiom,
! [B: $tType,I7: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [I6: set @ I7,F2: I7 > A > B,F11: I7 > A > B,X3: A,S3: set @ A] :
( ! [I4: I7] :
( ( member @ I7 @ I4 @ I6 )
=> ( has_derivative @ A @ B @ ( F2 @ I4 ) @ ( F11 @ I4 ) @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) )
=> ( has_derivative @ A @ B
@ ^ [X: A] :
( groups7121269368397514597t_prod @ I7 @ B
@ ^ [I3: I7] : ( F2 @ I3 @ X )
@ I6 )
@ ^ [Y2: A] :
( groups7311177749621191930dd_sum @ I7 @ B
@ ^ [I3: I7] :
( times_times @ B @ ( F11 @ I3 @ Y2 )
@ ( groups7121269368397514597t_prod @ I7 @ B
@ ^ [J3: I7] : ( F2 @ J3 @ X3 )
@ ( minus_minus @ ( set @ I7 ) @ I6 @ ( insert @ I7 @ I3 @ ( bot_bot @ ( set @ I7 ) ) ) ) ) )
@ I6 )
@ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ).
% has_derivative_prod
thf(fact_5446_has__derivative__powr,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G7: A > real,X3: A,X8: set @ A,F2: A > real,F11: A > real] :
( ( has_derivative @ A @ real @ G @ G7 @ ( topolo174197925503356063within @ A @ X3 @ X8 ) )
=> ( ( has_derivative @ A @ real @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ X8 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) )
=> ( ( member @ A @ X3 @ X8 )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( powr @ real @ ( G @ X ) @ ( F2 @ X ) )
@ ^ [H: A] : ( times_times @ real @ ( powr @ real @ ( G @ X3 ) @ ( F2 @ X3 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( F11 @ H ) @ ( ln_ln @ real @ ( G @ X3 ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G7 @ H ) @ ( F2 @ X3 ) ) @ ( G @ X3 ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ X8 ) ) ) ) ) ) ) ).
% has_derivative_powr
thf(fact_5447_has__derivative__real__sqrt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X3: A,G7: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) )
=> ( ( has_derivative @ A @ real @ G @ G7 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( sqrt @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G7 @ X ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G @ X3 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% has_derivative_real_sqrt
thf(fact_5448_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K5: A,X3: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N ) @ ( power_power @ A @ K5 @ N ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( filterlim @ A @ A
@ ^ [H: A] :
( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X3 @ H ) @ N ) @ ( power_power @ A @ X3 @ N ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ X3 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_aux
thf(fact_5449_isCont__powser,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K5: A,X3: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ K5 @ N ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] :
( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ) ) ).
% isCont_powser
thf(fact_5450_isCont__powser_H,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ Aa )
& ( real_V3459762299906320749_field @ Aa ) )
=> ! [A2: A,F2: A > Aa,C2: nat > Aa,K5: Aa] :
( ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( summable @ Aa
@ ^ [N: nat] : ( times_times @ Aa @ ( C2 @ N ) @ ( power_power @ Aa @ K5 @ N ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ Aa @ ( F2 @ A2 ) ) @ ( real_V7770717601297561774m_norm @ Aa @ K5 ) )
=> ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] :
( suminf @ Aa
@ ^ [N: nat] : ( times_times @ Aa @ ( C2 @ N ) @ ( power_power @ Aa @ ( F2 @ X ) @ N ) ) ) ) ) ) ) ) ).
% isCont_powser'
thf(fact_5451_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_5452_finite__greaterThanLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% finite_greaterThanLessThan_int
thf(fact_5453_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_5454_tendsto__mult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: B > A,L: A,F5: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X: B] : ( times_times @ A @ C2 @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L ) )
@ F5 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% tendsto_mult_left_iff
thf(fact_5455_tendsto__mult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: B > A,L: A,F5: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C2 ) )
@ F5 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% tendsto_mult_right_iff
thf(fact_5456_power__tendsto__0__iff,axiom,
! [A: $tType,N2: nat,F2: A > real,F5: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( filterlim @ A @ real
@ ^ [X: A] : ( power_power @ real @ ( F2 @ X ) @ N2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ).
% power_tendsto_0_iff
thf(fact_5457_isCont__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F2: A > B,G: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% isCont_Pair
thf(fact_5458_isCont__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F2: A > B,G: B > C,L: C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ L ) @ ( topolo174197925503356063within @ B @ ( F2 @ A2 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X5: A] :
( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X5 @ A2 ) ) @ D4 ) )
=> ( ( F2 @ X5 )
!= ( F2 @ A2 ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ C @ L )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% isCont_LIM_compose2
thf(fact_5459_isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X3: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ X3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X3 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% isCont_iff
thf(fact_5460_LIM__not__zero,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( zero @ Aa )
& ( topological_t2_space @ Aa ) )
=> ! [K: Aa,A2: A] :
( ( K
!= ( zero_zero @ Aa ) )
=> ~ ( filterlim @ A @ Aa
@ ^ [X: A] : K
@ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_not_zero
thf(fact_5461_tendsto__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: A > B,L: filter @ B,X3: A,S3: set @ A,T4: set @ A] :
( ( filterlim @ A @ B @ F2 @ L @ ( topolo174197925503356063within @ A @ X3 @ S3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 )
=> ( filterlim @ A @ B @ F2 @ L @ ( topolo174197925503356063within @ A @ X3 @ T4 ) ) ) ) ) ).
% tendsto_within_subset
thf(fact_5462_tendsto__arcosh,axiom,
! [B: $tType,F2: B > real,A2: real,F5: filter @ B] :
( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( filterlim @ B @ real
@ ^ [X: B] : ( arcosh @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F5 ) ) ) ).
% tendsto_arcosh
thf(fact_5463_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
@ ^ [X: B] : A2
@ ( topolo7230453075368039082e_nhds @ A @ B2 )
@ F5 )
= ( A2 = B2 ) ) ) ) ).
% tendsto_const_iff
thf(fact_5464_tendsto__unique,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ B,F2: B > A,A2: A,B2: A] :
( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 )
=> ( A2 = B2 ) ) ) ) ) ).
% tendsto_unique
thf(fact_5465_tendsto__bot,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,A2: A] : ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( bot_bot @ ( filter @ B ) ) ) ) ).
% tendsto_bot
thf(fact_5466_nhds__neq__bot,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A] :
( ( topolo7230453075368039082e_nhds @ A @ A2 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% nhds_neq_bot
thf(fact_5467_continuous__bot,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B] : ( topolo3448309680560233919inuous @ A @ B @ ( bot_bot @ ( filter @ A ) ) @ F2 ) ) ).
% continuous_bot
thf(fact_5468_continuous__trivial__limit,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [Net: filter @ A,F2: A > B] :
( ( Net
= ( bot_bot @ ( filter @ A ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ Net @ F2 ) ) ) ).
% continuous_trivial_limit
thf(fact_5469_LIM__offset__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A2: A,L5: B] :
( ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero_cancel
thf(fact_5470_LIM__offset__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L5: B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero
thf(fact_5471_LIM__isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_isCont_iff
thf(fact_5472_tendsto__null__power,axiom,
! [B: $tType,A: $tType] :
( ( real_V2822296259951069270ebra_1 @ B )
=> ! [F2: A > B,F5: filter @ A,N2: nat] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( filterlim @ A @ B
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N2 )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% tendsto_null_power
thf(fact_5473_LIM__offset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L5: B,A2: A,K: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X: A] : ( F2 @ ( plus_plus @ A @ X @ K ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ K ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset
thf(fact_5474_tendsto__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,A2: B,F5: filter @ A,G: A > C,B2: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F5 )
=> ( ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ B2 ) @ F5 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ B2 ) )
@ F5 ) ) ) ) ).
% tendsto_Pair
thf(fact_5475_continuous__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F5: filter @ A,F2: A > B,G: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ F5 @ G )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F5
@ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_Pair
thf(fact_5476_Lim__transform__eq,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,G: B > A,F5: filter @ B,A2: A] :
( ( filterlim @ B @ A
@ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
= ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) ) ) ) ).
% Lim_transform_eq
thf(fact_5477_LIM__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ).
% LIM_zero_cancel
thf(fact_5478_Lim__transform2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,A2: A,F5: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( filterlim @ B @ A
@ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) ) ) ) ).
% Lim_transform2
thf(fact_5479_Lim__transform,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: B > A,A2: A,F5: filter @ B,F2: B > A] :
( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( filterlim @ B @ A
@ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) ) ) ) ).
% Lim_transform
thf(fact_5480_LIM__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ).
% LIM_zero_iff
thf(fact_5481_LIM__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( filterlim @ A @ B
@ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ).
% LIM_zero
thf(fact_5482_tendsto__mult__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D > A,F5: filter @ D,G: D > A] :
( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( ( filterlim @ D @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X: D] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_mult_zero
thf(fact_5483_tendsto__mult__left__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D > A,F5: filter @ D,C2: A] :
( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X: D] : ( times_times @ A @ ( F2 @ X ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_mult_left_zero
thf(fact_5484_tendsto__mult__right__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D > A,F5: filter @ D,C2: A] :
( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X: D] : ( times_times @ A @ C2 @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_mult_right_zero
thf(fact_5485_tendsto__add__zero,axiom,
! [B: $tType,D: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F2: D > B,F5: filter @ D,G: D > B] :
( ( filterlim @ D @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( filterlim @ D @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ D @ B
@ ^ [X: D] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% tendsto_add_zero
thf(fact_5486_tendsto__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A2: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( inverse_inverse @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_inverse
thf(fact_5487_tendsto__norm__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 ) ) ) ).
% tendsto_norm_zero
thf(fact_5488_tendsto__norm__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ).
% tendsto_norm_zero_iff
thf(fact_5489_tendsto__norm__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ).
% tendsto_norm_zero_cancel
thf(fact_5490_tendsto__sgn,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,L: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
=> ( ( L
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( sgn_sgn @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L ) )
@ F5 ) ) ) ) ).
% tendsto_sgn
thf(fact_5491_tendsto__divide__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,F5: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( divide_divide @ A @ ( F2 @ X ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_divide_zero
thf(fact_5492_tendsto__divide,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,A2: A,F5: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A2 @ B2 ) )
@ F5 ) ) ) ) ) ).
% tendsto_divide
thf(fact_5493_continuous__add,axiom,
! [B: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [F5: filter @ D,F2: D > B,G: D > B] :
( ( topolo3448309680560233919inuous @ D @ B @ F5 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D @ B @ F5 @ G )
=> ( topolo3448309680560233919inuous @ D @ B @ F5
@ ^ [X: D] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_add
thf(fact_5494_tendsto__add,axiom,
! [A: $tType,B: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [F2: B > A,A2: A,F5: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A2 @ B2 ) )
@ F5 ) ) ) ) ).
% tendsto_add
thf(fact_5495_tendsto__add__const__iff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [C2: A,F2: B > A,D2: A,F5: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ C2 @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ C2 @ D2 ) )
@ F5 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ D2 ) @ F5 ) ) ) ).
% tendsto_add_const_iff
thf(fact_5496_tendsto__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: A > A,A2: A,F5: filter @ A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( ( cos @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X: A] : ( tan @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tan @ A @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_tan
thf(fact_5497_tendsto__null__sum,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I6: set @ B,F2: A > B > C,F5: filter @ A] :
( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( F2 @ X @ I4 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) )
=> ( filterlim @ A @ C
@ ^ [I3: A] : ( groups7311177749621191930dd_sum @ B @ C @ ( F2 @ I3 ) @ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) ) ) ).
% tendsto_null_sum
thf(fact_5498_tendsto__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,A2: A,F5: filter @ C] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( ( cosh @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ C @ A
@ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tanh @ A @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_tanh
thf(fact_5499_tendsto__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: A > A,A2: A,F5: filter @ A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( ( sin @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X: A] : ( cot @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cot @ A @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_cot
thf(fact_5500_tendsto__log,axiom,
! [A: $tType,F2: A > real,A2: real,F5: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( filterlim @ A @ real @ G @ ( 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
@ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( log @ A2 @ B2 ) )
@ F5 ) ) ) ) ) ) ).
% tendsto_log
thf(fact_5501_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
= ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_5502_tendsto__artanh,axiom,
! [A: $tType,F2: A > real,A2: real,F5: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( 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
@ ^ [X: A] : ( artanh @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( artanh @ real @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_artanh
thf(fact_5503_LIM__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A2: A,F2: A > D,L5: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( filterlim @ A @ D @ F2 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ D
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_offset_zero_iff
thf(fact_5504_IVT,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,A2: A,Y: B,B2: A] :
( ( ord_less_eq @ B @ ( F2 @ A2 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ! [X5: A] :
( ( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) @ F2 ) )
=> ? [X5: A] :
( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 )
& ( ( F2 @ X5 )
= Y ) ) ) ) ) ) ) ).
% IVT
thf(fact_5505_IVT2,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,B2: A,Y: B,A2: A] :
( ( ord_less_eq @ B @ ( F2 @ B2 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F2 @ A2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ! [X5: A] :
( ( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) @ F2 ) )
=> ? [X5: A] :
( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 )
& ( ( F2 @ X5 )
= Y ) ) ) ) ) ) ) ).
% IVT2
thf(fact_5506_LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [R: real,A2: A,F2: A > B,G: A > B,L: B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( ! [X5: A] :
( ( X5 != A2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X5 @ A2 ) ) @ R )
=> ( ( F2 @ X5 )
= ( G @ X5 ) ) ) )
=> ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_equal2
thf(fact_5507_LIM__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L5: B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( 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 )
& ! [X: A] :
( ( ( X != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ A2 ) ) @ S8 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X ) @ L5 ) ) @ R5 ) ) ) ) ) ) ) ).
% LIM_eq
thf(fact_5508_LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F2: A > B,L5: B] :
( ! [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
=> ? [S9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S9 )
& ! [X5: A] :
( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X5 @ A2 ) ) @ S9 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X5 ) @ L5 ) ) @ R4 ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_I
thf(fact_5509_LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L5: B,A2: A,R2: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [S2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S2 )
& ! [X2: A] :
( ( ( X2 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ A2 ) ) @ S2 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X2 ) @ L5 ) ) @ R2 ) ) ) ) ) ) ).
% LIM_D
thf(fact_5510_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > A,A2: A,D6: A] :
( ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ A2 @ H ) ) @ ( F2 @ A2 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [X: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ X ) @ ( F2 @ A2 ) ) @ ( minus_minus @ A @ X @ A2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_5511_LIM__fun__less__zero,axiom,
! [F2: real > real,L: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
& ! [X2: real] :
( ( ( X2 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X2 ) ) @ R4 ) )
=> ( ord_less @ real @ ( F2 @ X2 ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_less_zero
thf(fact_5512_LIM__fun__not__zero,axiom,
! [F2: real > real,L: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( L
!= ( zero_zero @ real ) )
=> ? [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
& ! [X2: real] :
( ( ( X2 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X2 ) ) @ R4 ) )
=> ( ( F2 @ X2 )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_not_zero
thf(fact_5513_LIM__fun__gt__zero,axiom,
! [F2: real > real,L: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
& ! [X2: real] :
( ( ( X2 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X2 ) ) @ R4 ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% LIM_fun_gt_zero
thf(fact_5514_LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,B2: B,A2: A,G: B > C,C2: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X5: A] :
( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X5 @ A2 ) ) @ D4 ) )
=> ( ( F2 @ X5 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose2
thf(fact_5515_continuous__at__within__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,S: set @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ G )
=> ( ( ( G @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_5516_isCont__add,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [A2: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% isCont_add
thf(fact_5517_continuous__at__within__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A2: A,S: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_5518_continuous__at__within__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,S: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_at_within_sgn
thf(fact_5519_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X3: A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X3 @ H ) ) @ ( F2 @ X3 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_5520_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X3: A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X3 @ H ) ) @ ( F2 @ X3 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_5521_lim__exp__minus__1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( filterlim @ A @ A
@ ^ [Z6: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z6 ) @ ( one_one @ A ) ) @ Z6 )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% lim_exp_minus_1
thf(fact_5522_lemma__termdiff4,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [K: real,F2: A > B,K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ! [H6: A] :
( ( H6
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H6 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ H6 ) ) @ ( times_times @ real @ K5 @ ( real_V7770717601297561774m_norm @ A @ H6 ) ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% lemma_termdiff4
thf(fact_5523_isCont__eq__Lb,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M8: A] :
( ! [X2: real] :
( ( ( ord_less_eq @ real @ A2 @ X2 )
& ( ord_less_eq @ real @ X2 @ B2 ) )
=> ( ord_less_eq @ A @ M8 @ ( F2 @ X2 ) ) )
& ? [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 )
& ( ( F2 @ X5 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Lb
thf(fact_5524_isCont__eq__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M8: A] :
( ! [X2: real] :
( ( ( ord_less_eq @ real @ A2 @ X2 )
& ( ord_less_eq @ real @ X2 @ B2 ) )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ M8 ) )
& ? [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 )
& ( ( F2 @ X5 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Ub
thf(fact_5525_isCont__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M8: A] :
! [X2: real] :
( ( ( ord_less_eq @ real @ A2 @ X2 )
& ( ord_less_eq @ real @ X2 @ B2 ) )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ M8 ) ) ) ) ) ).
% isCont_bounded
thf(fact_5526_isCont__inverse__function2,axiom,
! [A2: real,X3: real,B2: real,G: real > real,F2: real > real] :
( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A2 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B2 )
=> ( ( G @ ( F2 @ 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 ) ) ) @ F2 ) ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X3 ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_5527_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X3: A] :
( ( has_derivative @ A @ A @ F2 @ ( times_times @ A @ D6 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X3 @ H ) ) @ ( F2 @ X3 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_5528_isCont__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ( G @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% isCont_divide
thf(fact_5529_isCont__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ).
% isCont_sgn
thf(fact_5530_filterlim__at__to__0,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,F5: filter @ B,A2: A] :
( ( filterlim @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [X: A] : ( F2 @ ( plus_plus @ A @ X @ A2 ) )
@ F5
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_5531_continuous__within__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,S: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ S ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ X3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ S )
@ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_within_tan
thf(fact_5532_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N2: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N2 @ ( minus_minus @ nat @ J @ ( suc @ I ) ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) ) @ N2 )
= ( suc @ ( plus_plus @ nat @ I @ N2 ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_5533_continuous__within__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,S: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ S ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ X3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ S )
@ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_within_cot
thf(fact_5534_continuous__at__within__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: C,A3: set @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X3 @ A3 ) @ F2 )
=> ( ( ( cosh @ A @ ( F2 @ X3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X3 @ A3 )
@ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_at_within_tanh
thf(fact_5535_isCont__has__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M8: A] :
( ! [X2: real] :
( ( ( ord_less_eq @ real @ A2 @ X2 )
& ( ord_less_eq @ real @ X2 @ B2 ) )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ M8 ) )
& ! [N8: A] :
( ( ord_less @ A @ N8 @ M8 )
=> ? [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 )
& ( ord_less @ A @ N8 @ ( F2 @ X5 ) ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_5536_isCont__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ ( tan @ A ) ) ) ) ).
% isCont_tan
thf(fact_5537_filterlim__shift__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,D2: A,F5: filter @ B,A2: A] :
( ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F2 @ ( plus_plus @ A @ D2 ) ) @ F5 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ D2 ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift_iff
thf(fact_5538_filterlim__shift,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,F5: filter @ B,A2: A,D2: A] :
( ( filterlim @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F2 @ ( plus_plus @ A @ D2 ) ) @ F5 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ D2 ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift
thf(fact_5539_isCont__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( sin @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ ( cot @ A ) ) ) ) ).
% isCont_cot
thf(fact_5540_isCont__tanh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cosh @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ ( tanh @ A ) ) ) ) ).
% isCont_tanh
thf(fact_5541_powser__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A2: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X5: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ S )
=> ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( A2 @ N ) @ ( power_power @ A @ X5 @ N ) )
@ ( F2 @ X5 ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0
thf(fact_5542_powser__limit__0__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A2: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X5: A] :
( ( X5
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ S )
=> ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( A2 @ N ) @ ( power_power @ A @ X5 @ N ) )
@ ( F2 @ X5 ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0_strong
thf(fact_5543_lemma__termdiff5,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_Vector_banach @ B ) )
=> ! [K: real,F2: nat > real,G: A > nat > B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ( summable @ real @ F2 )
=> ( ! [H6: A,N3: nat] :
( ( H6
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H6 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( G @ H6 @ N3 ) ) @ ( times_times @ real @ ( F2 @ N3 ) @ ( real_V7770717601297561774m_norm @ A @ H6 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( suminf @ B @ ( G @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% lemma_termdiff5
thf(fact_5544_isCont__tan_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ A2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ).
% isCont_tan'
thf(fact_5545_isCont__arcosh,axiom,
! [X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ ( arcosh @ real ) ) ) ).
% isCont_arcosh
thf(fact_5546_continuous__at__within__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,S: set @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S ) @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ A2 ) )
=> ( ( ( F2 @ A2 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A2 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ).
% continuous_at_within_log
thf(fact_5547_isCont__cot_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ A2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ).
% isCont_cot'
thf(fact_5548_DERIV__inverse__function,axiom,
! [F2: real > real,D6: real,G: real > real,X3: real,A2: real,B2: real] :
( ( has_field_derivative @ real @ F2 @ D6 @ ( topolo174197925503356063within @ real @ ( G @ X3 ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( D6
!= ( zero_zero @ real ) )
=> ( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ( ! [Y4: real] :
( ( ord_less @ real @ A2 @ Y4 )
=> ( ( ord_less @ real @ Y4 @ B2 )
=> ( ( F2 @ ( G @ Y4 ) )
= Y4 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ G )
=> ( has_field_derivative @ real @ G @ ( inverse_inverse @ real @ D6 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_inverse_function
thf(fact_5549_isCont__arccos,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ arccos ) ) ) ).
% isCont_arccos
thf(fact_5550_isCont__arcsin,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ arcsin ) ) ) ).
% isCont_arcsin
thf(fact_5551_LIM__less__bound,axiom,
! [B2: real,X3: real,F2: real > real] :
( ( ord_less @ real @ B2 @ X3 )
=> ( ! [X5: real] :
( ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ B2 @ X3 ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X5 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) ) ) ) ) ).
% LIM_less_bound
thf(fact_5552_isCont__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ A2 ) )
=> ( ( ( F2 @ A2 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A2 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ).
% isCont_log
thf(fact_5553_isCont__artanh,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ ( artanh @ real ) ) ) ) ).
% isCont_artanh
thf(fact_5554_isCont__inverse__function,axiom,
! [D2: real,X3: real,G: real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z3 @ X3 ) ) @ D2 )
=> ( ( G @ ( F2 @ Z3 ) )
= Z3 ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z3 @ X3 ) ) @ D2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X3 ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ).
% isCont_inverse_function
thf(fact_5555_GMVT_H,axiom,
! [A2: real,B2: real,F2: real > real,G: real > real,G7: real > real,F11: 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 ) ) ) @ F2 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A2 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ G ) ) )
=> ( ! [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
=> ( ( ord_less @ real @ Z3 @ B2 )
=> ( has_field_derivative @ real @ G @ ( 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 @ F2 @ ( F11 @ Z3 ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [C3: real] :
( ( ord_less @ real @ A2 @ C3 )
& ( ord_less @ real @ C3 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) @ ( G7 @ C3 ) )
= ( times_times @ real @ ( minus_minus @ real @ ( G @ B2 ) @ ( G @ A2 ) ) @ ( F11 @ C3 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_5556_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 ) ) )
=> ! [N6: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_5557_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 ) )
=> ! [N6: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) @ ( one_one @ nat ) ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_5558_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
@ ^ [N: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_5559_trivial__limit__sequentially,axiom,
( ( at_top @ nat )
!= ( bot_bot @ ( filter @ nat ) ) ) ).
% trivial_limit_sequentially
thf(fact_5560_tendsto__zero__mult__left__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N: nat] : ( times_times @ A @ C2 @ ( A2 @ N ) )
@ ( 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_5561_tendsto__zero__mult__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N: nat] : ( times_times @ A @ ( A2 @ N ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_right_iff
thf(fact_5562_tendsto__zero__divide__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N: nat] : ( divide_divide @ A @ ( A2 @ N ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_5563_filterlim__Suc,axiom,
filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ).
% filterlim_Suc
thf(fact_5564_filterlim__sequentially__Suc,axiom,
! [A: $tType,F2: nat > A,F5: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X: nat] : ( F2 @ ( suc @ X ) )
@ F5
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F2 @ F5 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_5565_trivial__limit__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_top @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_top_linorder
thf(fact_5566_approx__from__above__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ? [U4: nat > A] :
( ! [N6: nat] : ( ord_less @ A @ X3 @ ( U4 @ N6 ) )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_above_dense_linorder
thf(fact_5567_approx__from__below__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ X3 )
=> ? [U4: nat > A] :
( ! [N6: nat] : ( ord_less @ A @ ( U4 @ N6 ) @ X3 )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_below_dense_linorder
thf(fact_5568_LIMSEQ__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_Suc
thf(fact_5569_LIMSEQ__imp__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_imp_Suc
thf(fact_5570_LIMSEQ__offset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,K: nat,A2: A] :
( ( filterlim @ nat @ A
@ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_offset
thf(fact_5571_LIMSEQ__ignore__initial__segment,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,A2: A,K: nat] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_5572_lim__mono,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [N7: nat,X8: nat > A,Y7: nat > A,X3: A,Y: A] :
( ! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ).
% lim_mono
thf(fact_5573_LIMSEQ__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,X3: A,Y7: nat > A,Y: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ).
% LIMSEQ_le
thf(fact_5574_Lim__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: nat > A,L: A,M7: nat,C5: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ M7 @ N3 )
=> ( ord_less_eq @ A @ ( F2 @ N3 ) @ C5 ) )
=> ( ord_less_eq @ A @ L @ C5 ) ) ) ) ).
% Lim_bounded
thf(fact_5575_Lim__bounded2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: nat > A,L: A,N7: nat,C5: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ A @ C5 @ ( F2 @ N3 ) ) )
=> ( ord_less_eq @ A @ C5 @ L ) ) ) ) ).
% Lim_bounded2
thf(fact_5576_LIMSEQ__le__const,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,X3: A,A2: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ A @ A2 @ ( X8 @ N3 ) ) )
=> ( ord_less_eq @ A @ A2 @ X3 ) ) ) ) ).
% LIMSEQ_le_const
thf(fact_5577_LIMSEQ__le__const2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,X3: A,A2: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ A2 ) )
=> ( ord_less_eq @ A @ X3 @ A2 ) ) ) ) ).
% LIMSEQ_le_const2
thf(fact_5578_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_5579_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_5580_Inf__as__limit,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [U4: nat > A] :
( ! [N6: nat] : ( member @ A @ ( U4 @ N6 ) @ A3 )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ ( complete_Inf_Inf @ A @ A3 ) ) @ ( at_top @ nat ) ) ) ) ) ).
% Inf_as_limit
thf(fact_5581_summable__LIMSEQ__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ_zero
thf(fact_5582_mult__nat__left__at__top,axiom,
! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( filterlim @ nat @ nat @ ( times_times @ nat @ C2 ) @ ( at_top @ nat ) @ ( at_top @ nat ) ) ) ).
% mult_nat_left_at_top
thf(fact_5583_mult__nat__right__at__top,axiom,
! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( filterlim @ nat @ nat
@ ^ [X: nat] : ( times_times @ nat @ X @ C2 )
@ ( at_top @ nat )
@ ( at_top @ nat ) ) ) ).
% mult_nat_right_at_top
thf(fact_5584_monoseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: nat > A,X3: A] :
( ( topological_monoseq @ A @ A2 )
=> ( ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ( ! [N6: nat] : ( ord_less_eq @ A @ ( A2 @ N6 ) @ X3 )
& ! [M6: nat,N6: nat] :
( ( ord_less_eq @ nat @ M6 @ N6 )
=> ( ord_less_eq @ A @ ( A2 @ M6 ) @ ( A2 @ N6 ) ) ) )
| ( ! [N6: nat] : ( ord_less_eq @ A @ X3 @ ( A2 @ N6 ) )
& ! [M6: nat,N6: nat] :
( ( ord_less_eq @ nat @ M6 @ N6 )
=> ( ord_less_eq @ A @ ( A2 @ N6 ) @ ( A2 @ M6 ) ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_5585_lim__const__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A] :
( filterlim @ nat @ A
@ ^ [N: nat] : ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_const_over_n
thf(fact_5586_lim__inverse__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_inverse_n
thf(fact_5587_LIMSEQ__linear,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X8: nat > A,X3: A,L: nat] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( X8 @ ( times_times @ nat @ N @ L ) )
@ ( topolo7230453075368039082e_nhds @ A @ X3 )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_linear
thf(fact_5588_telescope__summable_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) ) ) ) ) ).
% telescope_summable'
thf(fact_5589_telescope__summable,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) ) ) ) ) ).
% telescope_summable
thf(fact_5590_nested__sequence__unique,axiom,
! [F2: nat > real,G: nat > real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ ( G @ N3 ) )
=> ( ( filterlim @ nat @ real
@ ^ [N: nat] : ( minus_minus @ real @ ( F2 @ N ) @ ( G @ N ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) )
=> ? [L4: real] :
( ! [N6: nat] : ( ord_less_eq @ real @ ( F2 @ N6 ) @ L4 )
& ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) )
& ! [N6: nat] : ( ord_less_eq @ real @ L4 @ ( G @ N6 ) )
& ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_5591_LIMSEQ__inverse__zero,axiom,
! [X8: nat > real] :
( ! [R4: real] :
? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less @ real @ R4 @ ( X8 @ N3 ) ) )
=> ( filterlim @ nat @ real
@ ^ [N: nat] : ( inverse_inverse @ real @ ( X8 @ N ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_zero
thf(fact_5592_LIMSEQ__root__const,axiom,
! [C2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( filterlim @ nat @ real
@ ^ [N: nat] : ( root @ N @ C2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_root_const
thf(fact_5593_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim @ nat @ real
@ ^ [N: nat] : ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_5594_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N: nat] : ( plus_plus @ real @ R2 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_5595_increasing__LIMSEQ,axiom,
! [F2: nat > real,L: real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ L )
=> ( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [N6: nat] : ( ord_less_eq @ real @ L @ ( plus_plus @ real @ ( F2 @ N6 ) @ E ) ) )
=> ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( at_top @ nat ) ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_5596_lim__1__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_1_over_n
thf(fact_5597_LIMSEQ__Suc__n__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( semiring_1_of_nat @ A @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_5598_LIMSEQ__n__over__Suc__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_5599_LIMSEQ__realpow__zero,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ X3 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_5600_telescope__sums_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
@ ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ C2 ) ) ) ) ).
% telescope_sums'
thf(fact_5601_telescope__sums,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) )
@ ( minus_minus @ A @ C2 @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% telescope_sums
thf(fact_5602_LIMSEQ__divide__realpow__zero,axiom,
! [X3: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( filterlim @ nat @ real
@ ^ [N: nat] : ( divide_divide @ real @ A2 @ ( power_power @ real @ X3 @ N ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_5603_LIMSEQ__abs__realpow__zero2,axiom,
! [C2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ C2 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ C2 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ).
% LIMSEQ_abs_realpow_zero2
thf(fact_5604_LIMSEQ__abs__realpow__zero,axiom,
! [C2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ C2 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ ( abs_abs @ real @ C2 ) ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ).
% LIMSEQ_abs_realpow_zero
thf(fact_5605_LIMSEQ__inverse__realpow__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( filterlim @ nat @ real
@ ^ [N: nat] : ( inverse_inverse @ real @ ( power_power @ real @ X3 @ N ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_5606_sums__def_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F3: nat > A,S8: A] :
( filterlim @ nat @ A
@ ^ [N: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ S8 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def'
thf(fact_5607_root__test__convergence,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,X3: real] :
( ( filterlim @ nat @ real
@ ^ [N: nat] : ( root @ N @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ X3 )
@ ( at_top @ nat ) )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% root_test_convergence
thf(fact_5608_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N: nat] : ( plus_plus @ real @ R2 @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_5609_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L5: A,R2: real] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No: nat] :
! [N6: nat] :
( ( ord_less_eq @ nat @ No @ N6 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N6 ) @ L5 ) ) @ R2 ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_5610_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L5: A] :
( ! [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
=> ? [No2: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No2 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N3 ) @ L5 ) ) @ R4 ) ) )
=> ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_5611_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ No3 @ N )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N ) @ L5 ) ) @ R5 ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_5612_LIMSEQ__power__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X3 ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_power_zero
thf(fact_5613_tendsto__at__iff__sequentially,axiom,
! [C: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > C,A2: C,X3: A,S: set @ A] :
( ( filterlim @ A @ C @ F2 @ ( topolo7230453075368039082e_nhds @ C @ A2 ) @ ( topolo174197925503356063within @ A @ X3 @ S ) )
= ( ! [X6: nat > A] :
( ! [I3: nat] : ( member @ A @ ( X6 @ I3 ) @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C @ ( comp @ A @ C @ nat @ F2 @ X6 ) @ ( topolo7230453075368039082e_nhds @ C @ A2 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% tendsto_at_iff_sequentially
thf(fact_5614_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: B > nat,F5: filter @ B,X3: A] :
( ( filterlim @ B @ nat @ F2 @ ( at_top @ nat ) @ F5 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y2: B] : ( power_power @ A @ X3 @ ( F2 @ Y2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_power_zero
thf(fact_5615_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N: nat] : ( times_times @ real @ R2 @ ( plus_plus @ real @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_5616_LIMSEQ__norm__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_norm_0
thf(fact_5617_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Df: A,Z2: A,S: nat > A,A2: A] :
( ( has_field_derivative @ A @ F2 @ Df @ ( topolo174197925503356063within @ A @ Z2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ nat @ A @ S @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( S @ N3 )
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ Z2 @ ( S @ N ) ) ) @ ( F2 @ Z2 ) ) @ ( S @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( Df = A2 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_5618_powser__times__n__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ X3 @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_5619_lim__n__over__pown,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ X3 @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_5620_summable,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
=> ( summable @ real
@ ^ [N: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) @ ( A2 @ N ) ) ) ) ) ) ).
% summable
thf(fact_5621_zeroseq__arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real
@ ^ [N: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% zeroseq_arctan_series
thf(fact_5622_summable__Leibniz_H_I3_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
=> ( filterlim @ nat @ real
@ ^ [N: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_5623_summable__Leibniz_H_I2_J,axiom,
! [A2: nat > real,N2: nat] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
=> ( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_5624_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] :
( ! [N6: nat] :
( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) ) )
@ L4 )
& ( filterlim @ nat @ real
@ ^ [N: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L4 )
@ ( at_top @ nat ) )
& ! [N6: nat] :
( ord_less_eq @ real @ L4
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) @ ( one_one @ nat ) ) ) ) )
& ( filterlim @ nat @ real
@ ^ [N: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L4 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_5625_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
@ ^ [N: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(5)
thf(fact_5626_summable__Leibniz_H_I4_J,axiom,
! [A2: nat > real,N2: 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
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_5627_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F11: A > B,X3: A] :
( ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F11 )
& ( filterlim @ A @ B
@ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X3 ) ) ) @ ( minus_minus @ B @ ( F2 @ Y2 ) @ ( plus_plus @ B @ ( F2 @ X3 ) @ ( F11 @ ( minus_minus @ A @ Y2 @ X3 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_5628_summable__Cauchy_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [M2: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) @ ( G @ M2 ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_Cauchy'
thf(fact_5629_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,D6: A > B,X3: A] :
( ( has_derivative @ A @ B @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ ( 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 @ ( F2 @ ( plus_plus @ A @ X3 @ H ) ) @ ( F2 @ X3 ) ) @ ( 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_5630_eventually__const,axiom,
! [A: $tType,F5: filter @ A,P: $o] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X: A] : P
@ F5 )
= P ) ) ).
% eventually_const
thf(fact_5631_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_5632_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat
@ ^ [N: nat] : ( P @ ( plus_plus @ nat @ N @ K ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_5633_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N5: A] :
! [N: A] :
( ( ord_less @ A @ N5 @ N )
=> ( P @ N ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_5634_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,P: A > $o] :
( ! [X5: A] :
( ( ord_less_eq @ A @ C2 @ X5 )
=> ( P @ X5 ) )
=> ( eventually @ A @ P @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_5635_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N5: A] :
! [N: A] :
( ( ord_less_eq @ A @ N5 @ N )
=> ( P @ N ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_5636_eventually__sequentiallyI,axiom,
! [C2: nat,P: nat > $o] :
( ! [X5: nat] :
( ( ord_less_eq @ nat @ C2 @ X5 )
=> ( P @ X5 ) )
=> ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_5637_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
= ( ? [N5: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N5 @ N )
=> ( P @ N ) ) ) ) ).
% eventually_sequentially
thf(fact_5638_eventually__bot,axiom,
! [A: $tType,P: A > $o] : ( eventually @ A @ P @ ( bot_bot @ ( filter @ A ) ) ) ).
% eventually_bot
thf(fact_5639_eventually__happens,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ( eventually @ A @ P @ Net )
=> ( ( Net
= ( bot_bot @ ( filter @ A ) ) )
| ? [X_1: A] : ( P @ X_1 ) ) ) ).
% eventually_happens
thf(fact_5640_eventually__happens_H,axiom,
! [A: $tType,F5: filter @ A,P: A > $o] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A @ P @ F5 )
=> ? [X_1: A] : ( P @ X_1 ) ) ) ).
% eventually_happens'
thf(fact_5641_False__imp__not__eventually,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ! [X5: A] :
~ ( P @ X5 )
=> ( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ~ ( eventually @ A @ P @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_5642_eventually__const__iff,axiom,
! [A: $tType,P: $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X: A] : P
@ F5 )
= ( P
| ( F5
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_5643_trivial__limit__def,axiom,
! [A: $tType,F5: filter @ A] :
( ( F5
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X: A] : $false
@ F5 ) ) ).
% trivial_limit_def
thf(fact_5644_bounded__linear__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X: A] : ( zero_zero @ B ) ) ) ).
% bounded_linear_zero
thf(fact_5645_bounded__linear__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% bounded_linear_add
thf(fact_5646_eventually__ex,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X: A] :
? [X6: B] : ( P @ X @ X6 )
@ F5 )
= ( ? [Y9: A > B] :
( eventually @ A
@ ^ [X: A] : ( P @ X @ ( Y9 @ X ) )
@ F5 ) ) ) ).
% eventually_ex
thf(fact_5647_eventually__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] : ( eventually @ A @ ( ord_less_eq @ A @ C2 ) @ ( at_top @ A ) ) ) ).
% eventually_ge_at_top
thf(fact_5648_eventually__gt__at__top,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] : ( eventually @ A @ ( ord_less @ A @ C2 ) @ ( at_top @ A ) ) ) ).
% eventually_gt_at_top
thf(fact_5649_eventually__ball__finite__distrib,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( eventually @ B
@ ^ [X: B] :
! [Y2: A] :
( ( member @ A @ Y2 @ A3 )
=> ( P @ X @ Y2 ) )
@ Net )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( eventually @ B
@ ^ [Y2: B] : ( P @ Y2 @ X )
@ Net ) ) ) ) ) ).
% eventually_ball_finite_distrib
thf(fact_5650_eventually__ball__finite,axiom,
! [A: $tType,B: $tType,A3: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( eventually @ B
@ ^ [Y2: B] : ( P @ Y2 @ X5 )
@ Net ) )
=> ( eventually @ B
@ ^ [X: B] :
! [Y2: A] :
( ( member @ A @ Y2 @ A3 )
=> ( P @ X @ Y2 ) )
@ Net ) ) ) ).
% eventually_ball_finite
thf(fact_5651_le__sequentially,axiom,
! [F5: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F5 @ ( at_top @ nat ) )
= ( ! [N5: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N5 ) @ F5 ) ) ) ).
% le_sequentially
thf(fact_5652_sequentially__offset,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [I3: nat] : ( P @ ( plus_plus @ nat @ I3 @ K ) )
@ ( at_top @ nat ) ) ) ).
% sequentially_offset
thf(fact_5653_LIM__at__top__divide,axiom,
! [A: $tType,F2: A > real,A2: real,F5: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
@ F5 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_top @ real )
@ F5 ) ) ) ) ) ).
% LIM_at_top_divide
thf(fact_5654_filterlim__inverse__at__top,axiom,
! [A: $tType,F2: A > real,F5: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F5 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( inverse_inverse @ real @ ( F2 @ X ) )
@ ( at_top @ real )
@ F5 ) ) ) ).
% filterlim_inverse_at_top
thf(fact_5655_filterlim__inverse__at__top__iff,axiom,
! [A: $tType,F2: A > real,F5: filter @ A] :
( ( eventually @ A
@ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F5 )
=> ( ( filterlim @ A @ real
@ ^ [X: A] : ( inverse_inverse @ real @ ( F2 @ X ) )
@ ( at_top @ real )
@ F5 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ).
% filterlim_inverse_at_top_iff
thf(fact_5656_filterlim__at__top__iff__inverse__0,axiom,
! [A: $tType,F2: A > real,F5: filter @ A] :
( ( eventually @ A
@ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F5 )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F5 )
= ( filterlim @ A @ real @ ( comp @ real @ real @ A @ ( inverse_inverse @ real ) @ F2 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ).
% filterlim_at_top_iff_inverse_0
thf(fact_5657_eventually__nhds__top,axiom,
! [A: $tType] :
( ( ( order_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: A,P: A > $o] :
( ( ord_less @ A @ B2 @ ( top_top @ A ) )
=> ( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ ( top_top @ A ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ B4 @ ( top_top @ A ) )
& ! [Z6: A] :
( ( ord_less @ A @ B4 @ Z6 )
=> ( P @ Z6 ) ) ) ) ) ) ) ).
% eventually_nhds_top
thf(fact_5658_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A] :
( ! [X5: A,Y4: A] :
( ( Q @ X5 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( ( F2 @ ( G @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( Q @ ( G @ X5 ) ) )
=> ( ( eventually @ A @ Q @ ( at_top @ A ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_5659_eventually__at__left__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X3: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X3 @ ( set_ord_lessThan @ A @ X3 ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ B4 @ X3 )
& ! [Y2: A] :
( ( ord_less @ A @ B4 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X3 )
=> ( P @ Y2 ) ) ) ) ) ) ) ).
% eventually_at_left_field
thf(fact_5660_eventually__at__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y: A,X3: A,P: A > $o] :
( ( ord_less @ A @ Y @ X3 )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X3 @ ( set_ord_lessThan @ A @ X3 ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ B4 @ X3 )
& ! [Y2: A] :
( ( ord_less @ A @ B4 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X3 )
=> ( P @ Y2 ) ) ) ) ) ) ) ) ).
% eventually_at_left
thf(fact_5661_tendsto__sandwich,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,G: B > A,Net: filter @ B,H2: B > A,C2: A] :
( ( eventually @ B
@ ^ [N: B] : ( ord_less_eq @ A @ ( F2 @ N ) @ ( G @ N ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N: B] : ( ord_less_eq @ A @ ( G @ N ) @ ( H2 @ N ) )
@ Net )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( ( filterlim @ B @ A @ H2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net ) ) ) ) ) ) ).
% tendsto_sandwich
thf(fact_5662_order__tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,X3: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ F5 )
= ( ! [L2: A] :
( ( ord_less @ A @ L2 @ X3 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ L2 @ ( F2 @ X ) )
@ F5 ) )
& ! [U2: A] :
( ( ord_less @ A @ X3 @ U2 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ ( F2 @ X ) @ U2 )
@ F5 ) ) ) ) ) ).
% order_tendsto_iff
thf(fact_5663_order__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [Y: A,F2: B > A,F5: filter @ B] :
( ! [A5: A] :
( ( ord_less @ A @ A5 @ Y )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ A5 @ ( F2 @ X ) )
@ F5 ) )
=> ( ! [A5: A] :
( ( ord_less @ A @ Y @ A5 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ ( F2 @ X ) @ A5 )
@ F5 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 ) ) ) ) ).
% order_tendstoI
thf(fact_5664_order__tendstoD_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y: A,F5: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 )
=> ( ( ord_less @ A @ A2 @ Y )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ A2 @ ( F2 @ X ) )
@ F5 ) ) ) ) ).
% order_tendstoD(1)
thf(fact_5665_order__tendstoD_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y: A,F5: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 )
=> ( ( ord_less @ A @ Y @ A2 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ ( F2 @ X ) @ A2 )
@ F5 ) ) ) ) ).
% order_tendstoD(2)
thf(fact_5666_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X: A] :
! [Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( P @ Y2 ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_5667_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X ) )
@ F5 ) ) ) ) ).
% filterlim_at_top
thf(fact_5668_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_5669_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,F5: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( at_top @ A ) @ F5 )
=> ( ( eventually @ B
@ ^ [X: B] : ( ord_less_eq @ A @ ( F2 @ X ) @ ( G @ X ) )
@ F5 )
=> ( filterlim @ B @ A @ G @ ( at_top @ A ) @ F5 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_5670_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X: A] : ( ord_less @ B @ Z9 @ ( F2 @ X ) )
@ F5 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_5671_bounded__linear_Otendsto__zero,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: C > A,F5: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( filterlim @ C @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ C @ B
@ ^ [X: C] : ( F2 @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% bounded_linear.tendsto_zero
thf(fact_5672_eventually__Inf__base,axiom,
! [A: $tType,B3: set @ ( filter @ A ),P: A > $o] :
( ( B3
!= ( bot_bot @ ( set @ ( filter @ A ) ) ) )
=> ( ! [F6: filter @ A] :
( ( member @ ( filter @ A ) @ F6 @ B3 )
=> ! [G5: filter @ A] :
( ( member @ ( filter @ A ) @ G5 @ B3 )
=> ? [X2: filter @ A] :
( ( member @ ( filter @ A ) @ X2 @ B3 )
& ( ord_less_eq @ ( filter @ A ) @ X2 @ ( inf_inf @ ( filter @ A ) @ F6 @ G5 ) ) ) ) )
=> ( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B3 ) )
= ( ? [X: filter @ A] :
( ( member @ ( filter @ A ) @ X @ B3 )
& ( eventually @ A @ P @ X ) ) ) ) ) ) ).
% eventually_Inf_base
thf(fact_5673_eventually__INF__finite,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: B > $o,F5: A > ( filter @ B )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F5 @ A3 ) ) )
= ( ? [Q7: A > B > $o] :
( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( eventually @ B @ ( Q7 @ X ) @ ( F5 @ X ) ) )
& ! [Y2: B] :
( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( Q7 @ X @ Y2 ) )
=> ( P @ Y2 ) ) ) ) ) ) ).
% eventually_INF_finite
thf(fact_5674_eventually__at__leftI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
=> ( P @ X5 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ).
% eventually_at_leftI
thf(fact_5675_has__derivative__within__singleton__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: A > B,X3: A] :
( ( has_derivative @ A @ B @ F2 @ G @ ( topolo174197925503356063within @ A @ X3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( real_V3181309239436604168linear @ A @ B @ G ) ) ) ).
% has_derivative_within_singleton_iff
thf(fact_5676_eventually__at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o,A2: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( eventually @ A
@ ^ [X: A] : ( P @ ( plus_plus @ A @ X @ A2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% eventually_at_to_0
thf(fact_5677_decreasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [L: A,F2: B > A,F5: filter @ B] :
( ( eventually @ B
@ ^ [N: B] : ( ord_less_eq @ A @ L @ ( F2 @ N ) )
@ F5 )
=> ( ! [X5: A] :
( ( ord_less @ A @ L @ X5 )
=> ( eventually @ B
@ ^ [N: B] : ( ord_less @ A @ ( F2 @ N ) @ X5 )
@ F5 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% decreasing_tendsto
thf(fact_5678_increasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,L: A,F5: filter @ B] :
( ( eventually @ B
@ ^ [N: B] : ( ord_less_eq @ A @ ( F2 @ N ) @ L )
@ F5 )
=> ( ! [X5: A] :
( ( ord_less @ A @ X5 @ L )
=> ( eventually @ B
@ ^ [N: B] : ( ord_less @ A @ X5 @ ( F2 @ N ) )
@ F5 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% increasing_tendsto
thf(fact_5679_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z9: B] :
( ( ord_less @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_5680_tendsto__le,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F5: filter @ B,F2: B > A,X3: A,G: B > A,Y: A] :
( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ F5 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 )
=> ( ( eventually @ B
@ ^ [X: B] : ( ord_less_eq @ A @ ( G @ X ) @ ( F2 @ X ) )
@ F5 )
=> ( ord_less_eq @ A @ Y @ X3 ) ) ) ) ) ) ).
% tendsto_le
thf(fact_5681_tendsto__lowerbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: B > A,X3: A,F5: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ F5 )
=> ( ( eventually @ B
@ ^ [I3: B] : ( ord_less_eq @ A @ A2 @ ( F2 @ I3 ) )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ A2 @ X3 ) ) ) ) ) ).
% tendsto_lowerbound
thf(fact_5682_tendsto__upperbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: B > A,X3: A,F5: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ F5 )
=> ( ( eventually @ B
@ ^ [I3: B] : ( ord_less_eq @ A @ ( F2 @ I3 ) @ A2 )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ X3 @ A2 ) ) ) ) ) ).
% tendsto_upperbound
thf(fact_5683_eventually__INF,axiom,
! [A: $tType,B: $tType,P: A > $o,F5: B > ( filter @ A ),B3: set @ B] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ B @ ( filter @ A ) @ F5 @ B3 ) ) )
= ( ? [X6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X6 @ B3 )
& ( finite_finite2 @ B @ X6 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ B @ ( filter @ A ) @ F5 @ X6 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_5684_filterlim__pow__at__top,axiom,
! [A: $tType,N2: nat,F2: A > real,F5: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( power_power @ real @ ( F2 @ X ) @ N2 )
@ ( at_top @ real )
@ F5 ) ) ) ).
% filterlim_pow_at_top
thf(fact_5685_bounded__linear_Opos__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ? [K8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K8 )
& ! [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ K8 ) ) ) ) ) ).
% bounded_linear.pos_bounded
thf(fact_5686_bounded__linear__intro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,K5: real] :
( ! [X5: A,Y4: A] :
( ( F2 @ ( plus_plus @ A @ X5 @ Y4 ) )
= ( plus_plus @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ! [R4: real,X5: A] :
( ( F2 @ ( real_V8093663219630862766scaleR @ A @ R4 @ X5 ) )
= ( real_V8093663219630862766scaleR @ B @ R4 @ ( F2 @ X5 ) ) )
=> ( ! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K5 ) )
=> ( real_V3181309239436604168linear @ A @ B @ F2 ) ) ) ) ) ).
% bounded_linear_intro
thf(fact_5687_tendsto__imp__filterlim__at__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L5: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ L5 )
@ F5 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L5 @ ( set_ord_lessThan @ B @ L5 ) ) @ F5 ) ) ) ) ).
% tendsto_imp_filterlim_at_left
thf(fact_5688_eventually__Inf,axiom,
! [A: $tType,P: A > $o,B3: set @ ( filter @ A )] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B3 ) )
= ( ? [X6: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X6 @ B3 )
& ( finite_finite2 @ ( filter @ A ) @ X6 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ X6 ) ) ) ) ) ).
% eventually_Inf
thf(fact_5689_filterlim__at__top__mult__tendsto__pos,axiom,
! [A: $tType,F2: A > real,C2: real,F5: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( times_times @ real @ ( G @ X ) @ ( F2 @ X ) )
@ ( at_top @ real )
@ F5 ) ) ) ) ).
% filterlim_at_top_mult_tendsto_pos
thf(fact_5690_filterlim__tendsto__pos__mult__at__top,axiom,
! [A: $tType,F2: A > real,C2: real,F5: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( times_times @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_top @ real )
@ F5 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_top
thf(fact_5691_tendsto__neg__powr,axiom,
! [A: $tType,S: real,F2: A > real,F5: filter @ A] :
( ( ord_less @ real @ S @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ S )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 ) ) ) ).
% tendsto_neg_powr
thf(fact_5692_filterlim__at__top__at__left,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A,A2: A] :
( ! [X5: A,Y4: A] :
( ( Q @ X5 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( ( F2 @ ( G @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( Q @ ( G @ X5 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) )
=> ( ! [B5: A] :
( ( Q @ B5 )
=> ( ord_less @ A @ B5 @ A2 ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_left
thf(fact_5693_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B3: set @ A,F5: A > ( filter @ B ),P: B > $o] :
( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ B3 )
=> ! [B5: A] :
( ( member @ A @ B5 @ B3 )
=> ? [X2: A] :
( ( member @ A @ X2 @ B3 )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X2 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ A5 ) @ ( F5 @ B5 ) ) ) ) ) )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F5 @ B3 ) ) )
= ( ? [X: A] :
( ( member @ A @ X @ B3 )
& ( eventually @ B @ P @ ( F5 @ X ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_5694_tendsto__0__le,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: filter @ A,G: A > C,K5: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G @ X ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) @ K5 ) )
@ F5 )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F5 ) ) ) ) ).
% tendsto_0_le
thf(fact_5695_filterlim__at__withinI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,C2: A,F5: filter @ B,A3: set @ A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 )
=> ( ( eventually @ B
@ ^ [X: B] : ( member @ A @ ( F2 @ X ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ F5 )
=> ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ C2 @ A3 ) @ F5 ) ) ) ) ).
% filterlim_at_withinI
thf(fact_5696_tendsto__powr_H,axiom,
! [A: $tType,F2: A > real,A2: real,F5: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( ( A2
!= ( zero_zero @ real ) )
| ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
& ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F5 ) ) )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F5 ) ) ) ) ).
% tendsto_powr'
thf(fact_5697_tendsto__powr2,axiom,
! [A: $tType,F2: A > real,A2: real,F5: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F5 ) ) ) ) ) ).
% tendsto_powr2
thf(fact_5698_tendsto__zero__powrI,axiom,
! [A: $tType,F2: A > real,F5: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 ) ) ) ) ) ).
% tendsto_zero_powrI
thf(fact_5699_DERIV__neg__imp__decreasing__at__top,axiom,
! [B2: real,F2: real > real,Flim: real] :
( ! [X5: real] :
( ( ord_less_eq @ real @ B2 @ X5 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_top @ real ) )
=> ( ord_less @ real @ Flim @ ( F2 @ B2 ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_5700_has__derivative__at__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F11: A > B,X3: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F11 )
& ( filterlim @ A @ B
@ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X3 ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y2 ) @ ( F2 @ X3 ) ) @ ( F11 @ ( minus_minus @ A @ Y2 @ X3 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% has_derivative_at_within
thf(fact_5701_has__derivativeI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F11: A > B,X3: A,F2: A > B,S: set @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F11 )
=> ( ( filterlim @ A @ B
@ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X3 ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y2 ) @ ( F2 @ X3 ) ) @ ( F11 @ ( minus_minus @ A @ Y2 @ X3 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% has_derivativeI
thf(fact_5702_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F11: A > B,X3: A] :
( ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F11 )
& ? [E4: A > B] :
( ! [H: A] :
( ( F2 @ ( plus_plus @ A @ X3 @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X3 ) @ ( F11 @ 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_5703_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F11: A > B,X3: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F11 )
& ( filterlim @ A @ B
@ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X3 ) ) ) @ ( minus_minus @ B @ ( F2 @ Y2 ) @ ( plus_plus @ B @ ( F2 @ X3 ) @ ( F11 @ ( minus_minus @ A @ Y2 @ X3 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% has_derivative_within
thf(fact_5704_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C2: nat > A,K: nat,N2: nat,B3: real] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( eventually @ A
@ ^ [Z6: A] :
( ord_less_eq @ real @ B3
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_5705_summable__bounded__partials,axiom,
! [A: $tType] :
( ( ( real_V8037385150606011577_space @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [X02: nat] :
! [A4: nat] :
( ( ord_less_eq @ nat @ X02 @ A4 )
=> ! [B4: nat] :
( ( ord_less @ nat @ A4 @ B4 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or3652927894154168847AtMost @ nat @ A4 @ B4 ) ) ) @ ( G @ A4 ) ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_bounded_partials
thf(fact_5706_has__derivative__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( has_derivative @ A @ B )
= ( ^ [F3: A > B,F13: A > B,F9: filter @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F13 )
& ( filterlim @ A @ B
@ ^ [Y2: A] :
( real_V8093663219630862766scaleR @ B
@ ( inverse_inverse @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( minus_minus @ A @ Y2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X: A] : X ) ) ) )
@ ( minus_minus @ B
@ ( minus_minus @ B @ ( F3 @ Y2 )
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X: A] : X ) ) )
@ ( F13
@ ( minus_minus @ A @ Y2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X: A] : X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F9 ) ) ) ) ) ).
% has_derivative_def
thf(fact_5707_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% finite_greaterThanAtMost
thf(fact_5708_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_5709_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K: A] :
( ( ord_less_eq @ A @ L @ K )
=> ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanAtMost_empty
thf(fact_5710_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or3652927894154168847AtMost @ A @ K @ L ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_5711_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_5712_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_5713_image__add__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( image @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_5714_cSup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ X3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X3 ) )
= X3 ) ) ) ).
% cSup_greaterThanAtMost
thf(fact_5715_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X3 @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanAtMost
thf(fact_5716_cInf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X3 ) )
= Y ) ) ) ).
% cInf_greaterThanAtMost
thf(fact_5717_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X3 @ Y ) )
= X3 ) ) ) ).
% Inf_greaterThanAtMost
thf(fact_5718_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A2 @ B2 )
= ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ D2 @ C2 ) )
| ( ( A2 = C2 )
& ( B2 = D2 ) ) ) ) ) ).
% Ioc_inj
thf(fact_5719_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
= ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_5720_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_5721_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_5722_ivl__disj__int__two_I6_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(6)
thf(fact_5723_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ord_less_eq @ A @ D2 @ C2 )
| ( ord_less_eq @ A @ B2 @ C2 )
| ( ord_less_eq @ A @ D2 @ A2 ) ) ) ) ).
% Ioc_disjoint
thf(fact_5724_ivl__disj__int__two_I8_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(8)
thf(fact_5725_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_5726_ivl__disj__int__two_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(2)
thf(fact_5727_Lim__ident__at,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X3: A,S: set @ A] :
( ( ( topolo174197925503356063within @ A @ X3 @ S )
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( topolo3827282254853284352ce_Lim @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ S )
@ ^ [X: A] : X )
= X3 ) ) ) ).
% Lim_ident_at
thf(fact_5728_eventually__at__left__real,axiom,
! [B2: real,A2: real] :
( ( ord_less @ real @ B2 @ A2 )
=> ( eventually @ real
@ ^ [X: real] : ( member @ real @ X @ ( set_or5935395276787703475ssThan @ real @ B2 @ A2 ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ).
% eventually_at_left_real
thf(fact_5729_sum_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).
% sum.head
thf(fact_5730_not__tendsto__and__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F5: filter @ B,F2: B > A,C2: A] :
( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 )
=> ~ ( filterlim @ B @ A @ F2 @ ( at_infinity @ A ) @ F5 ) ) ) ) ).
% not_tendsto_and_filterlim_at_infinity
thf(fact_5731_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).
% prod.head
thf(fact_5732_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_5733_tendsto__Lim,axiom,
! [A: $tType,B: $tType] :
( ( topological_t2_space @ B )
=> ! [Net: filter @ A,F2: A > B,L: B] :
( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ Net )
=> ( ( topolo3827282254853284352ce_Lim @ A @ B @ Net @ F2 )
= L ) ) ) ) ).
% tendsto_Lim
thf(fact_5734_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_5735_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_5736_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A4: A,B4: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A4 @ B4 ) @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_5737_tendsto__add__filterlim__at__infinity_H,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F5: filter @ A,G: A > B,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F5 )
=> ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F5 )
=> ( filterlim @ A @ B
@ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_infinity @ B )
@ F5 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity'
thf(fact_5738_tendsto__add__filterlim__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,C2: B,F5: filter @ A,G: A > B] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F5 )
=> ( ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F5 )
=> ( filterlim @ A @ B
@ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_infinity @ B )
@ F5 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity
thf(fact_5739_continuous__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F5: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F5 @ G )
=> ( ( ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X: A] : X ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_divide
thf(fact_5740_continuous__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F5: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X: A] : X ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_inverse
thf(fact_5741_continuous__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F5: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X: A] : X ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_sgn
thf(fact_5742_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_5743_filterlim__at__infinity__imp__filterlim__at__top,axiom,
! [A: $tType,F2: A > real,F5: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_infinity @ real ) @ F5 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F5 )
=> ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F5 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_top
thf(fact_5744_tendsto__mult__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,C2: A,F5: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A @ G @ ( at_infinity @ A ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_infinity @ A )
@ F5 ) ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_5745_tendsto__divide__0,axiom,
! [A: $tType,C: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: C > A,C2: A,F5: filter @ C,G: C > A] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 )
=> ( ( filterlim @ C @ A @ G @ ( at_infinity @ A ) @ F5 )
=> ( filterlim @ C @ A
@ ^ [X: C] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_divide_0
thf(fact_5746_filterlim__power__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [F2: A > B,F5: filter @ A,N2: nat] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( filterlim @ A @ B
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N2 )
@ ( at_infinity @ B )
@ F5 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_5747_continuous__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F5 @ F2 )
=> ( ( ( cos @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X: A] : X ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F5
@ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_tan
thf(fact_5748_continuous__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F5 @ F2 )
=> ( ( ( sin @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X: A] : X ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F5
@ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_cot
thf(fact_5749_continuous__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F5 @ F2 )
=> ( ( ( cosh @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ C @ C @ F5
@ ^ [X: C] : X ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ F5
@ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_tanh
thf(fact_5750_continuous__arcosh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F5 @ F2 )
=> ( ( ord_less @ real @ ( one_one @ real )
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X: A] : X ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F5
@ ^ [X: A] : ( arcosh @ real @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_arcosh
thf(fact_5751_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_5752_eventually__at__infinity__pos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P4: A > $o] :
( ( eventually @ A @ P4 @ ( at_infinity @ A ) )
= ( ? [B4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B4 )
& ! [X: A] :
( ( ord_less_eq @ real @ B4 @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( P4 @ X ) ) ) ) ) ) ).
% eventually_at_infinity_pos
thf(fact_5753_filterlim__inverse__at__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [G: A > B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X: A] : ( inverse_inverse @ B @ ( G @ X ) )
@ ( topolo174197925503356063within @ B @ ( zero_zero @ B ) @ ( top_top @ ( set @ B ) ) )
@ F5 )
= ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F5 ) ) ) ).
% filterlim_inverse_at_iff
thf(fact_5754_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N2: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N2 @ ( minus_minus @ nat @ J @ I ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) ) @ N2 )
= ( suc @ ( plus_plus @ nat @ I @ N2 ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_5755_continuous__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F5 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F5 @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X: A] : X ) ) )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X: A] : X ) )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X: A] : X ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F5
@ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ).
% continuous_log
thf(fact_5756_filterlim__divide__at__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,C2: A,F5: filter @ A,G: A > A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 )
=> ( ( filterlim @ A @ A @ G @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) @ F5 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X: A] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_infinity @ A )
@ F5 ) ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_5757_filterlim__at__infinity,axiom,
! [C: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C2: real,F2: C > A,F5: filter @ C] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ C @ A @ F2 @ ( at_infinity @ A ) @ F5 )
= ( ! [R5: real] :
( ( ord_less @ real @ C2 @ R5 )
=> ( eventually @ C
@ ^ [X: C] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X ) ) )
@ F5 ) ) ) ) ) ) ).
% filterlim_at_infinity
thf(fact_5758_filterlim__realpow__sequentially__gt1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X3: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X3 ) @ ( at_infinity @ A ) @ ( at_top @ nat ) ) ) ) ).
% filterlim_realpow_sequentially_gt1
thf(fact_5759_lim__at__infinity__0,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L: A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) )
= ( filterlim @ A @ A @ ( comp @ A @ A @ A @ F2 @ ( inverse_inverse @ A ) ) @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% lim_at_infinity_0
thf(fact_5760_lim__zero__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L: A] :
( ( filterlim @ A @ A
@ ^ [X: A] : ( F2 @ ( divide_divide @ A @ ( one_one @ A ) @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) ) ) ) ).
% lim_zero_infinity
thf(fact_5761_has__derivative__at__within__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X3: A,S3: set @ A,F2: A > B,F11: A > B] :
( ( member @ A @ X3 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S3 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F11 )
& ? [E4: A > B] :
( ! [H: A] :
( ( member @ A @ ( plus_plus @ A @ X3 @ H ) @ S3 )
=> ( ( F2 @ ( plus_plus @ A @ X3 @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X3 ) @ ( F11 @ 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_5762_has__derivativeI__sandwich,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [E3: real,F11: A > B,S: set @ A,X3: A,F2: A > B,H7: A > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( ( real_V3181309239436604168linear @ A @ B @ F11 )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ S )
=> ( ( Y4 != X3 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y4 @ X3 ) @ E3 )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) @ ( F11 @ ( minus_minus @ A @ Y4 @ X3 ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y4 @ X3 ) ) ) @ ( H7 @ Y4 ) ) ) ) )
=> ( ( filterlim @ A @ real @ H7 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ) ).
% has_derivativeI_sandwich
thf(fact_5763_filterlim__pow__at__bot__even,axiom,
! [N2: nat,F2: real > real,F5: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F5 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( power_power @ real @ ( F2 @ X ) @ N2 )
@ ( at_top @ real )
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_5764_finite__greaterThanAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% finite_greaterThanAtMost_int
thf(fact_5765_open__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo1002775350975398744n_open @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% open_empty
thf(fact_5766_dist__add__cancel2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ B2 @ A2 ) @ ( plus_plus @ A @ C2 @ A2 ) )
= ( real_V557655796197034286t_dist @ A @ B2 @ C2 ) ) ) ).
% dist_add_cancel2
thf(fact_5767_dist__add__cancel,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ A2 @ C2 ) )
= ( real_V557655796197034286t_dist @ A @ B2 @ C2 ) ) ) ).
% dist_add_cancel
thf(fact_5768_dist__0__norm,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A] :
( ( real_V557655796197034286t_dist @ A @ ( zero_zero @ A ) @ X3 )
= ( real_V7770717601297561774m_norm @ A @ X3 ) ) ) ).
% dist_0_norm
thf(fact_5769_zero__less__dist__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X3 @ Y ) )
= ( X3 != Y ) ) ) ).
% zero_less_dist_iff
thf(fact_5770_open__Inter,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ S3 )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ S3 )
=> ( topolo1002775350975398744n_open @ A @ X5 ) )
=> ( topolo1002775350975398744n_open @ A @ ( complete_Inf_Inf @ ( set @ A ) @ S3 ) ) ) ) ) ).
% open_Inter
thf(fact_5771_open__INT,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: set @ B,B3: B > ( set @ A )] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( topolo1002775350975398744n_open @ A @ ( B3 @ X5 ) ) )
=> ( topolo1002775350975398744n_open @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) ) ) ) ) ).
% open_INT
thf(fact_5772_not__open__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X3: A] :
~ ( topolo1002775350975398744n_open @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% not_open_singleton
thf(fact_5773_separation__t2,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
= ( ? [U5: set @ A,V6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U5 )
& ( topolo1002775350975398744n_open @ A @ V6 )
& ( member @ A @ X3 @ U5 )
& ( member @ A @ Y @ V6 )
& ( ( inf_inf @ ( set @ A ) @ U5 @ V6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% separation_t2
thf(fact_5774_hausdorff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
=> ? [U6: set @ A,V7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U6 )
& ( topolo1002775350975398744n_open @ A @ V7 )
& ( member @ A @ X3 @ U6 )
& ( member @ A @ Y @ V7 )
& ( ( inf_inf @ ( set @ A ) @ U6 @ V7 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% hausdorff
thf(fact_5775_trivial__limit__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_bot @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_bot_linorder
thf(fact_5776_first__countable__basis,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X3: A] :
? [A8: nat > ( set @ A )] :
( ! [I5: nat] :
( ( member @ A @ X3 @ ( A8 @ I5 ) )
& ( topolo1002775350975398744n_open @ A @ ( A8 @ I5 ) ) )
& ! [S10: set @ A] :
( ( ( topolo1002775350975398744n_open @ A @ S10 )
& ( member @ A @ X3 @ S10 ) )
=> ? [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( A8 @ I4 ) @ S10 ) ) ) ) ).
% first_countable_basis
thf(fact_5777_open__subopen,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S7: set @ A] :
! [X: A] :
( ( member @ A @ X @ S7 )
=> ? [T8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T8 )
& ( member @ A @ X @ T8 )
& ( ord_less_eq @ ( set @ A ) @ T8 @ S7 ) ) ) ) ) ) ).
% open_subopen
thf(fact_5778_openI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ? [T10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T10 )
& ( member @ A @ X5 @ T10 )
& ( ord_less_eq @ ( set @ A ) @ T10 @ S3 ) ) )
=> ( topolo1002775350975398744n_open @ A @ S3 ) ) ) ).
% openI
thf(fact_5779_norm__conv__dist,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A )
= ( ^ [X: A] : ( real_V557655796197034286t_dist @ A @ X @ ( zero_zero @ A ) ) ) ) ) ).
% norm_conv_dist
thf(fact_5780_open__Collect__ex,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: B > A > $o] :
( ! [I4: B] : ( topolo1002775350975398744n_open @ A @ ( collect @ A @ ( P @ I4 ) ) )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X: A] :
? [I3: B] : ( P @ I3 @ X ) ) ) ) ) ).
% open_Collect_ex
thf(fact_5781_dist__triangle__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,Z2: A,Y: A,E3: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Z2 ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z2 ) ) @ E3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y ) @ E3 ) ) ) ).
% dist_triangle_lt
thf(fact_5782_dist__triangle__less__add,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X1: A,Y: A,E1: real,X22: A,E22: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ Y ) @ E1 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ Y ) @ E22 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X22 ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% dist_triangle_less_add
thf(fact_5783_open__ball,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,D2: real] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [Y2: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y2 ) @ D2 ) ) ) ) ).
% open_ball
thf(fact_5784_dist__commute__lessI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y: A,X3: A,E3: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X3 ) @ E3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y ) @ E3 ) ) ) ).
% dist_commute_lessI
thf(fact_5785_open__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S7: set @ A] :
! [X: A] :
( ( member @ A @ X @ S7 )
=> ? [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
& ! [Y2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y2 @ X ) @ E4 )
=> ( member @ A @ Y2 @ S7 ) ) ) ) ) ) ) ).
% open_dist
thf(fact_5786_Inf__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A3: set @ A,X3: A] :
( ( topolo1002775350975398744n_open @ A @ A3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less @ A @ X3 @ X5 ) )
=> ~ ( member @ A @ ( complete_Inf_Inf @ A @ A3 ) @ A3 ) ) ) ) ).
% Inf_notin_open
thf(fact_5787_Sup__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A3: set @ A,X3: A] :
( ( topolo1002775350975398744n_open @ A @ A3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less @ A @ X5 @ X3 ) )
=> ~ ( member @ A @ ( complete_Sup_Sup @ A @ A3 ) @ A3 ) ) ) ) ).
% Sup_notin_open
thf(fact_5788_dist__pos__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X3 @ Y ) ) ) ) ).
% dist_pos_lt
thf(fact_5789_dist__not__less__zero,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,Y: A] :
~ ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y ) @ ( zero_zero @ real ) ) ) ).
% dist_not_less_zero
thf(fact_5790_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] :
? [X: A,Y2: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X @ Y2 ) )
& ( X != Y2 ) ) ) ) ) ).
% open_diagonal_complement
thf(fact_5791_at__within__open__subset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A,S3: set @ A,T4: set @ A] :
( ( member @ A @ A2 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ T4 )
=> ( ( topolo174197925503356063within @ A @ A2 @ T4 )
= ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% at_within_open_subset
thf(fact_5792_open__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A,X3: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( member @ A @ X3 @ S3 )
=> ( ( ord_less @ A @ X3 @ Y )
=> ? [B5: A] :
( ( ord_less @ A @ X3 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ X3 @ B5 ) @ S3 ) ) ) ) ) ) ).
% open_right
thf(fact_5793_open__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A,X3: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( member @ A @ X3 @ S3 )
=> ( ( ord_less @ A @ Y @ X3 )
=> ? [B5: A] :
( ( ord_less @ A @ B5 @ X3 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ B5 @ X3 ) @ S3 ) ) ) ) ) ) ).
% open_left
thf(fact_5794_open__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X: A,Y2: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X @ Y2 ) )
& ( ord_less @ A @ X @ Y2 ) ) ) ) ) ).
% open_subdiagonal
thf(fact_5795_open__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X: A,Y2: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X @ Y2 ) )
& ( ord_less @ A @ Y2 @ X ) ) ) ) ) ).
% open_superdiagonal
thf(fact_5796_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N5: A] :
! [N: A] :
( ( ord_less_eq @ A @ N @ N5 )
=> ( P @ N ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_5797_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N5: A] :
! [N: A] :
( ( ord_less @ A @ N @ N5 )
=> ( P @ N ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_5798_countable__basis__at__decseq,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X3: A] :
~ ! [A8: nat > ( set @ A )] :
( ! [I5: nat] : ( topolo1002775350975398744n_open @ A @ ( A8 @ I5 ) )
=> ( ! [I5: nat] : ( member @ A @ X3 @ ( A8 @ I5 ) )
=> ~ ! [S10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S10 )
=> ( ( member @ A @ X3 @ S10 )
=> ( eventually @ nat
@ ^ [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( A8 @ I3 ) @ S10 )
@ ( at_top @ nat ) ) ) ) ) ) ) ).
% countable_basis_at_decseq
thf(fact_5799_eventually__at,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A2: A,S3: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
= ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X: A] :
( ( member @ A @ X @ S3 )
=> ( ( ( X != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ D5 ) )
=> ( P @ X ) ) ) ) ) ) ) ).
% eventually_at
thf(fact_5800_eventually__nhds__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A2: A] :
( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ A2 ) )
= ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ D5 )
=> ( P @ X ) ) ) ) ) ) ).
% eventually_nhds_metric
thf(fact_5801_has__field__derivative__transform__within,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F11: A,A2: A,S3: set @ A,D2: real,G: A > A] :
( ( has_field_derivative @ A @ F2 @ F11 @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ( member @ A @ A2 @ S3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ D2 )
=> ( ( F2 @ X5 )
= ( G @ X5 ) ) ) )
=> ( has_field_derivative @ A @ G @ F11 @ ( topolo174197925503356063within @ A @ A2 @ S3 ) ) ) ) ) ) ) ).
% has_field_derivative_transform_within
thf(fact_5802_has__derivative__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F11: A > B,X3: A,S: set @ A,D2: real,G: A > B] :
( ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ( member @ A @ X3 @ S )
=> ( ! [X17: A] :
( ( member @ A @ X17 @ S )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X17 @ X3 ) @ D2 )
=> ( ( F2 @ X17 )
= ( G @ X17 ) ) ) )
=> ( has_derivative @ A @ B @ G @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ) ).
% has_derivative_transform_within
thf(fact_5803_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ A @ X @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_5804_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X: A] : ( ord_less @ A @ X @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_5805_Cauchy__def,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X6: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M9: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M9 @ M2 )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ M9 @ N )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M2 ) @ ( X6 @ N ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_def
thf(fact_5806_Cauchy__altdef2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [S8: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N5: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N5 @ N )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S8 @ N ) @ ( S8 @ N5 ) ) @ E4 ) ) ) ) ) ) ).
% Cauchy_altdef2
thf(fact_5807_metric__CauchyD,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,E3: real] :
( ( topolo3814608138187158403Cauchy @ A @ X8 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M8: nat] :
! [M6: nat] :
( ( ord_less_eq @ nat @ M8 @ M6 )
=> ! [N6: nat] :
( ( ord_less_eq @ nat @ M8 @ N6 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M6 ) @ ( X8 @ N6 ) ) @ E3 ) ) ) ) ) ) ).
% metric_CauchyD
thf(fact_5808_metric__CauchyI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M10 @ M3 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M10 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% metric_CauchyI
thf(fact_5809_lim__explicit,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,F0: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ F0 ) @ ( at_top @ nat ) )
= ( ! [S7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S7 )
=> ( ( member @ A @ F0 @ S7 )
=> ? [N5: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N5 @ N )
=> ( member @ A @ ( F2 @ N ) @ S7 ) ) ) ) ) ) ) ).
% lim_explicit
thf(fact_5810_Lim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L: B,X3: A,S3: set @ A,D2: real,G: A > B] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X3 @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [X17: A] :
( ( member @ A @ X17 @ S3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X17 @ X3 ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X17 @ X3 ) @ D2 )
=> ( ( F2 @ X17 )
= ( G @ X17 ) ) ) ) )
=> ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ) ) ).
% Lim_transform_within
thf(fact_5811_dist__triangle__half__r,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y: A,X1: A,E3: real,X22: 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 @ X22 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X22 ) @ E3 ) ) ) ) ).
% dist_triangle_half_r
thf(fact_5812_dist__triangle__half__l,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X1: A,Y: A,E3: real,X22: 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 @ X22 @ Y ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X22 ) @ E3 ) ) ) ) ).
% dist_triangle_half_l
thf(fact_5813_eventually__at__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A2: A,S3: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
= ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X: A] :
( ( member @ A @ X @ S3 )
=> ( ( ( X != A2 )
& ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ D5 ) )
=> ( P @ X ) ) ) ) ) ) ) ).
% eventually_at_le
thf(fact_5814_dist__triangle__third,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X1: A,X22: A,E3: real,X32: A,X42: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X22 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ 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_5815_at__within__nhd,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X3: A,S3: set @ A,T4: set @ A,U3: set @ A] :
( ( member @ A @ X3 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ T4 @ S3 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ U3 @ S3 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( topolo174197925503356063within @ A @ X3 @ T4 )
= ( topolo174197925503356063within @ A @ X3 @ U3 ) ) ) ) ) ) ).
% at_within_nhd
thf(fact_5816_filterlim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [G: A > B,G6: filter @ B,X3: A,S3: set @ A,F5: filter @ B,D2: real,F2: A > B] :
( ( filterlim @ A @ B @ G @ G6 @ ( topolo174197925503356063within @ A @ X3 @ S3 ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ G6 @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [X17: A] :
( ( member @ A @ X17 @ S3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X17 @ X3 ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X17 @ X3 ) @ D2 )
=> ( ( F2 @ X17 )
= ( G @ X17 ) ) ) ) )
=> ( filterlim @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ) ) ) ).
% filterlim_transform_within
thf(fact_5817_Cauchy__altdef,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [F3: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M9: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M9 @ M2 )
=> ! [N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ M2 ) @ ( F3 @ N ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_5818_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M10 @ M3 )
=> ! [N3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% CauchyI'
thf(fact_5819_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_5820_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
! [A: $tType,F2: A > real,F5: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_infinity @ real ) @ F5 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less @ real @ ( F2 @ X ) @ ( zero_zero @ real ) )
@ F5 )
=> ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F5 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_bot
thf(fact_5821_tendstoD,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L: A,F5: filter @ B,E3: real] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ L ) @ E3 )
@ F5 ) ) ) ) ).
% tendstoD
thf(fact_5822_tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L: A,F5: filter @ B] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ L ) @ E )
@ F5 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ).
% tendstoI
thf(fact_5823_tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
= ( ! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ L ) @ E4 )
@ F5 ) ) ) ) ) ).
% tendsto_iff
thf(fact_5824_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ Z9 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_5825_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ Z9 )
@ F5 ) ) ) ) ).
% filterlim_at_bot
thf(fact_5826_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ Z9 )
@ F5 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_5827_metric__LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [G: A > B,L: B,A2: A,R: real,F2: A > B] :
( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( ! [X5: A] :
( ( X5 != A2 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ R )
=> ( ( F2 @ X5 )
= ( G @ X5 ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_equal2
thf(fact_5828_metric__LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [A2: A,F2: A > B,L5: B] :
( ! [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
=> ? [S9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S9 )
& ! [X5: A] :
( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ S9 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X5 ) @ L5 ) @ R4 ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% metric_LIM_I
thf(fact_5829_metric__LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F2: A > B,L5: B,A2: A,R2: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [S2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S2 )
& ! [X2: A] :
( ( ( X2 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A2 ) @ S2 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X2 ) @ L5 ) @ R2 ) ) ) ) ) ) ).
% metric_LIM_D
thf(fact_5830_LIM__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F2: A > B,L5: B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( 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 )
& ! [X: A] :
( ( ( X != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ S8 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X ) @ L5 ) @ R5 ) ) ) ) ) ) ) ).
% LIM_def
thf(fact_5831_metric__LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: A,R2: real] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No: nat] :
! [N6: nat] :
( ( ord_less_eq @ nat @ No @ N6 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N6 ) @ L5 ) @ R2 ) ) ) ) ) ).
% metric_LIMSEQ_D
thf(fact_5832_metric__LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: A] :
( ! [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
=> ? [No2: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No2 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N3 ) @ L5 ) @ R4 ) ) )
=> ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% metric_LIMSEQ_I
thf(fact_5833_lim__sequentially,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ No3 @ N )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N ) @ L5 ) @ R5 ) ) ) ) ) ) ).
% lim_sequentially
thf(fact_5834_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X6: nat > A] :
! [J3: nat] :
? [M9: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M9 @ M2 )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ M9 @ N )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M2 ) @ ( X6 @ N ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_5835_metric__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,B2: B,A2: A,G: B > C,C2: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X5: A] :
( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ D4 ) )
=> ( ( F2 @ X5 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_compose2
thf(fact_5836_filterlim__tendsto__pos__mult__at__bot,axiom,
! [A: $tType,F2: A > real,C2: real,F5: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G @ ( at_bot @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( times_times @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_bot @ real )
@ F5 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_bot
thf(fact_5837_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z9: B] :
( ( ord_less @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ Z9 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_5838_metric__isCont__LIM__compose2,axiom,
! [D: $tType,C: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ D ) )
=> ! [A2: A,F2: A > C,G: C > D,L: D] :
( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( filterlim @ C @ D @ G @ ( topolo7230453075368039082e_nhds @ D @ L ) @ ( topolo174197925503356063within @ C @ ( F2 @ A2 ) @ ( top_top @ ( set @ C ) ) ) )
=> ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X5: A] :
( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ D4 ) )
=> ( ( F2 @ X5 )
!= ( F2 @ A2 ) ) ) )
=> ( filterlim @ A @ D
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ D @ L )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_isCont_LIM_compose2
thf(fact_5839_tendsto__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A2: A,S3: set @ A,F2: A > D,L5: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( member @ A @ A2 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( filterlim @ A @ D @ F2 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
= ( filterlim @ A @ D
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% tendsto_offset_zero_iff
thf(fact_5840_filterlim__inverse__at__bot,axiom,
! [A: $tType,F2: A > real,F5: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less @ real @ ( F2 @ X ) @ ( zero_zero @ real ) )
@ F5 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( inverse_inverse @ real @ ( F2 @ X ) )
@ ( at_bot @ real )
@ F5 ) ) ) ).
% filterlim_inverse_at_bot
thf(fact_5841_filterlim__tendsto__neg__mult__at__bot,axiom,
! [A: $tType,F2: A > real,C2: real,F5: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F5 )
=> ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( times_times @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_bot @ real )
@ F5 ) ) ) ) ).
% filterlim_tendsto_neg_mult_at_bot
thf(fact_5842_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 )
@ ( image @ A @ ( set @ A )
@ ^ [X: A] :
( collect @ A
@ ^ [Y2: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ E4 ) )
@ K3 ) ) ) ) ) ) ) ) ).
% totally_bounded_metric
thf(fact_5843_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ No3 )
& ! [N: nat] :
( ( ord_less_eq @ nat @ No3 @ N )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N ) @ L5 ) @ R5 ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_5844_DERIV__pos__imp__increasing__at__bot,axiom,
! [B2: real,F2: real > real,Flim: real] :
( ! [X5: real] :
( ( ord_less_eq @ real @ X5 @ B2 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) )
=> ( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_bot @ real ) )
=> ( ord_less @ real @ Flim @ ( F2 @ B2 ) ) ) ) ).
% DERIV_pos_imp_increasing_at_bot
thf(fact_5845_filterlim__pow__at__bot__odd,axiom,
! [N2: nat,F2: real > real,F5: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F5 )
=> ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( power_power @ real @ ( F2 @ X ) @ N2 )
@ ( at_bot @ real )
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_5846_Bfun__metric__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F3: A > B,F9: filter @ A] :
? [Y2: B,K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X ) @ Y2 ) @ K6 )
@ F9 ) ) ) ) ) ).
% Bfun_metric_def
thf(fact_5847_Bfun__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F3: A > B,F9: filter @ A] :
? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X ) ) @ K6 )
@ F9 ) ) ) ) ) ).
% Bfun_def
thf(fact_5848_BfunE,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( bfun @ A @ B @ F2 @ F5 )
=> ~ ! [B10: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B10 )
=> ~ ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) @ B10 )
@ F5 ) ) ) ) ).
% BfunE
thf(fact_5849_Bseq__add__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( bfun @ nat @ A
@ ^ [X: nat] : ( plus_plus @ A @ ( F2 @ X ) @ C2 )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ).
% Bseq_add_iff
thf(fact_5850_Bseq__add,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X: nat] : ( plus_plus @ A @ ( F2 @ X ) @ C2 )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_add
thf(fact_5851_Bseq__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( bfun @ nat @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ).
% Bseq_Suc_iff
thf(fact_5852_Bseq__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,K: nat] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [N: nat] : ( X8 @ ( plus_plus @ nat @ N @ K ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_ignore_initial_segment
thf(fact_5853_Bseq__offset,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,K: nat] :
( ( bfun @ nat @ A
@ ^ [N: nat] : ( X8 @ ( plus_plus @ nat @ N @ K ) )
@ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) ) ) ).
% Bseq_offset
thf(fact_5854_Bseq__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( bfun @ nat @ A
@ ^ [X: nat] : ( times_times @ A @ C2 @ ( F2 @ X ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_cmult_iff
thf(fact_5855_BseqD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
=> ? [K8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K8 )
& ! [N6: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N6 ) ) @ K8 ) ) ) ) ).
% BseqD
thf(fact_5856_BseqE,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
=> ~ ! [K8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K8 )
=> ~ ! [N6: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N6 ) ) @ K8 ) ) ) ) ).
% BseqE
thf(fact_5857_BseqI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [K5: real,X8: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N3 ) ) @ K5 )
=> ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) ) ) ) ).
% BseqI
thf(fact_5858_Bseq__def,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N ) ) @ K6 ) ) ) ) ) ).
% Bseq_def
thf(fact_5859_Bseq__iff1a,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [N5: nat] :
! [N: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ).
% Bseq_iff1a
thf(fact_5860_Bseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [N5: nat] :
! [N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ).
% Bseq_iff
thf(fact_5861_Bseq__iff2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [X: A] :
! [N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X8 @ N ) @ ( uminus_uminus @ A @ X ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff2
thf(fact_5862_Bseq__iff3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [N5: nat] :
! [N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X8 @ N ) @ ( uminus_uminus @ A @ ( X8 @ N5 ) ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff3
thf(fact_5863_Bfun__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A2: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( bfun @ B @ A
@ ^ [X: B] : ( inverse_inverse @ A @ ( F2 @ X ) )
@ F5 ) ) ) ) ).
% Bfun_inverse
thf(fact_5864_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G: B > A,Y7: set @ B,X8: set @ A,F5: filter @ B,F2: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ Y7 ) @ X8 )
=> ( ( eventually @ B
@ ^ [X: B] : ( member @ B @ X @ Y7 )
@ F5 )
=> ( ( map_filter_on @ A @ C @ X8 @ F2 @ ( map_filter_on @ B @ A @ Y7 @ G @ F5 ) )
= ( map_filter_on @ B @ C @ Y7 @ ( comp @ A @ C @ B @ F2 @ G ) @ F5 ) ) ) ) ).
% map_filter_on_comp
thf(fact_5865_cauchy__filter__metric,axiom,
! [A: $tType] :
( ( ( real_V768167426530841204y_dist @ A )
& ( topolo7287701948861334536_space @ A ) )
=> ( ( topolo6773858410816713723filter @ A )
= ( ^ [F9: filter @ A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [P3: A > $o] :
( ( eventually @ A @ P3 @ F9 )
& ! [X: A,Y2: A] :
( ( ( P3 @ X )
& ( P3 @ Y2 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ E4 ) ) ) ) ) ) ) ).
% cauchy_filter_metric
thf(fact_5866_GMVT,axiom,
! [A2: real,B2: real,F2: real > real,G: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ( ! [X5: real] :
( ( ( ord_less @ real @ A2 @ X5 )
& ( ord_less @ real @ X5 @ B2 ) )
=> ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ G ) )
=> ( ! [X5: real] :
( ( ( ord_less @ real @ A2 @ X5 )
& ( ord_less @ real @ X5 @ B2 ) )
=> ( differentiable @ real @ real @ G @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [G_c: real,F_c: real,C3: real] :
( ( has_field_derivative @ real @ G @ G_c @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
& ( has_field_derivative @ real @ F2 @ F_c @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ A2 @ C3 )
& ( ord_less @ real @ C3 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) @ G_c )
= ( times_times @ real @ ( minus_minus @ real @ ( G @ B2 ) @ ( G @ A2 ) ) @ F_c ) ) ) ) ) ) ) ) ).
% GMVT
thf(fact_5867_differentiable__cmult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Q4: B > A,C2: A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ ( Q4 @ T3 ) @ C2 )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q4 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_right_iff
thf(fact_5868_differentiable__cmult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: A,Q4: B > A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ C2 @ ( Q4 @ T3 ) )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q4 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_left_iff
thf(fact_5869_differentiable__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: filter @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ F5 )
=> ( ( differentiable @ A @ B @ G @ F5 )
=> ( differentiable @ A @ B
@ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) )
@ F5 ) ) ) ) ).
% differentiable_add
thf(fact_5870_differentiable__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,X3: A,S: set @ A,T2: set @ A] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ) ).
% differentiable_within_subset
thf(fact_5871_differentiable__sum,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [S: set @ A,F2: A > B > C,Net: filter @ B] :
( ( finite_finite2 @ A @ S )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( differentiable @ B @ C @ ( F2 @ X5 ) @ Net ) )
=> ( differentiable @ B @ C
@ ^ [X: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [A4: A] : ( F2 @ A4 @ X )
@ S )
@ Net ) ) ) ) ).
% differentiable_sum
thf(fact_5872_differentiable__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X3: A,S: set @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( G @ X3 )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ).
% differentiable_divide
thf(fact_5873_differentiable__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X3: A,S: set @ A] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( F2 @ X3 )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% differentiable_inverse
thf(fact_5874_MVT,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X5: real] :
( ( ord_less @ real @ A2 @ X5 )
=> ( ( ord_less @ real @ X5 @ B2 )
=> ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [L4: real,Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( has_field_derivative @ real @ F2 @ L4 @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) )
& ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ L4 ) ) ) ) ) ) ).
% MVT
thf(fact_5875_eventually__filtercomap__at__topological,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ B )
=> ! [P: A > $o,F2: A > B,A3: B,B3: set @ B] :
( ( eventually @ A @ P @ ( filtercomap @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ A3 @ B3 ) ) )
= ( ? [S7: set @ B] :
( ( topolo1002775350975398744n_open @ B @ S7 )
& ( member @ B @ A3 @ S7 )
& ! [X: A] :
( ( member @ B @ ( F2 @ X ) @ ( minus_minus @ ( set @ B ) @ ( inf_inf @ ( set @ B ) @ S7 @ B3 ) @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
=> ( P @ X ) ) ) ) ) ) ).
% eventually_filtercomap_at_topological
thf(fact_5876_at__within__eq,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [X: A,S8: set @ A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image @ ( set @ A ) @ ( filter @ A )
@ ^ [S7: set @ A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S7 @ S8 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [S7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S7 )
& ( member @ A @ X @ S7 ) ) ) ) ) ) ) ) ).
% at_within_eq
thf(fact_5877_filtercomap__bot,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( filtercomap @ A @ B @ F2 @ ( bot_bot @ ( filter @ B ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ).
% filtercomap_bot
thf(fact_5878_principal__le__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ ( principal @ A @ A3 ) @ ( principal @ A @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% principal_le_iff
thf(fact_5879_continuous__on__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( F2 @ X5 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_sgn
thf(fact_5880_continuous__on__add,axiom,
! [B: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [S: set @ D,F2: D > B,G: D > B] :
( ( topolo81223032696312382ous_on @ D @ B @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ D @ B @ S @ G )
=> ( topolo81223032696312382ous_on @ D @ B @ S
@ ^ [X: D] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_on_add
thf(fact_5881_continuous__on__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [S: set @ A,F2: A > B,G: A > C] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ C @ S @ G )
=> ( topolo81223032696312382ous_on @ A @ ( product_prod @ B @ C ) @ S
@ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_on_Pair
thf(fact_5882_continuous__on__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S: set @ A,F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ G )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( G @ X5 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_on_divide
thf(fact_5883_IVT2_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,B2: A,Y: B,A2: A] :
( ( ord_less_eq @ B @ ( F2 @ B2 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F2 @ A2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ? [X5: A] :
( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 )
& ( ( F2 @ X5 )
= Y ) ) ) ) ) ) ) ).
% IVT2'
thf(fact_5884_IVT_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,A2: A,Y: B,B2: A] :
( ( ord_less_eq @ B @ ( F2 @ A2 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ? [X5: A] :
( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 )
& ( ( F2 @ X5 )
= Y ) ) ) ) ) ) ) ).
% IVT'
thf(fact_5885_continuous__on__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S: set @ A,F2: A > B,T2: set @ A] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( topolo81223032696312382ous_on @ A @ B @ T2 @ F2 ) ) ) ) ).
% continuous_on_subset
thf(fact_5886_continuous__on__sing,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X3: A,F2: A > B] : ( topolo81223032696312382ous_on @ A @ B @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ F2 ) ) ).
% continuous_on_sing
thf(fact_5887_continuous__on__empty,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B] : ( topolo81223032696312382ous_on @ A @ B @ ( bot_bot @ ( set @ A ) ) @ F2 ) ) ).
% continuous_on_empty
thf(fact_5888_continuous__on__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( F2 @ X5 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_inverse
thf(fact_5889_continuous__on__compose2,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [T2: set @ A,G: A > B,S: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ A @ B @ T2 @ G )
=> ( ( topolo81223032696312382ous_on @ C @ A @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ S ) @ T2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S
@ ^ [X: C] : ( G @ ( F2 @ X ) ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_5890_bot__eq__principal__empty,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( principal @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bot_eq_principal_empty
thf(fact_5891_principal__eq__bot__iff,axiom,
! [A: $tType,X8: set @ A] :
( ( ( principal @ A @ X8 )
= ( bot_bot @ ( filter @ A ) ) )
= ( X8
= ( bot_bot @ ( set @ A ) ) ) ) ).
% principal_eq_bot_iff
thf(fact_5892_filtercomap__neq__bot,axiom,
! [A: $tType,B: $tType,F5: filter @ A,F2: B > A] :
( ! [P8: A > $o] :
( ( eventually @ A @ P8 @ F5 )
=> ? [X2: B] : ( P8 @ ( F2 @ X2 ) ) )
=> ( ( filtercomap @ B @ A @ F2 @ F5 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtercomap_neq_bot
thf(fact_5893_continuous__onI__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( dense_order @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( topolo1002775350975398744n_open @ B @ ( image @ A @ B @ F2 @ A3 ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( member @ A @ Y4 @ A3 )
=> ( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ A3 @ F2 ) ) ) ) ).
% continuous_onI_mono
thf(fact_5894_open__Collect__less,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% open_Collect_less
thf(fact_5895_nhds__discrete,axiom,
! [A: $tType] :
( ( topolo8865339358273720382pology @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [X: A] : ( principal @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% nhds_discrete
thf(fact_5896_continuous__on__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ A,F2: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S @ F2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( cos @ A @ ( F2 @ X5 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S
@ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_tan
thf(fact_5897_open__Collect__less__Int,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real,G: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ real @ S @ G )
=> ? [A8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A8 )
& ( ( inf_inf @ ( set @ A ) @ A8 @ S )
= ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ S )
& ( ord_less @ real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ).
% open_Collect_less_Int
thf(fact_5898_continuous__on__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ A,F2: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S @ F2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( sin @ A @ ( F2 @ X5 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S
@ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_cot
thf(fact_5899_continuous__on__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A3 @ F2 )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ A3 )
=> ( ( cosh @ A @ ( F2 @ X5 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ C @ A @ A3
@ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_tanh
thf(fact_5900_eventually__filtercomap__at__top__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F2: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_top @ A ) ) )
= ( ? [N5: A] :
! [X: B] :
( ( ord_less_eq @ A @ N5 @ ( F2 @ X ) )
=> ( P @ X ) ) ) ) ) ).
% eventually_filtercomap_at_top_linorder
thf(fact_5901_eventually__filtercomap__at__top__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: B > $o,F2: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_top @ A ) ) )
= ( ? [N5: A] :
! [X: B] :
( ( ord_less @ A @ N5 @ ( F2 @ X ) )
=> ( P @ X ) ) ) ) ) ).
% eventually_filtercomap_at_top_dense
thf(fact_5902_filtercomap__neq__bot__surj,axiom,
! [A: $tType,B: $tType,F5: filter @ A,F2: B > A] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( filtercomap @ B @ A @ F2 @ F5 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% filtercomap_neq_bot_surj
thf(fact_5903_eventually__filtercomap__at__bot__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F2: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_bot @ A ) ) )
= ( ? [N5: A] :
! [X: B] :
( ( ord_less_eq @ A @ ( F2 @ X ) @ N5 )
=> ( P @ X ) ) ) ) ) ).
% eventually_filtercomap_at_bot_linorder
thf(fact_5904_eventually__filtercomap__at__bot__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: B > $o,F2: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_bot @ A ) ) )
= ( ? [N5: A] :
! [X: B] :
( ( ord_less @ A @ ( F2 @ X ) @ N5 )
=> ( P @ X ) ) ) ) ) ).
% eventually_filtercomap_at_bot_dense
thf(fact_5905_open__Collect__positive,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ? [A8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A8 )
& ( ( inf_inf @ ( set @ A ) @ A8 @ S )
= ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ S )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) ) ) ) ) ) ) ) ).
% open_Collect_positive
thf(fact_5906_continuous__on__powr_H,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S: set @ C,F2: C > real,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S @ G )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ S )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X5 ) )
& ( ( ( F2 @ X5 )
= ( zero_zero @ real ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X5 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S
@ ^ [X: C] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_on_powr'
thf(fact_5907_continuous__on__log,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real,G: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ real @ S @ G )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X5 ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( F2 @ X5 )
!= ( one_one @ real ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X5 ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ).
% continuous_on_log
thf(fact_5908_tendsto__principal__singleton,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,X3: B] : ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( F2 @ X3 ) ) @ ( principal @ B @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% tendsto_principal_singleton
thf(fact_5909_nhds__discrete__open,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X3: A] :
( ( topolo1002775350975398744n_open @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( topolo7230453075368039082e_nhds @ A @ X3 )
= ( principal @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% nhds_discrete_open
thf(fact_5910_DERIV__atLeastAtMost__imp__continuous__on,axiom,
! [A: $tType] :
( ( ( ord @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,B2: A,F2: A > A] :
( ! [X5: A] :
( ( ord_less_eq @ A @ A2 @ X5 )
=> ( ( ord_less_eq @ A @ X5 @ B2 )
=> ? [Y5: A] : ( has_field_derivative @ A @ F2 @ Y5 @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 ) ) ) ).
% DERIV_atLeastAtMost_imp_continuous_on
thf(fact_5911_Rolle__deriv,axiom,
! [A2: real,B2: real,F2: real > real,F11: real > real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ( F2 @ A2 )
= ( F2 @ B2 ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X5: real] :
( ( ord_less @ real @ A2 @ X5 )
=> ( ( ord_less @ real @ X5 @ B2 )
=> ( has_derivative @ real @ real @ F2 @ ( F11 @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( ( F11 @ Z3 )
= ( ^ [V5: real] : ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% Rolle_deriv
thf(fact_5912_mvt,axiom,
! [A2: real,B2: real,F2: real > real,F11: real > real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X5: real] :
( ( ord_less @ real @ A2 @ X5 )
=> ( ( ord_less @ real @ X5 @ B2 )
=> ( has_derivative @ real @ real @ F2 @ ( F11 @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ~ ! [Xi: real] :
( ( ord_less @ real @ A2 @ Xi )
=> ( ( ord_less @ real @ Xi @ B2 )
=> ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
!= ( F11 @ Xi @ ( minus_minus @ real @ B2 @ A2 ) ) ) ) ) ) ) ) ).
% mvt
thf(fact_5913_continuous__on__Icc__at__leftD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,B2: A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ B2 ) ) @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ).
% continuous_on_Icc_at_leftD
thf(fact_5914_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,I6: set @ A,F5: A > ( set @ B ),F2: B > C,G6: D > ( set @ C ),J4: set @ D] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F5 @ I4 ) @ ( F5 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F5 @ J2 ) @ ( F5 @ I4 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F2
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image @ D @ ( filter @ C )
@ ^ [J3: D] : ( principal @ C @ ( G6 @ J3 ) )
@ J4 ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image @ A @ ( filter @ B )
@ ^ [I3: A] : ( principal @ B @ ( F5 @ I3 ) )
@ I6 ) ) )
= ( ! [X: D] :
( ( member @ D @ X @ J4 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ I6 )
& ! [Z6: B] :
( ( member @ B @ Z6 @ ( F5 @ Y2 ) )
=> ( member @ C @ ( F2 @ Z6 ) @ ( G6 @ X ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_5915_DERIV__pos__imp__increasing__open,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ord_less @ real @ A2 @ X5 )
=> ( ( ord_less @ real @ X5 @ B2 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ord_less @ real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ) ).
% DERIV_pos_imp_increasing_open
thf(fact_5916_DERIV__neg__imp__decreasing__open,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ord_less @ real @ A2 @ X5 )
=> ( ( ord_less @ real @ X5 @ B2 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ord_less @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ) ).
% DERIV_neg_imp_decreasing_open
thf(fact_5917_DERIV__isconst__end,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X5: real] :
( ( ord_less @ real @ A2 @ X5 )
=> ( ( ord_less @ real @ X5 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ( F2 @ B2 )
= ( F2 @ A2 ) ) ) ) ) ).
% DERIV_isconst_end
thf(fact_5918_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X8: set @ A,F2: A > ( set @ B )] :
( ( finite_finite2 @ A @ X8 )
=> ( ( complete_Inf_Inf @ ( filter @ B )
@ ( image @ A @ ( filter @ B )
@ ^ [X: A] : ( principal @ B @ ( F2 @ X ) )
@ X8 ) )
= ( principal @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ X8 ) ) ) ) ) ).
% INF_principal_finite
thf(fact_5919_at__within__def,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [A4: A,S8: set @ A] : ( inf_inf @ ( filter @ A ) @ ( topolo7230453075368039082e_nhds @ A @ A4 ) @ ( principal @ A @ ( minus_minus @ ( set @ A ) @ S8 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% at_within_def
thf(fact_5920_DERIV__isconst2,axiom,
! [A2: real,B2: real,F2: real > real,X3: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X5: real] :
( ( ord_less @ real @ A2 @ X5 )
=> ( ( ord_less @ real @ X5 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ( ( F2 @ X3 )
= ( F2 @ A2 ) ) ) ) ) ) ) ).
% DERIV_isconst2
thf(fact_5921_at__left__eq,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ X3 )
=> ( ( topolo174197925503356063within @ A @ X3 @ ( set_ord_lessThan @ A @ X3 ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A4: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ A4 @ X3 ) )
@ ( set_ord_lessThan @ A @ X3 ) ) ) ) ) ) ).
% at_left_eq
thf(fact_5922_Rolle,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ( F2 @ A2 )
= ( F2 @ B2 ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X5: real] :
( ( ord_less @ real @ A2 @ X5 )
=> ( ( ord_less @ real @ X5 @ B2 )
=> ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% Rolle
thf(fact_5923_complete__uniform,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo2479028161051973599mplete @ A )
= ( ^ [S7: set @ A] :
! [F9: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F9 @ ( principal @ A @ S7 ) )
=> ( ( F9
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( topolo6773858410816713723filter @ A @ F9 )
=> ? [X: A] :
( ( member @ A @ X @ S7 )
& ( ord_less_eq @ ( filter @ A ) @ F9 @ ( topolo7230453075368039082e_nhds @ A @ X ) ) ) ) ) ) ) ) ) ).
% complete_uniform
thf(fact_5924_at__within__order,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X3: A,S: set @ A] :
( ( ( top_top @ ( set @ A ) )
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( topolo174197925503356063within @ A @ X3 @ S )
= ( inf_inf @ ( filter @ A )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A4: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A4 ) @ S ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( set_ord_greaterThan @ A @ X3 ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A4: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A4 ) @ S ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( set_ord_lessThan @ A @ X3 ) ) ) ) ) ) ) ).
% at_within_order
thf(fact_5925_continuous__on__IccI,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A2: A,B2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ B2 ) ) @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) )
=> ( ! [X5: A] :
( ( ord_less @ A @ A2 @ X5 )
=> ( ( ord_less @ A @ X5 @ B2 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X5 ) ) @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 ) ) ) ) ) ) ).
% continuous_on_IccI
thf(fact_5926_greaterThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) )
= ( ord_less @ A @ K @ I ) ) ) ).
% greaterThan_iff
thf(fact_5927_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X3 ) @ ( set_ord_greaterThan @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% greaterThan_subset_iff
thf(fact_5928_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A] :
( ( ord_less @ A @ X3 @ ( top_top @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X3 ) )
= ( top_top @ A ) ) ) ) ).
% Sup_greaterThanAtLeast
thf(fact_5929_greaterThan__non__empty,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X3: A] :
( ( set_ord_greaterThan @ A @ X3 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% greaterThan_non_empty
thf(fact_5930_infinite__Ioi,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [A2: A] :
~ ( finite_finite2 @ A @ ( set_ord_greaterThan @ A @ A2 ) ) ) ).
% infinite_Ioi
thf(fact_5931_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less @ A @ L2 ) ) ) ) ) ).
% greaterThan_def
thf(fact_5932_trivial__limit__at__right__real,axiom,
! [A: $tType] :
( ( ( dense_order @ A )
& ( no_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X3: A] :
( ( topolo174197925503356063within @ A @ X3 @ ( set_ord_greaterThan @ A @ X3 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_right_real
thf(fact_5933_INT__greaterThan__UNIV,axiom,
( ( complete_Inf_Inf @ ( set @ nat ) @ ( image @ nat @ ( set @ nat ) @ ( set_ord_greaterThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% INT_greaterThan_UNIV
thf(fact_5934_eventually__at__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X3: A,Y: A,P: A > $o] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X3 @ ( set_ord_greaterThan @ A @ X3 ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ X3 @ B4 )
& ! [Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ B4 )
=> ( P @ Y2 ) ) ) ) ) ) ) ) ).
% eventually_at_right
thf(fact_5935_eventually__at__right__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X3: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X3 @ ( set_ord_greaterThan @ A @ X3 ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ X3 @ B4 )
& ! [Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ B4 )
=> ( P @ Y2 ) ) ) ) ) ) ) ).
% eventually_at_right_field
thf(fact_5936_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_5937_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_5938_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_5939_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_5940_eventually__at__right__less,axiom,
! [A: $tType] :
( ( ( no_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X3: A] : ( eventually @ A @ ( ord_less @ A @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( set_ord_greaterThan @ A @ X3 ) ) ) ) ).
% eventually_at_right_less
thf(fact_5941_less__separate,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ? [A5: A,B5: A] :
( ( member @ A @ X3 @ ( set_ord_lessThan @ A @ A5 ) )
& ( member @ A @ Y @ ( set_ord_greaterThan @ A @ B5 ) )
& ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A5 ) @ ( set_ord_greaterThan @ A @ B5 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% less_separate
thf(fact_5942_eventually__at__rightI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
=> ( P @ X5 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% eventually_at_rightI
thf(fact_5943_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_5944_eventually__at__right__real,axiom,
! [A2: real,B2: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( eventually @ real
@ ^ [X: real] : ( member @ real @ X @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ).
% eventually_at_right_real
thf(fact_5945_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_ord_greaterThan @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K ) @ ( insert @ nat @ ( suc @ K ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% greaterThan_Suc
thf(fact_5946_filterlim__times__pos,axiom,
! [A: $tType,B: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: B > A,P4: A,F14: filter @ B,C2: A,L: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ P4 @ ( set_ord_greaterThan @ A @ P4 ) ) @ F14 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( L
= ( times_times @ A @ C2 @ P4 ) )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( times_times @ A @ C2 @ ( F2 @ X ) )
@ ( topolo174197925503356063within @ A @ L @ ( set_ord_greaterThan @ A @ L ) )
@ F14 ) ) ) ) ) ).
% filterlim_times_pos
thf(fact_5947_tendsto__imp__filterlim__at__right,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L5: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less @ B @ L5 @ ( F2 @ X ) )
@ F5 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L5 @ ( set_ord_greaterThan @ B @ L5 ) ) @ F5 ) ) ) ) ).
% tendsto_imp_filterlim_at_right
thf(fact_5948_continuous__on__Icc__at__rightD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,B2: A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% continuous_on_Icc_at_rightD
thf(fact_5949_filterlim__at__bot__at__right,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A,A2: A] :
( ! [X5: A,Y4: A] :
( ( Q @ X5 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( ( F2 @ ( G @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( Q @ ( G @ X5 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( ! [B5: A] :
( ( Q @ B5 )
=> ( ord_less @ A @ A2 @ B5 ) )
=> ( ( eventually @ B @ P @ ( at_bot @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_bot_at_right
thf(fact_5950_nhds__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [X: A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image @ real @ ( filter @ A )
@ ^ [E4: real] :
( principal @ A
@ ( collect @ A
@ ^ [Y2: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y2 @ X ) @ E4 ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% nhds_metric
thf(fact_5951_at__right__eq,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( topolo174197925503356063within @ A @ X3 @ ( set_ord_greaterThan @ A @ X3 ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A4: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ X3 @ A4 ) )
@ ( set_ord_greaterThan @ A @ X3 ) ) ) ) ) ) ).
% at_right_eq
thf(fact_5952_isCont__If__ge,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,G: A > B,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) @ G )
=> ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( G @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( if @ B @ ( ord_less_eq @ A @ X @ A2 ) @ ( G @ X ) @ ( F2 @ X ) ) ) ) ) ) ).
% isCont_If_ge
thf(fact_5953_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S3: set @ A] :
( ! [A5: A,B5: A,X5: A] :
( ( member @ A @ A5 @ S3 )
=> ( ( member @ A @ B5 @ S3 )
=> ( ( ord_less_eq @ A @ A5 @ X5 )
=> ( ( ord_less_eq @ A @ X5 @ B5 )
=> ( member @ A @ X5 @ S3 ) ) ) ) )
=> ? [A5: A,B5: A] :
( ( S3
= ( bot_bot @ ( set @ A ) ) )
| ( S3
= ( top_top @ ( set @ A ) ) )
| ( S3
= ( set_ord_lessThan @ A @ B5 ) )
| ( S3
= ( set_ord_atMost @ A @ B5 ) )
| ( S3
= ( set_ord_greaterThan @ A @ A5 ) )
| ( S3
= ( set_ord_atLeast @ A @ A5 ) )
| ( S3
= ( set_or5935395276787703475ssThan @ A @ A5 @ B5 ) )
| ( S3
= ( set_or3652927894154168847AtMost @ A @ A5 @ B5 ) )
| ( S3
= ( set_or7035219750837199246ssThan @ A @ A5 @ B5 ) )
| ( S3
= ( set_or1337092689740270186AtMost @ A @ A5 @ B5 ) ) ) ) ) ).
% interval_cases
thf(fact_5954_sequentially__imp__eventually__at__right,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ! [F4: nat > A] :
( ! [N6: nat] : ( ord_less @ A @ A2 @ ( F4 @ N6 ) )
=> ( ! [N6: nat] : ( ord_less @ A @ ( F4 @ N6 ) @ B2 )
=> ( ( order_antimono @ nat @ A @ F4 )
=> ( ( filterlim @ nat @ A @ F4 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N: nat] : ( P @ ( F4 @ N ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_right
thf(fact_5955_tendsto__at__right__sequentially,axiom,
! [C: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: B,B2: B,X8: B > C,L5: C] :
( ( ord_less @ B @ A2 @ B2 )
=> ( ! [S5: nat > B] :
( ! [N6: nat] : ( ord_less @ B @ A2 @ ( S5 @ N6 ) )
=> ( ! [N6: nat] : ( ord_less @ B @ ( S5 @ N6 ) @ B2 )
=> ( ( order_antimono @ nat @ B @ S5 )
=> ( ( filterlim @ nat @ B @ S5 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C
@ ^ [N: nat] : ( X8 @ ( S5 @ N ) )
@ ( topolo7230453075368039082e_nhds @ C @ L5 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ C @ X8 @ ( topolo7230453075368039082e_nhds @ C @ L5 ) @ ( topolo174197925503356063within @ B @ A2 @ ( set_ord_greaterThan @ B @ A2 ) ) ) ) ) ) ).
% tendsto_at_right_sequentially
thf(fact_5956_atLeast__0,axiom,
( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
= ( top_top @ ( set @ nat ) ) ) ).
% atLeast_0
thf(fact_5957_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
= ( ord_less_eq @ A @ K @ I ) ) ) ).
% atLeast_iff
thf(fact_5958_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_5959_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X3 ) @ ( set_ord_atLeast @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% atLeast_subset_iff
thf(fact_5960_image__add__atLeast,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A] :
( ( image @ A @ A @ ( plus_plus @ A @ K ) @ ( set_ord_atLeast @ A @ I ) )
= ( set_ord_atLeast @ A @ ( plus_plus @ A @ K @ I ) ) ) ) ).
% image_add_atLeast
thf(fact_5961_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_5962_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( set_ord_greaterThan @ nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_5963_atLeast__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ( set_ord_atLeast @ A @ X3 )
= ( top_top @ ( set @ A ) ) )
= ( X3
= ( bot_bot @ A ) ) ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_5964_infinite__Ici,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [A2: A] :
~ ( finite_finite2 @ A @ ( set_ord_atLeast @ A @ A2 ) ) ) ).
% infinite_Ici
thf(fact_5965_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less_eq @ A @ L2 ) ) ) ) ) ).
% atLeast_def
thf(fact_5966_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X3 ) ) ) ) ) ).
% antimonoD
thf(fact_5967_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X3 ) ) ) ) ) ).
% antimonoE
thf(fact_5968_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ Y4 ) @ ( F2 @ X5 ) ) )
=> ( order_antimono @ A @ B @ F2 ) ) ) ).
% antimonoI
thf(fact_5969_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F3: A > B] :
! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) ) ) ) ) ) ).
% antimono_def
thf(fact_5970_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_5971_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_5972_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_5973_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_5974_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_5975_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_5976_decseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F3: nat > A] :
! [N: nat] : ( ord_less_eq @ A @ ( F3 @ ( suc @ N ) ) @ ( F3 @ N ) ) ) ) ) ).
% decseq_Suc_iff
thf(fact_5977_decseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( order_antimono @ nat @ A @ X8 ) ) ) ).
% decseq_SucI
thf(fact_5978_decseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: nat > A,I: nat] :
( ( order_antimono @ nat @ A @ A3 )
=> ( ord_less_eq @ A @ ( A3 @ ( suc @ I ) ) @ ( A3 @ I ) ) ) ) ).
% decseq_SucD
thf(fact_5979_decseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,I: nat,J: nat] :
( ( order_antimono @ nat @ A @ F2 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( F2 @ J ) @ ( F2 @ I ) ) ) ) ) ).
% decseqD
thf(fact_5980_decseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X6: nat > A] :
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ A @ ( X6 @ N ) @ ( X6 @ M2 ) ) ) ) ) ) ).
% decseq_def
thf(fact_5981_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_5982_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_5983_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_5984_atMost__Int__atLeast,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ N2 ) @ ( set_ord_atLeast @ A @ N2 ) )
= ( insert @ A @ N2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atMost_Int_atLeast
thf(fact_5985_decseq__ge,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,L5: A,N2: nat] :
( ( order_antimono @ nat @ A @ X8 )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ L5 @ ( X8 @ N2 ) ) ) ) ) ).
% decseq_ge
thf(fact_5986_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K ) @ ( insert @ nat @ K @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast_Suc
thf(fact_5987_Gcd__eq__Max,axiom,
! [M7: set @ nat] :
( ( finite_finite2 @ nat @ M7 )
=> ( ( M7
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( gcd_Gcd @ nat @ M7 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image @ nat @ ( set @ nat )
@ ^ [M2: nat] :
( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M2 ) )
@ M7 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_5988_continuous__at__Sup__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,S3: set @ A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Sup_Sup @ A @ S3 ) @ ( set_ord_lessThan @ A @ ( complete_Sup_Sup @ A @ S3 ) ) ) @ F2 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S3 )
=> ( ( F2 @ ( complete_Sup_Sup @ A @ S3 ) )
= ( complete_Inf_Inf @ B @ ( image @ A @ B @ F2 @ S3 ) ) ) ) ) ) ) ) ).
% continuous_at_Sup_antimono
thf(fact_5989_bdd__above_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: set @ A,M7: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ A @ X5 @ M7 ) )
=> ( condit941137186595557371_above @ A @ A3 ) ) ) ).
% bdd_above.I
thf(fact_5990_bdd__above__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit941137186595557371_above @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_above_empty
thf(fact_5991_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Max_singleton
thf(fact_5992_Max__divisors__self__nat,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ N2 ) ) )
= N2 ) ) ).
% Max_divisors_self_nat
thf(fact_5993_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X3 )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_5994_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X3 )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_5995_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ B,C2: A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image @ B @ A
@ ^ [Uu3: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% Max_const
thf(fact_5996_bdd__above__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I6: set @ B,A3: B > ( set @ A )] :
( ( finite_finite2 @ B @ I6 )
=> ( ( condit941137186595557371_above @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ I6 )
=> ( condit941137186595557371_above @ A @ ( A3 @ X ) ) ) ) ) ) ) ).
% bdd_above_UN
thf(fact_5997_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ A3 ) )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ).
% Max_insert
thf(fact_5998_bdd__above_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A3: set @ B,F2: B > A,M7: A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ M7 ) )
=> ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ).
% bdd_above.I2
thf(fact_5999_Max_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ).
% Max.in_idem
thf(fact_6000_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ A3 ) ) ) ) ).
% Max_in
thf(fact_6001_bdd__above__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: set @ A,A3: set @ A] :
( ( condit941137186595557371_above @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( condit941137186595557371_above @ A @ A3 ) ) ) ) ).
% bdd_above_mono
thf(fact_6002_bdd__above_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: set @ A] :
( ( condit941137186595557371_above @ A @ A3 )
=> ~ ! [M8: A] :
~ ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ M8 ) ) ) ) ).
% bdd_above.E
thf(fact_6003_bdd__above_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit941137186595557371_above @ A )
= ( ^ [A6: set @ A] :
? [M9: A] :
! [X: A] :
( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ X @ M9 ) ) ) ) ) ).
% bdd_above.unfold
thf(fact_6004_cSup__upper,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A,X8: set @ A] :
( ( member @ A @ X3 @ X8 )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% cSup_upper
thf(fact_6005_cSup__upper2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A,X8: set @ A,Y: A] :
( ( member @ A @ X3 @ X8 )
=> ( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ord_less_eq @ A @ Y @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ).
% cSup_upper2
thf(fact_6006_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ).
% Max_ge
thf(fact_6007_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ A3 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= X3 ) ) ) ) ) ).
% Max_eqI
thf(fact_6008_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ B3 )
& ( ord_less_eq @ A @ X5 @ Xa ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B3 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ A3 )
& ( ord_less_eq @ A @ X5 @ Xa ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( lattic643756798349783984er_Max @ A @ B3 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_6009_Max_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ A @ A2 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ).
% Max.coboundedI
thf(fact_6010_bdd__above__finite,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( condit941137186595557371_above @ A @ A3 ) ) ) ).
% bdd_above_finite
thf(fact_6011_bdd__above__nat,axiom,
( ( condit941137186595557371_above @ nat )
= ( finite_finite2 @ nat ) ) ).
% bdd_above_nat
thf(fact_6012_cSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A3: set @ B,X3: B,U: A] :
( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( member @ B @ X3 @ A3 )
=> ( ( ord_less_eq @ A @ U @ ( F2 @ X3 ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_6013_cSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: B,A3: set @ B,F2: B > A] :
( ( member @ B @ X3 @ A3 )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% cSUP_upper
thf(fact_6014_cSup__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B3: set @ A,A3: set @ A] :
( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B3 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less_eq @ A @ B5 @ X2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B3 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_6015_cSup__le__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S3: set @ A,A2: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S3 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S3 ) @ A2 )
= ( ! [X: A] :
( ( member @ A @ X @ S3 )
=> ( ord_less_eq @ A @ X @ A2 ) ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_6016_less__cSup__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Y: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X8 ) )
= ( ? [X: A] :
( ( member @ A @ X @ X8 )
& ( ord_less @ A @ Y @ X ) ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_6017_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A3 )
= M )
= ( ( member @ A @ M @ A3 )
& ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ M ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_6018_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ? [X: A] :
( ( member @ A @ X @ A3 )
& ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_6019_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ( member @ A @ M @ A3 )
& ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ M ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_6020_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X3 )
=> ! [A11: A] :
( ( member @ A @ A11 @ A3 )
=> ( ord_less_eq @ A @ A11 @ X3 ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_6021_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A3 )
=> ( ord_less_eq @ A @ A5 @ X3 ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X3 ) ) ) ) ) ).
% Max.boundedI
thf(fact_6022_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ? [X: A] :
( ( member @ A @ X @ A3 )
& ( ord_less @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_6023_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ A3 )
=> ( ord_less_eq @ A @ B5 @ A2 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ A2 @ A3 ) )
= A2 ) ) ) ) ).
% Max_insert2
thf(fact_6024_Max__Sup,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% Max_Sup
thf(fact_6025_cSup__eq__Max,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= ( lattic643756798349783984er_Max @ A @ X8 ) ) ) ) ) ).
% cSup_eq_Max
thf(fact_6026_cSUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A3: set @ B,Y: A,I: B] :
( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ Y )
=> ( ( member @ B @ I @ A3 )
=> ( ord_less @ A @ ( F2 @ I ) @ Y ) ) ) ) ) ).
% cSUP_lessD
thf(fact_6027_Max_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Max.infinite
thf(fact_6028_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,G: C > A,B3: set @ C,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ C @ A @ G @ B3 ) )
=> ( ! [N3: B] :
( ( member @ B @ N3 @ A3 )
=> ? [X2: C] :
( ( member @ C @ X2 @ B3 )
& ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ X2 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B3 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_6029_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,U: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U )
= ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X ) @ U ) ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_6030_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_6031_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M7: set @ A,N7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M7 @ N7 )
=> ( ( M7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M7 ) @ ( lattic643756798349783984er_Max @ A @ N7 ) ) ) ) ) ) ).
% Max_mono
thf(fact_6032_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ ( lattic643756798349783984er_Max @ A @ B3 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_6033_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N7: set @ A] :
( ! [X5: A,Y4: A] :
( ( H2 @ ( ord_max @ A @ X5 @ Y4 ) )
= ( ord_max @ A @ ( H2 @ X5 ) @ ( H2 @ Y4 ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798349783984er_Max @ A @ N7 ) )
= ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ H2 @ N7 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_6034_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B3 ) @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ).
% Max.subset
thf(fact_6035_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ A3 ) )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_6036_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] : ( member @ A @ ( ord_max @ A @ X5 @ Y4 ) @ ( insert @ A @ X5 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Max.closed
thf(fact_6037_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: set @ B,F2: B > A,A2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A3 )
& ( ord_less @ A @ A2 @ ( F2 @ X ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_6038_card__le__Suc__Max,axiom,
! [S3: set @ nat] :
( ( finite_finite2 @ nat @ S3 )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ S3 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S3 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_6039_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X6: set @ nat] :
( if @ nat
@ ( X6
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X6 ) ) ) ) ).
% Sup_nat_def
thf(fact_6040_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M2: nat,N: nat] :
( if @ nat
@ ( N
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N ) @ M2 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_6041_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S3: set @ B,F2: B > A,K: A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( F2 @ X ) @ K )
@ S3 ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image @ B @ A @ F2 @ S3 ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_6042_gcd__is__Max__divisors__nat,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( gcd_gcd @ nat @ M @ N2 )
= ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D5: nat] :
( ( dvd_dvd @ nat @ D5 @ M )
& ( dvd_dvd @ nat @ D5 @ N2 ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_6043_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,G: B > A,B3: set @ B,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ G @ B3 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_6044_cSup__cInf,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S3: set @ A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S3 )
=> ( ( complete_Sup_Sup @ A @ S3 )
= ( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [X: A] :
! [Y2: A] :
( ( member @ A @ Y2 @ S3 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_6045_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_6046_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ A3 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ A3 ) )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_6047_cSUP__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A3: set @ C,B3: C > ( set @ D ),F2: D > B] :
( ( A3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ A3 )
=> ( ( B3 @ X5 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit941137186595557371_above @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image @ C @ ( set @ B )
@ ^ [X: C] : ( image @ D @ B @ F2 @ ( B3 @ X ) )
@ A3 ) ) )
=> ( ( complete_Sup_Sup @ B @ ( image @ D @ B @ F2 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ B3 @ A3 ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image @ C @ B
@ ^ [X: C] : ( complete_Sup_Sup @ B @ ( image @ D @ B @ F2 @ ( B3 @ X ) ) )
@ A3 ) ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_6048_sum__le__card__Max,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( times_times @ nat @ ( finite_card @ A @ A3 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ A @ nat @ F2 @ A3 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_6049_continuous__at__Inf__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,S3: set @ A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Inf_Inf @ A @ S3 ) @ ( set_ord_greaterThan @ A @ ( complete_Inf_Inf @ A @ S3 ) ) ) @ F2 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S3 )
=> ( ( F2 @ ( complete_Inf_Inf @ A @ S3 ) )
= ( complete_Sup_Sup @ B @ ( image @ A @ B @ F2 @ S3 ) ) ) ) ) ) ) ) ).
% continuous_at_Inf_antimono
thf(fact_6050_dual__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Min @ A
@ ^ [X: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X ) )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% dual_Min
thf(fact_6051_bdd__below_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: set @ A,M7: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ A @ M7 @ X5 ) )
=> ( condit1013018076250108175_below @ A @ A3 ) ) ) ).
% bdd_below.I
thf(fact_6052_bdd__belowI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: set @ A,M: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ A @ M @ X5 ) )
=> ( condit1013018076250108175_below @ A @ A3 ) ) ) ).
% bdd_belowI
thf(fact_6053_bdd__below__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit1013018076250108175_below @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_below_empty
thf(fact_6054_Max__divisors__self__int,axiom,
! [N2: int] :
( ( N2
!= ( zero_zero @ int ) )
=> ( ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ N2 ) ) )
= ( abs_abs @ int @ N2 ) ) ) ).
% Max_divisors_self_int
thf(fact_6055_bdd__below__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I6: set @ B,A3: B > ( set @ A )] :
( ( finite_finite2 @ B @ I6 )
=> ( ( condit1013018076250108175_below @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ I6 )
=> ( condit1013018076250108175_below @ A @ ( A3 @ X ) ) ) ) ) ) ) ).
% bdd_below_UN
thf(fact_6056_linorder_OMin_Ocong,axiom,
! [A: $tType] :
( ( lattices_Min @ A )
= ( lattices_Min @ A ) ) ).
% linorder.Min.cong
thf(fact_6057_bdd__below__finite,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( condit1013018076250108175_below @ A @ A3 ) ) ) ).
% bdd_below_finite
thf(fact_6058_bdd__below_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit1013018076250108175_below @ A )
= ( ^ [A6: set @ A] :
? [M9: A] :
! [X: A] :
( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ M9 @ X ) ) ) ) ) ).
% bdd_below.unfold
thf(fact_6059_bdd__below_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: set @ A] :
( ( condit1013018076250108175_below @ A @ A3 )
=> ~ ! [M8: A] :
~ ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ M8 @ X2 ) ) ) ) ).
% bdd_below.E
thf(fact_6060_bdd__below__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: set @ A,A3: set @ A] :
( ( condit1013018076250108175_below @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( condit1013018076250108175_below @ A @ A3 ) ) ) ) ).
% bdd_below_mono
thf(fact_6061_bdd__below_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A3: set @ B,M7: A,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ M7 @ ( F2 @ X5 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ).
% bdd_below.I2
thf(fact_6062_bdd__belowI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A3: set @ B,M: A,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ M @ ( F2 @ X5 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ).
% bdd_belowI2
thf(fact_6063_cInf__lower,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A,X8: set @ A] :
( ( member @ A @ X3 @ X8 )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X3 ) ) ) ) ).
% cInf_lower
thf(fact_6064_cInf__lower2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A,X8: set @ A,Y: A] :
( ( member @ A @ X3 @ X8 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y ) ) ) ) ) ).
% cInf_lower2
thf(fact_6065_cINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A3: set @ B,X3: B] :
( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( F2 @ X3 ) ) ) ) ) ).
% cINF_lower
thf(fact_6066_cINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A3: set @ B,X3: B,U: A] :
( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( member @ B @ X3 @ A3 )
=> ( ( ord_less_eq @ A @ ( F2 @ X3 ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U ) ) ) ) ) ).
% cINF_lower2
thf(fact_6067_le__cInf__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S3: set @ A,A2: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S3 )
=> ( ( ord_less_eq @ A @ A2 @ ( complete_Inf_Inf @ A @ S3 ) )
= ( ! [X: A] :
( ( member @ A @ X @ S3 )
=> ( ord_less_eq @ A @ A2 @ X ) ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_6068_cInf__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B3: set @ A,A3: set @ A] :
( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B3 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less_eq @ A @ X2 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) ) ) ) ) ).
% cInf_mono
thf(fact_6069_cInf__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Y: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y )
= ( ? [X: A] :
( ( member @ A @ X @ X8 )
& ( ord_less @ A @ X @ Y ) ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_6070_less__cINF__D,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A3: set @ B,Y: A,I: B] :
( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
=> ( ( member @ B @ I @ A3 )
=> ( ord_less @ A @ Y @ ( F2 @ I ) ) ) ) ) ) ).
% less_cINF_D
thf(fact_6071_gcd__is__Max__divisors__int,axiom,
! [N2: int,M: int] :
( ( N2
!= ( zero_zero @ int ) )
=> ( ( gcd_gcd @ int @ M @ N2 )
= ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D5: int] :
( ( dvd_dvd @ int @ D5 @ M )
& ( dvd_dvd @ int @ D5 @ N2 ) ) ) ) ) ) ).
% gcd_is_Max_divisors_int
thf(fact_6072_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,U: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( ord_less_eq @ A @ U @ ( F2 @ X ) ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_6073_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B3: set @ B,F2: C > A,A3: set @ C,G: B > A] :
( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ C @ A @ F2 @ A3 ) )
=> ( ! [M3: B] :
( ( member @ B @ M3 @ B3 )
=> ? [X2: C] :
( ( member @ C @ X2 @ A3 )
& ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G @ M3 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_6074_cInf__superset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B3 ) @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_6075_cInf__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ A2 @ X8 ) )
= ( inf_inf @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ).
% cInf_insert
thf(fact_6076_cInf__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( condit1013018076250108175_below @ A @ X8 )
=> ( ( ( X8
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ A2 @ X8 ) )
= A2 ) )
& ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ A2 @ X8 ) )
= ( inf_inf @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_6077_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: set @ B,F2: B > A,A2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ A2 )
= ( ? [X: B] :
( ( member @ B @ X @ A3 )
& ( ord_less @ A @ ( F2 @ X ) @ A2 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_6078_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ G @ A3 ) )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A3 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [A4: B] : ( inf_inf @ A @ ( F2 @ A4 ) @ ( G @ A4 ) )
@ A3 ) ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_6079_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,G: B > A,B3: set @ B,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ G @ B3 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ B3 )
=> ( ord_less_eq @ A @ ( G @ X5 ) @ ( F2 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_6080_less__eq__cInf__inter,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( condit1013018076250108175_below @ A @ A3 )
=> ( ( condit1013018076250108175_below @ A @ B3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_6081_cINF__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,A2: B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( insert @ B @ A2 @ A3 ) ) )
= ( inf_inf @ A @ ( F2 @ A2 ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ).
% cINF_insert
thf(fact_6082_cInf__le__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_6083_cInf__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S3: set @ A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S3 )
=> ( ( complete_Inf_Inf @ A @ S3 )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [X: A] :
! [Y2: A] :
( ( member @ A @ Y2 @ S3 )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_6084_cINF__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A3: set @ C,B3: C > ( set @ D ),F2: D > B] :
( ( A3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ A3 )
=> ( ( B3 @ X5 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit1013018076250108175_below @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image @ C @ ( set @ B )
@ ^ [X: C] : ( image @ D @ B @ F2 @ ( B3 @ X ) )
@ A3 ) ) )
=> ( ( complete_Inf_Inf @ B @ ( image @ D @ B @ F2 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ B3 @ A3 ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image @ C @ B
@ ^ [X: C] : ( complete_Inf_Inf @ B @ ( image @ D @ B @ F2 @ ( B3 @ X ) ) )
@ A3 ) ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_6085_uniformity__dist,axiom,
! [A: $tType] :
( ( real_V768167426530841204y_dist @ A )
=> ( ( topolo7806501430040627800ormity @ A )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ A @ A ) )
@ ( image @ real @ ( filter @ ( product_prod @ A @ A ) )
@ ^ [E4: real] :
( principal @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X: A,Y2: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ E4 ) ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ).
% uniformity_dist
thf(fact_6086_compactE__image,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,C5: set @ B,F2: B > ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ! [T5: B] :
( ( member @ B @ T5 @ C5 )
=> ( topolo1002775350975398744n_open @ A @ ( F2 @ T5 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ C5 ) ) )
=> ~ ! [C9: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C9 @ C5 )
=> ( ( finite_finite2 @ B @ C9 )
=> ~ ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ C9 ) ) ) ) ) ) ) ) ) ).
% compactE_image
thf(fact_6087_uniformity__bot,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo7806501430040627800ormity @ A )
!= ( bot_bot @ ( filter @ ( product_prod @ A @ A ) ) ) ) ) ).
% uniformity_bot
thf(fact_6088_compact__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo2193935891317330818ompact @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% compact_empty
thf(fact_6089_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 ) )
=> ~ ! [X2: A,Y5: A] :
( ( D7 @ ( product_Pair @ A @ A @ X2 @ Y5 ) )
=> ! [Z4: A] :
( ( D7 @ ( product_Pair @ A @ A @ Y5 @ Z4 ) )
=> ( E5 @ ( product_Pair @ A @ A @ X2 @ Z4 ) ) ) ) ) ) ) ).
% uniformity_transE
thf(fact_6090_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 ) )
& ! [X2: A,Y5: A,Z4: A] :
( ( D7 @ ( product_Pair @ A @ A @ X2 @ Y5 ) )
=> ( ( D7 @ ( product_Pair @ A @ A @ Y5 @ Z4 ) )
=> ( E5 @ ( product_Pair @ A @ A @ X2 @ Z4 ) ) ) ) ) ) ) ).
% uniformity_trans
thf(fact_6091_uniformity__refl,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o,X3: A] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ( E5 @ ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% uniformity_refl
thf(fact_6092_compact__attains__inf,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ S3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ S3 )
=> ( ord_less_eq @ A @ X5 @ Xa ) ) ) ) ) ) ).
% compact_attains_inf
thf(fact_6093_compact__attains__sup,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ S3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ S3 )
=> ( ord_less_eq @ A @ Xa @ X5 ) ) ) ) ) ) ).
% compact_attains_sup
thf(fact_6094_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
@ ^ [X: A,Y2: A] : ( E5 @ ( product_Pair @ A @ A @ Y2 @ X ) ) )
@ ( topolo7806501430040627800ormity @ A ) ) ) ) ).
% uniformity_sym
thf(fact_6095_continuous__attains__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ? [X5: A] :
( ( member @ A @ X5 @ S )
& ! [Xa: A] :
( ( member @ A @ Xa @ S )
=> ( ord_less_eq @ B @ ( F2 @ Xa ) @ ( F2 @ X5 ) ) ) ) ) ) ) ) ).
% continuous_attains_sup
thf(fact_6096_continuous__attains__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ? [X5: A] :
( ( member @ A @ X5 @ S )
& ! [Xa: A] :
( ( member @ A @ Xa @ S )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Xa ) ) ) ) ) ) ) ) ).
% continuous_attains_inf
thf(fact_6097_Cauchy__uniform__iff,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X6: nat > A] :
! [P3: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ P3 @ ( topolo7806501430040627800ormity @ A ) )
=> ? [N5: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N5 @ N )
=> ! [M2: nat] :
( ( ord_less_eq @ nat @ N5 @ M2 )
=> ( P3 @ ( product_Pair @ A @ A @ ( X6 @ N ) @ ( X6 @ M2 ) ) ) ) ) ) ) ) ) ).
% Cauchy_uniform_iff
thf(fact_6098_uniformity__complex__def,axiom,
( ( topolo7806501430040627800ormity @ complex )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ complex @ complex ) )
@ ( image @ real @ ( filter @ ( product_prod @ complex @ complex ) )
@ ^ [E4: real] :
( principal @ ( product_prod @ complex @ complex )
@ ( collect @ ( product_prod @ complex @ complex )
@ ( product_case_prod @ complex @ complex @ $o
@ ^ [X: complex,Y2: complex] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ complex @ X @ Y2 ) @ E4 ) ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% uniformity_complex_def
thf(fact_6099_uniformity__real__def,axiom,
( ( topolo7806501430040627800ormity @ real )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ real @ real ) )
@ ( image @ real @ ( filter @ ( product_prod @ real @ real ) )
@ ^ [E4: real] :
( principal @ ( product_prod @ real @ real )
@ ( collect @ ( product_prod @ real @ real )
@ ( product_case_prod @ real @ real @ $o
@ ^ [X: real,Y2: real] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ real @ X @ Y2 ) @ E4 ) ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% uniformity_real_def
thf(fact_6100_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 ) )
=> ? [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
& ! [X: A] :
( ( member @ A @ X @ S7 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ X6 )
& ( E6 @ ( product_Pair @ A @ A @ Y2 @ X ) ) ) ) ) ) ) ) ) ).
% totally_bounded_def
thf(fact_6101_eventually__uniformity__metric,axiom,
! [A: $tType] :
( ( real_V768167426530841204y_dist @ A )
=> ! [P: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ P @ ( topolo7806501430040627800ormity @ A ) )
= ( ? [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
& ! [X: A,Y2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ E4 )
=> ( P @ ( product_Pair @ A @ A @ X @ Y2 ) ) ) ) ) ) ) ).
% eventually_uniformity_metric
thf(fact_6102_tendsto__iff__uniformity,axiom,
! [A: $tType,B: $tType] :
( ( topolo7287701948861334536_space @ B )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
= ( ! [E6: ( product_prod @ B @ B ) > $o] :
( ( eventually @ ( product_prod @ B @ B ) @ E6 @ ( topolo7806501430040627800ormity @ B ) )
=> ( eventually @ A
@ ^ [X: A] : ( E6 @ ( product_Pair @ B @ B @ ( F2 @ X ) @ L ) )
@ F5 ) ) ) ) ) ).
% tendsto_iff_uniformity
thf(fact_6103_compactE,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,T11: set @ ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ T11 ) )
=> ( ! [B10: set @ A] :
( ( member @ ( set @ A ) @ B10 @ T11 )
=> ( topolo1002775350975398744n_open @ A @ B10 ) )
=> ~ ! [T12: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ T12 @ T11 )
=> ( ( finite_finite2 @ ( set @ A ) @ T12 )
=> ~ ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ T12 ) ) ) ) ) ) ) ) ).
% compactE
thf(fact_6104_compactI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A] :
( ! [C8: set @ ( set @ A )] :
( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C8 )
=> ( topolo1002775350975398744n_open @ A @ X2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ ( complete_Sup_Sup @ ( set @ A ) @ C8 ) )
=> ? [C10: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C10 @ C8 )
& ( finite_finite2 @ ( set @ A ) @ C10 )
& ( ord_less_eq @ ( set @ A ) @ S @ ( complete_Sup_Sup @ ( set @ A ) @ C10 ) ) ) ) )
=> ( topolo2193935891317330818ompact @ A @ S ) ) ) ).
% compactI
thf(fact_6105_compact__eq__Heine__Borel,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo2193935891317330818ompact @ A )
= ( ^ [S7: set @ A] :
! [C7: set @ ( set @ A )] :
( ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ C7 )
=> ( topolo1002775350975398744n_open @ A @ X ) )
& ( ord_less_eq @ ( set @ A ) @ S7 @ ( complete_Sup_Sup @ ( set @ A ) @ C7 ) ) )
=> ? [D8: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ D8 @ C7 )
& ( finite_finite2 @ ( set @ A ) @ D8 )
& ( ord_less_eq @ ( set @ A ) @ S7 @ ( complete_Sup_Sup @ ( set @ A ) @ D8 ) ) ) ) ) ) ) ).
% compact_eq_Heine_Borel
thf(fact_6106_prod__filter__INF2,axiom,
! [B: $tType,C: $tType,A: $tType,J4: set @ A,A3: filter @ B,B3: A > ( filter @ C )] :
( ( J4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ A3 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image @ A @ ( filter @ C ) @ B3 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I3: A] : ( prod_filter @ B @ C @ A3 @ ( B3 @ I3 ) )
@ J4 ) ) ) ) ).
% prod_filter_INF2
thf(fact_6107_prod__filter__INF1,axiom,
! [B: $tType,C: $tType,A: $tType,I6: set @ A,A3: A > ( filter @ B ),B3: filter @ C] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ A3 @ I6 ) ) @ B3 )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I3: A] : ( prod_filter @ B @ C @ ( A3 @ I3 ) @ B3 )
@ I6 ) ) ) ) ).
% prod_filter_INF1
thf(fact_6108_prod__filter__eq__bot,axiom,
! [A: $tType,B: $tType,A3: filter @ A,B3: filter @ B] :
( ( ( prod_filter @ A @ B @ A3 @ B3 )
= ( bot_bot @ ( filter @ ( product_prod @ A @ B ) ) ) )
= ( ( A3
= ( bot_bot @ ( filter @ A ) ) )
| ( B3
= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% prod_filter_eq_bot
thf(fact_6109_eventually__prod__same,axiom,
! [A: $tType,P: ( product_prod @ A @ A ) > $o,F5: filter @ A] :
( ( eventually @ ( product_prod @ A @ A ) @ P @ ( prod_filter @ A @ A @ F5 @ F5 ) )
= ( ? [Q7: A > $o] :
( ( eventually @ A @ Q7 @ F5 )
& ! [X: A,Y2: A] :
( ( Q7 @ X )
=> ( ( Q7 @ Y2 )
=> ( P @ ( product_Pair @ A @ A @ X @ Y2 ) ) ) ) ) ) ) ).
% eventually_prod_same
thf(fact_6110_eventually__prod__filter,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,F5: filter @ A,G6: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ P @ ( prod_filter @ A @ B @ F5 @ G6 ) )
= ( ? [Pf: A > $o,Pg: B > $o] :
( ( eventually @ A @ Pf @ F5 )
& ( eventually @ B @ Pg @ G6 )
& ! [X: A,Y2: B] :
( ( Pf @ X )
=> ( ( Pg @ Y2 )
=> ( P @ ( product_Pair @ A @ B @ X @ Y2 ) ) ) ) ) ) ) ).
% eventually_prod_filter
thf(fact_6111_prod__filter__mono__iff,axiom,
! [A: $tType,B: $tType,A3: filter @ A,B3: filter @ B,C5: filter @ A,D6: filter @ B] :
( ( A3
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( B3
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ A3 @ B3 ) @ ( prod_filter @ A @ B @ C5 @ D6 ) )
= ( ( ord_less_eq @ ( filter @ A ) @ A3 @ C5 )
& ( ord_less_eq @ ( filter @ B ) @ B3 @ D6 ) ) ) ) ) ).
% prod_filter_mono_iff
thf(fact_6112_eventually__prod__sequentially,axiom,
! [P: ( product_prod @ nat @ nat ) > $o] :
( ( eventually @ ( product_prod @ nat @ nat ) @ P @ ( prod_filter @ nat @ nat @ ( at_top @ nat ) @ ( at_top @ nat ) ) )
= ( ? [N5: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ N5 @ M2 )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ N5 @ N )
=> ( P @ ( product_Pair @ nat @ nat @ N @ M2 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_6113_eventually__prod2,axiom,
! [A: $tType,B: $tType,A3: filter @ A,P: B > $o,B3: filter @ B] :
( ( A3
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X: A] : P )
@ ( prod_filter @ A @ B @ A3 @ B3 ) )
= ( eventually @ B @ P @ B3 ) ) ) ).
% eventually_prod2
thf(fact_6114_eventually__prod1,axiom,
! [A: $tType,B: $tType,B3: filter @ A,P: B > $o,A3: filter @ B] :
( ( B3
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ^ [X: B,Y2: A] : ( P @ X ) )
@ ( prod_filter @ B @ A @ A3 @ B3 ) )
= ( eventually @ B @ P @ A3 ) ) ) ).
% eventually_prod1
thf(fact_6115_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_6116_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B,G6: filter @ B,F5: filter @ A,G: A > C,H7: filter @ C] :
( ( filterlim @ A @ B @ F2 @ G6 @ F5 )
=> ( ( filterlim @ A @ C @ G @ H7 @ F5 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) )
@ ( prod_filter @ B @ C @ G6 @ H7 )
@ F5 ) ) ) ).
% filterlim_Pair
thf(fact_6117_tendsto__add__Pair,axiom,
! [A: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [A2: A,B2: A] :
( filterlim @ ( product_prod @ A @ A ) @ A
@ ^ [X: product_prod @ A @ A] : ( plus_plus @ A @ ( product_fst @ A @ A @ X ) @ ( product_snd @ A @ A @ X ) )
@ ( 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_6118_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 )
@ ^ [X: A,Y2: A] :
( product_case_prod @ A @ A @ $o
@ ^ [Y6: A,Z6: A] :
( ( Y2 = Y6 )
=> ( E5 @ ( product_Pair @ A @ A @ X @ Z6 ) ) ) ) ) )
@ ( prod_filter @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ ( topolo7806501430040627800ormity @ A ) @ ( topolo7806501430040627800ormity @ A ) ) ) ) ) ).
% uniformity_trans'
thf(fact_6119_prod__filter__INF,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,I6: set @ A,J4: set @ B,A3: A > ( filter @ C ),B3: B > ( filter @ D )] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( J4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( prod_filter @ C @ D @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image @ A @ ( filter @ C ) @ A3 @ I6 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image @ B @ ( filter @ D ) @ B3 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image @ A @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [I3: A] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image @ B @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [J3: B] : ( prod_filter @ C @ D @ ( A3 @ I3 ) @ ( B3 @ J3 ) )
@ J4 ) )
@ I6 ) ) ) ) ) ).
% prod_filter_INF
thf(fact_6120_sequentially__imp__eventually__at__left,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: A,A2: A,P: A > $o] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ! [F4: nat > A] :
( ! [N6: nat] : ( ord_less @ A @ B2 @ ( F4 @ N6 ) )
=> ( ! [N6: nat] : ( ord_less @ A @ ( F4 @ N6 ) @ A2 )
=> ( ( order_mono @ nat @ A @ F4 )
=> ( ( filterlim @ nat @ A @ F4 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N: nat] : ( P @ ( F4 @ N ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_left
thf(fact_6121_relpow__finite__bounded1,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R )
@ ( collect @ nat
@ ^ [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_6122_finite__relpow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),N2: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ).
% finite_relpow
thf(fact_6123_mono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_mono @ nat @ A
@ ^ [I3: nat] : ( compow @ ( A > A ) @ I3 @ Q @ ( bot_bot @ A ) ) ) ) ) ).
% mono_funpow
thf(fact_6124_relpow__Suc__I2,axiom,
! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A ),Z2: A,N2: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) ) ) ) ).
% relpow_Suc_I2
thf(fact_6125_relpow__Suc__E2,axiom,
! [A: $tType,X3: A,Z2: A,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ).
% relpow_Suc_E2
thf(fact_6126_relpow__Suc__D2,axiom,
! [A: $tType,X3: A,Z2: A,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) )
=> ? [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ).
% relpow_Suc_D2
thf(fact_6127_relpow__Suc__I,axiom,
! [A: $tType,X3: A,Y: A,N2: nat,R: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) ) ) ) ).
% relpow_Suc_I
thf(fact_6128_relpow__Suc__E,axiom,
! [A: $tType,X3: A,Z2: A,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R ) ) ) ).
% relpow_Suc_E
thf(fact_6129_relpow__0__E,axiom,
! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R ) )
=> ( X3 = Y ) ) ).
% relpow_0_E
thf(fact_6130_relpow__0__I,axiom,
! [A: $tType,X3: A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R ) ) ).
% relpow_0_I
thf(fact_6131_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ).
% mono_invE
thf(fact_6132_funpow__mono,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: A > A,A3: A,B3: A,N2: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N2 @ F2 @ A3 ) @ ( compow @ ( A > A ) @ N2 @ F2 @ B3 ) ) ) ) ) ).
% funpow_mono
thf(fact_6133_incseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [F3: nat > A] :
! [N: nat] : ( ord_less_eq @ A @ ( F3 @ N ) @ ( F3 @ ( suc @ N ) ) ) ) ) ) ).
% incseq_Suc_iff
thf(fact_6134_incseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( order_mono @ nat @ A @ X8 ) ) ) ).
% incseq_SucI
thf(fact_6135_incseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: nat > A,I: nat] :
( ( order_mono @ nat @ A @ A3 )
=> ( ord_less_eq @ A @ ( A3 @ I ) @ ( A3 @ ( suc @ I ) ) ) ) ) ).
% incseq_SucD
thf(fact_6136_monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ) ).
% monoD
thf(fact_6137_monoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ) ).
% monoE
thf(fact_6138_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( order_mono @ A @ B @ F2 ) ) ) ).
% monoI
thf(fact_6139_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F3: A > B] :
! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) ) ) ) ) ).
% mono_def
thf(fact_6140_incseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,I: nat,J: nat] :
( ( order_mono @ nat @ A @ F2 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ) ).
% incseqD
thf(fact_6141_incseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [X6: nat > A] :
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ A @ ( X6 @ M2 ) @ ( X6 @ N ) ) ) ) ) ) ).
% incseq_def
thf(fact_6142_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F2: A > B,A3: A,B3: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( F2 @ ( inf_inf @ A @ A3 @ B3 ) ) @ ( inf_inf @ B @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) ) ) ) ).
% mono_inf
thf(fact_6143_relpow__Suc__D2_H,axiom,
! [A: $tType,N2: nat,R: set @ ( product_prod @ A @ A ),X2: A,Y5: A,Z4: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z4 ) @ R ) )
=> ? [W: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ W ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W @ Z4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_6144_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A2 ) ) ) ).
% mono_add
thf(fact_6145_mono__Suc,axiom,
order_mono @ nat @ nat @ suc ).
% mono_Suc
thf(fact_6146_mono__pow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,N2: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( order_mono @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) ) ) ) ).
% mono_pow
thf(fact_6147_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) )
=> ( ord_less @ A @ X3 @ Y ) ) ) ) ).
% mono_strict_invE
thf(fact_6148_cclfp__lowerbound,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F2: A > A,A3: A] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ A3 ) @ A3 )
=> ( ord_less_eq @ A @ ( order_532582986084564980_cclfp @ A @ F2 ) @ A3 ) ) ) ) ).
% cclfp_lowerbound
thf(fact_6149_mono__times__nat,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( order_mono @ nat @ nat @ ( times_times @ nat @ N2 ) ) ) ).
% mono_times_nat
thf(fact_6150_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_6151_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [F2: A > B,M: A,N2: A,M5: B,N4: B] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ( image @ A @ B @ F2 @ ( set_or7035219750837199246ssThan @ A @ M @ N2 ) )
= ( set_or7035219750837199246ssThan @ B @ M5 @ N4 ) )
=> ( ( ord_less @ A @ M @ N2 )
=> ( ( F2 @ M )
= M5 ) ) ) ) ) ).
% mono_image_least
thf(fact_6152_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [F2: A > A,P4: A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ P4 @ ( F2 @ P4 ) )
=> ( ord_less_eq @ A @ P4 @ ( compow @ ( A > A ) @ K @ F2 @ ( top_top @ A ) ) ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_6153_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F2: A > A,P4: A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ P4 ) @ P4 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) @ P4 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_6154_funpow__mono2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: A > A,I: nat,J: nat,X3: A,Y: A] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ X3 @ ( F2 @ X3 ) )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ I @ F2 @ X3 ) @ ( compow @ ( A > A ) @ J @ F2 @ Y ) ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_6155_relpowp__relpow__eq,axiom,
! [A: $tType,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( A > A > $o ) @ N2
@ ^ [X: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R ) )
= ( ^ [X: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ).
% relpowp_relpow_eq
thf(fact_6156_mono__Sup,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F2 @ A3 ) ) @ ( F2 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% mono_Sup
thf(fact_6157_mono__SUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A3: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image @ C @ B
@ ^ [X: C] : ( F2 @ ( A3 @ X ) )
@ I6 ) )
@ ( F2 @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ A3 @ I6 ) ) ) ) ) ) ).
% mono_SUP
thf(fact_6158_mono__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ A3 ) ) @ ( complete_Inf_Inf @ B @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ) ).
% mono_Inf
thf(fact_6159_mono__INF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A3: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ A3 @ I6 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image @ C @ B
@ ^ [X: C] : ( F2 @ ( A3 @ X ) )
@ I6 ) ) ) ) ) ).
% mono_INF
thf(fact_6160_relpow__E,axiom,
! [A: $tType,X3: A,Z2: A,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) )
=> ( ( ( N2
= ( zero_zero @ nat ) )
=> ( X3 != Z2 ) )
=> ~ ! [Y4: A,M3: nat] :
( ( N2
= ( suc @ M3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R ) ) ) ) ) ).
% relpow_E
thf(fact_6161_relpow__E2,axiom,
! [A: $tType,X3: A,Z2: A,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) )
=> ( ( ( N2
= ( zero_zero @ nat ) )
=> ( X3 != Z2 ) )
=> ~ ! [Y4: A,M3: nat] :
( ( N2
= ( suc @ M3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R ) ) ) ) ) ) ).
% relpow_E2
thf(fact_6162_antimono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_antimono @ nat @ A
@ ^ [I3: nat] : ( compow @ ( A > A ) @ I3 @ Q @ ( top_top @ A ) ) ) ) ) ).
% antimono_funpow
thf(fact_6163_relpow__empty,axiom,
! [A: $tType,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% relpow_empty
thf(fact_6164_incseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,L5: A,N2: nat] :
( ( order_mono @ nat @ A @ X8 )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( X8 @ N2 ) @ L5 ) ) ) ) ).
% incseq_le
thf(fact_6165_funpow__increasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [M: nat,N2: nat,F2: A > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N2 @ F2 @ ( top_top @ A ) ) @ ( compow @ ( A > A ) @ M @ F2 @ ( top_top @ A ) ) ) ) ) ) ).
% funpow_increasing
thf(fact_6166_funpow__decreasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [M: nat,N2: nat,F2: A > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M @ F2 @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N2 @ F2 @ ( bot_bot @ A ) ) ) ) ) ) ).
% funpow_decreasing
thf(fact_6167_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F2 @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( lattic643756798349783984er_Max @ B @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_6168_relpow__fun__conv,axiom,
! [A: $tType,A2: A,B2: A,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) )
= ( ? [F3: nat > A] :
( ( ( F3 @ ( zero_zero @ nat ) )
= A2 )
& ( ( F3 @ N2 )
= B2 )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_6169_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F2 @ A3 ) ) @ ( F2 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ) ).
% mono_cSup
thf(fact_6170_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A3: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit941137186595557371_above @ A @ ( image @ C @ A @ A3 @ I6 ) )
=> ( ( I6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image @ C @ B
@ ^ [X: C] : ( F2 @ ( A3 @ X ) )
@ I6 ) )
@ ( F2 @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ A3 @ I6 ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_6171_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ A3 ) ) @ ( complete_Inf_Inf @ B @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ) ) ) ).
% mono_cInf
thf(fact_6172_mono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A3: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit1013018076250108175_below @ A @ ( image @ C @ A @ A3 @ I6 ) )
=> ( ( I6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ A3 @ I6 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image @ C @ B
@ ^ [X: C] : ( F2 @ ( A3 @ X ) )
@ I6 ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_6173_relpow__finite__bounded,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R )
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ).
% relpow_finite_bounded
thf(fact_6174_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( order_mono @ nat @ nat
@ ^ [M2: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K @ M2 ) @ M2 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_6175_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A] :
( ( finite_finite2 @ A @ ( image @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( order_mono @ nat @ A @ F2 )
=> ( ! [N3: nat] :
( ( ( F2 @ N3 )
= ( F2 @ ( suc @ N3 ) ) )
=> ( ( F2 @ ( suc @ N3 ) )
= ( F2 @ ( suc @ ( suc @ N3 ) ) ) ) )
=> ? [N9: nat] :
( ! [N6: nat] :
( ( ord_less_eq @ nat @ N6 @ N9 )
=> ! [M6: nat] :
( ( ord_less_eq @ nat @ M6 @ N9 )
=> ( ( ord_less @ nat @ M6 @ N6 )
=> ( ord_less @ A @ ( F2 @ M6 ) @ ( F2 @ N6 ) ) ) ) )
& ! [N6: nat] :
( ( ord_less_eq @ nat @ N9 @ N6 )
=> ( ( F2 @ N9 )
= ( F2 @ N6 ) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_6176_tendsto__at__left__sequentially,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [B2: B,A2: B,X8: B > A,L5: A] :
( ( ord_less @ B @ B2 @ A2 )
=> ( ! [S5: nat > B] :
( ! [N6: nat] : ( ord_less @ B @ ( S5 @ N6 ) @ A2 )
=> ( ! [N6: nat] : ( ord_less @ B @ B2 @ ( S5 @ N6 ) )
=> ( ( order_mono @ nat @ B @ S5 )
=> ( ( filterlim @ nat @ B @ S5 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( X8 @ ( S5 @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ L5 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( topolo174197925503356063within @ B @ A2 @ ( set_ord_lessThan @ B @ A2 ) ) ) ) ) ) ).
% tendsto_at_left_sequentially
thf(fact_6177_continuous__at__Sup__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,S3: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Sup_Sup @ A @ S3 ) @ ( set_ord_lessThan @ A @ ( complete_Sup_Sup @ A @ S3 ) ) ) @ F2 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S3 )
=> ( ( F2 @ ( complete_Sup_Sup @ A @ S3 ) )
= ( complete_Sup_Sup @ B @ ( image @ A @ B @ F2 @ S3 ) ) ) ) ) ) ) ) ).
% continuous_at_Sup_mono
thf(fact_6178_continuous__at__Inf__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,S3: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Inf_Inf @ A @ S3 ) @ ( set_ord_greaterThan @ A @ ( complete_Inf_Inf @ A @ S3 ) ) ) @ F2 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S3 )
=> ( ( F2 @ ( complete_Inf_Inf @ A @ S3 ) )
= ( complete_Inf_Inf @ B @ ( image @ A @ B @ F2 @ S3 ) ) ) ) ) ) ) ) ).
% continuous_at_Inf_mono
thf(fact_6179_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N: nat,R6: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [I3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I3 @ R6 )
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I3 )
& ( ord_less_eq @ nat @ I3 @ ( suc @ N ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_6180_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 ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R )
@ ( collect @ nat
@ ^ [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_6181_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_6182_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_6183_ntrancl__Zero,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( zero_zero @ nat ) @ R )
= R ) ).
% ntrancl_Zero
thf(fact_6184_mono__Int,axiom,
! [B: $tType,A: $tType,F2: ( set @ A ) > ( set @ B ),A3: set @ A,B3: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F2 )
=> ( ord_less_eq @ ( set @ B ) @ ( F2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) @ ( inf_inf @ ( set @ B ) @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) ) ) ).
% mono_Int
thf(fact_6185_rtranclp_Omono,axiom,
! [A: $tType,R2: A > A > $o] :
( order_mono @ ( A > A > $o ) @ ( A > A > $o )
@ ^ [P6: A > A > $o,X15: A,X24: A] :
( ? [A4: A] :
( ( X15 = A4 )
& ( X24 = A4 ) )
| ? [A4: A,B4: A,C4: A] :
( ( X15 = A4 )
& ( X24 = C4 )
& ( P6 @ A4 @ B4 )
& ( R2 @ B4 @ C4 ) ) ) ) ).
% rtranclp.mono
thf(fact_6186_tranclp_Omono,axiom,
! [A: $tType,R2: A > A > $o] :
( order_mono @ ( A > A > $o ) @ ( A > A > $o )
@ ^ [P6: A > A > $o,X15: A,X24: A] :
( ? [A4: A,B4: A] :
( ( X15 = A4 )
& ( X24 = B4 )
& ( R2 @ A4 @ B4 ) )
| ? [A4: A,B4: A,C4: A] :
( ( X15 = A4 )
& ( X24 = C4 )
& ( P6 @ A4 @ B4 )
& ( R2 @ B4 @ C4 ) ) ) ) ).
% tranclp.mono
thf(fact_6187_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 ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ! [A5: A,B5: 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 @ A5 @ B5 ) ) @ R2 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A,B5: 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 @ A5 @ B5 ) ) @ ( 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 @ A5 @ B5 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 )
=> ( ( P @ A5 @ B5 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_6188_converse__trancl__induct,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ B2 ) @ R2 )
=> ( P @ Y4 ) )
=> ( ! [Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P @ Z3 )
=> ( P @ Y4 ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% converse_trancl_induct
thf(fact_6189_trancl__trans__induct,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),P: A > A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ R2 )
=> ( P @ X5 @ Y4 ) )
=> ( ! [X5: A,Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P @ X5 @ Y4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P @ Y4 @ Z3 )
=> ( P @ X5 @ Z3 ) ) ) ) )
=> ( P @ X3 @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_6190_trancl__into__trancl2,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_6191_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C2: 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 @ C2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_6192_irrefl__trancl__rD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ! [X5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( X3 != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_6193_converse__tranclE,axiom,
! [A: $tType,X3: A,Z2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ R2 )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ) ).
% converse_tranclE
thf(fact_6194_r__r__into__trancl,axiom,
! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% r_r_into_trancl
thf(fact_6195_trancl__induct,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ Y4 ) @ R2 )
=> ( P @ Y4 ) )
=> ( ! [Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ Y4 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 )
=> ( ( P @ Y4 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% trancl_induct
thf(fact_6196_trancl__trans,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_trans
thf(fact_6197_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 )
=> ~ ! [C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C3 @ B2 ) @ R2 ) ) ) ) ).
% tranclE
thf(fact_6198_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_6199_trancl_Osimps,axiom,
! [A: $tType,A12: A,A23: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ ( transitive_trancl @ A @ R2 ) )
= ( ? [A4: A,B4: A] :
( ( A12 = A4 )
& ( A23 = B4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R2 ) )
| ? [A4: A,B4: A,C4: A] :
( ( A12 = A4 )
& ( A23 = C4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ ( transitive_trancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C4 ) @ R2 ) ) ) ) ).
% trancl.simps
thf(fact_6200_trancl_Ocases,axiom,
! [A: $tType,A12: A,A23: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ R2 )
=> ~ ! [B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B5 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A23 ) @ R2 ) ) ) ) ).
% trancl.cases
thf(fact_6201_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_6202_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_6203_trancl__power,axiom,
! [A: $tType,P4: product_prod @ A @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P4 @ ( transitive_trancl @ A @ R ) )
= ( ? [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( member @ ( product_prod @ A @ A ) @ P4 @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ) ).
% trancl_power
thf(fact_6204_less__eq,axiom,
! [M: nat,N2: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N2 ) @ ( transitive_trancl @ nat @ pred_nat ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% less_eq
thf(fact_6205_finite_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( set @ A ) > $o ) @ ( ( set @ A ) > $o )
@ ^ [P6: ( set @ A ) > $o,X: set @ A] :
( ( X
= ( bot_bot @ ( set @ A ) ) )
| ? [A6: set @ A,A4: A] :
( ( X
= ( insert @ A @ A4 @ A6 ) )
& ( P6 @ A6 ) ) ) ) ).
% finite.mono
thf(fact_6206_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= Ys )
= ( ? [F3: nat > nat] :
( ( order_mono @ nat @ nat @ F3 )
& ( ( image @ nat @ nat @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys ) ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys @ ( F3 @ I3 ) ) ) )
& ! [I3: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) )
= ( ( F3 @ I3 )
= ( F3 @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_6207_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 )
@ ^ [X: A] :
( product_case_prod @ B @ C @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [Y2: B] : ( product_Pair @ ( product_prod @ A @ B ) @ C @ ( product_Pair @ A @ B @ X @ Y2 ) ) ) )
@ ( prod_filter @ A @ ( product_prod @ B @ C ) @ F5 @ ( prod_filter @ B @ C @ G6 @ H7 ) ) ) ) ).
% prod_filter_assoc
thf(fact_6208_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_6209_remdups__adj__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( remdups_adj @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% remdups_adj_set
thf(fact_6210_filtermap__bot,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( filtermap @ B @ A @ F2 @ ( bot_bot @ ( filter @ B ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ).
% filtermap_bot
thf(fact_6211_mono__compose,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ C ) )
=> ! [Q: A > B > C,F2: D > B] :
( ( order_mono @ A @ ( B > C ) @ Q )
=> ( order_mono @ A @ ( D > C )
@ ^ [I3: A,X: D] : ( Q @ I3 @ ( F2 @ X ) ) ) ) ) ).
% mono_compose
thf(fact_6212_filtermap__bot__iff,axiom,
! [A: $tType,B: $tType,F2: B > A,F5: filter @ B] :
( ( ( filtermap @ B @ A @ F2 @ F5 )
= ( bot_bot @ ( filter @ A ) ) )
= ( F5
= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtermap_bot_iff
thf(fact_6213_remdups__adj_Osimps_I1_J,axiom,
! [A: $tType] :
( ( remdups_adj @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remdups_adj.simps(1)
thf(fact_6214_remdups__adj__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( remdups_adj @ A @ Xs )
= Xs ) ) ).
% remdups_adj_distinct
thf(fact_6215_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_6216_filtermap__Pair,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > A,G: C > B,F5: filter @ C] :
( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) )
@ ( filtermap @ C @ ( product_prod @ A @ B )
@ ^ [X: C] : ( product_Pair @ A @ B @ ( F2 @ X ) @ ( G @ X ) )
@ F5 )
@ ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F2 @ F5 ) @ ( filtermap @ C @ B @ G @ F5 ) ) ) ).
% filtermap_Pair
thf(fact_6217_filtermap__nhds__times,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filtermap @ A @ A @ ( times_times @ A @ C2 ) @ ( topolo7230453075368039082e_nhds @ A @ A2 ) )
= ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ A2 ) ) ) ) ) ).
% filtermap_nhds_times
thf(fact_6218_remdups__adj__adjacent,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) )
=> ( ( nth @ A @ ( remdups_adj @ A @ Xs ) @ I )
!= ( nth @ A @ ( remdups_adj @ A @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_6219_at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A] :
( ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
= ( filtermap @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ X @ A2 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_to_0
thf(fact_6220_filtermap__times__pos__at__right,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [C2: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( filtermap @ A @ A @ ( times_times @ A @ C2 ) @ ( topolo174197925503356063within @ A @ P4 @ ( set_ord_greaterThan @ A @ P4 ) ) )
= ( topolo174197925503356063within @ A @ ( times_times @ A @ C2 @ P4 ) @ ( set_ord_greaterThan @ A @ ( times_times @ A @ C2 @ P4 ) ) ) ) ) ) ).
% filtermap_times_pos_at_right
thf(fact_6221_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_6222_prod__filter__principal__singleton,axiom,
! [A: $tType,B: $tType,X3: A,F5: filter @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ F5 )
= ( filtermap @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ F5 ) ) ).
% prod_filter_principal_singleton
thf(fact_6223_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_6224_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F5: filter @ A,X3: B] :
( ( prod_filter @ A @ B @ F5 @ ( principal @ B @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( filtermap @ A @ ( product_prod @ A @ B )
@ ^ [A4: A] : ( product_Pair @ A @ B @ A4 @ X3 )
@ F5 ) ) ).
% prod_filter_principal_singleton2
thf(fact_6225_cauchy__filter__metric__filtermap,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V768167426530841204y_dist @ B )
& ( topolo7287701948861334536_space @ B ) )
=> ! [F2: A > B,F5: filter @ A] :
( ( topolo6773858410816713723filter @ B @ ( filtermap @ A @ B @ F2 @ F5 ) )
= ( ! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [P3: A > $o] :
( ( eventually @ A @ P3 @ F5 )
& ! [X: A,Y2: A] :
( ( ( P3 @ X )
& ( P3 @ Y2 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X ) @ ( F2 @ Y2 ) ) @ E4 ) ) ) ) ) ) ) ).
% cauchy_filter_metric_filtermap
thf(fact_6226_iteratesp_Omono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F2: A > A] :
( order_mono @ ( A > $o ) @ ( A > $o )
@ ^ [P6: A > $o,X: A] :
( ? [Y2: A] :
( ( X
= ( F2 @ Y2 ) )
& ( P6 @ Y2 ) )
| ? [M9: set @ A] :
( ( X
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y2: A] :
( ( member @ A @ Y2 @ M9 )
=> ( P6 @ Y2 ) ) ) ) ) ) ).
% iteratesp.mono
thf(fact_6227_compact__imp__fip__image,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,I6: set @ B,F2: B > ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( topolo7761053866217962861closed @ A @ ( F2 @ I4 ) ) )
=> ( ! [I9: set @ B] :
( ( finite_finite2 @ B @ I9 )
=> ( ( ord_less_eq @ ( set @ B ) @ I9 @ I6 )
=> ( ( inf_inf @ ( set @ A ) @ S @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ I9 ) ) )
!= ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ S @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ I6 ) ) )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% compact_imp_fip_image
thf(fact_6228_closed__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo7761053866217962861closed @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% closed_empty
thf(fact_6229_closed__singleton,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [A2: A] : ( topolo7761053866217962861closed @ A @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% closed_singleton
thf(fact_6230_closed__Union,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ S3 )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ S3 )
=> ( topolo7761053866217962861closed @ A @ X5 ) )
=> ( topolo7761053866217962861closed @ A @ ( complete_Sup_Sup @ ( set @ A ) @ S3 ) ) ) ) ) ).
% closed_Union
thf(fact_6231_closed__UN,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: set @ B,B3: B > ( set @ A )] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( topolo7761053866217962861closed @ A @ ( B3 @ X5 ) ) )
=> ( topolo7761053866217962861closed @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) ) ) ) ) ).
% closed_UN
thf(fact_6232_filtermap__sequentually__ne__bot,axiom,
! [A: $tType,F2: nat > A] :
( ( filtermap @ nat @ A @ F2 @ ( at_top @ nat ) )
!= ( bot_bot @ ( filter @ A ) ) ) ).
% filtermap_sequentually_ne_bot
thf(fact_6233_closed__diagonal,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Y2: product_prod @ A @ A] :
? [X: A] :
( Y2
= ( product_Pair @ A @ A @ X @ X ) ) ) ) ) ).
% closed_diagonal
thf(fact_6234_chain__subset,axiom,
! [A: $tType,Ord: A > A > $o,A3: set @ A,B3: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( comple1602240252501008431_chain @ A @ Ord @ B3 ) ) ) ).
% chain_subset
thf(fact_6235_ccpo__Sup__upper,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A3: set @ A,X3: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_6236_ccpo__Sup__least,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A3: set @ A,Z2: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ A @ X5 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ Z2 ) ) ) ) ).
% ccpo_Sup_least
thf(fact_6237_finite__imp__closed,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( topolo7761053866217962861closed @ A @ S3 ) ) ) ).
% finite_imp_closed
thf(fact_6238_chain__empty,axiom,
! [A: $tType,Ord: A > A > $o] : ( comple1602240252501008431_chain @ A @ Ord @ ( bot_bot @ ( set @ A ) ) ) ).
% chain_empty
thf(fact_6239_closed__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X: A,Y2: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X @ Y2 ) )
& ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ).
% closed_subdiagonal
thf(fact_6240_closed__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X: A,Y2: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X @ Y2 ) )
& ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ).
% closed_superdiagonal
thf(fact_6241_chain__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X3: A] : ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% chain_singleton
thf(fact_6242_closed__Collect__le,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G )
=> ( topolo7761053866217962861closed @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% closed_Collect_le
thf(fact_6243_t4__space,axiom,
! [A: $tType] :
( ( topological_t4_space @ A )
=> ! [S3: set @ A,T4: set @ A] :
( ( topolo7761053866217962861closed @ A @ S3 )
=> ( ( topolo7761053866217962861closed @ A @ T4 )
=> ( ( ( inf_inf @ ( set @ A ) @ S3 @ T4 )
= ( bot_bot @ ( set @ A ) ) )
=> ? [U6: set @ A,V7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U6 )
& ( topolo1002775350975398744n_open @ A @ V7 )
& ( ord_less_eq @ ( set @ A ) @ S3 @ U6 )
& ( ord_less_eq @ ( set @ A ) @ T4 @ V7 )
& ( ( inf_inf @ ( set @ A ) @ U6 @ V7 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% t4_space
thf(fact_6244_t3__space,axiom,
! [A: $tType] :
( ( topological_t3_space @ A )
=> ! [S3: set @ A,Y: A] :
( ( topolo7761053866217962861closed @ A @ S3 )
=> ( ~ ( member @ A @ Y @ S3 )
=> ? [U6: set @ A,V7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U6 )
& ( topolo1002775350975398744n_open @ A @ V7 )
& ( member @ A @ Y @ U6 )
& ( ord_less_eq @ ( set @ A ) @ S3 @ V7 )
& ( ( inf_inf @ ( set @ A ) @ U6 @ V7 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% t3_space
thf(fact_6245_nhds__closed,axiom,
! [A: $tType] :
( ( topological_t3_space @ A )
=> ! [X3: A,A3: set @ A] :
( ( member @ A @ X3 @ A3 )
=> ( ( topolo1002775350975398744n_open @ A @ A3 )
=> ? [A15: set @ A] :
( ( member @ A @ X3 @ A15 )
& ( topolo7761053866217962861closed @ A @ A15 )
& ( ord_less_eq @ ( set @ A ) @ A15 @ A3 )
& ( eventually @ A
@ ^ [Y2: A] : ( member @ A @ Y2 @ A15 )
@ ( topolo7230453075368039082e_nhds @ A @ X3 ) ) ) ) ) ) ).
% nhds_closed
thf(fact_6246_continuous__on__closed__Union,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [I6: set @ A,U3: A > ( set @ B ),F2: B > C] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( topolo7761053866217962861closed @ B @ ( U3 @ I4 ) ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( topolo81223032696312382ous_on @ B @ C @ ( U3 @ I4 ) @ F2 ) )
=> ( topolo81223032696312382ous_on @ B @ C @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ U3 @ I6 ) ) @ F2 ) ) ) ) ) ).
% continuous_on_closed_Union
thf(fact_6247_Lim__in__closed__set,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F2: B > A,F5: filter @ B,L: A] :
( ( topolo7761053866217962861closed @ A @ S3 )
=> ( ( eventually @ B
@ ^ [X: B] : ( member @ A @ ( F2 @ X ) @ S3 )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
=> ( member @ A @ L @ S3 ) ) ) ) ) ) ).
% Lim_in_closed_set
thf(fact_6248_compact__imp__fip,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F5: set @ ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ! [T5: set @ A] :
( ( member @ ( set @ A ) @ T5 @ F5 )
=> ( topolo7761053866217962861closed @ A @ T5 ) )
=> ( ! [F15: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F15 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F15 @ F5 )
=> ( ( inf_inf @ ( set @ A ) @ S3 @ ( complete_Inf_Inf @ ( set @ A ) @ F15 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ S3 @ ( complete_Inf_Inf @ ( set @ A ) @ F5 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% compact_imp_fip
thf(fact_6249_compact__fip,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo2193935891317330818ompact @ A )
= ( ^ [U5: set @ A] :
! [A6: set @ ( set @ A )] :
( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A6 )
=> ( topolo7761053866217962861closed @ A @ X ) )
=> ( ! [B6: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B6 @ A6 )
=> ( ( finite_finite2 @ ( set @ A ) @ B6 )
=> ( ( inf_inf @ ( set @ A ) @ U5 @ ( complete_Inf_Inf @ ( set @ A ) @ B6 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ U5 @ ( complete_Inf_Inf @ ( set @ A ) @ A6 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% compact_fip
thf(fact_6250_in__chain__finite,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A3: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A3 ) @ A3 ) ) ) ) ) ).
% in_chain_finite
thf(fact_6251_inj__sgn__power,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( inj_on @ real @ real
@ ^ [Y2: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N2 ) )
@ ( top_top @ ( set @ real ) ) ) ) ).
% inj_sgn_power
thf(fact_6252_inj__on__empty,axiom,
! [B: $tType,A: $tType,F2: A > B] : ( inj_on @ A @ B @ F2 @ ( bot_bot @ ( set @ A ) ) ) ).
% inj_on_empty
thf(fact_6253_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_6254_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( inj_on @ A @ A
@ ^ [B4: A] : ( divide_divide @ A @ B4 @ A2 )
@ ( top_top @ ( set @ A ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_6255_inj__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ A @ C @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_apfst
thf(fact_6256_inj__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F2: B > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ B @ C @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% inj_apsnd
thf(fact_6257_inj__on__insert,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: A,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert @ A @ A2 @ A3 ) )
= ( ( inj_on @ A @ B @ F2 @ A3 )
& ~ ( member @ B @ ( F2 @ A2 ) @ ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_6258_inj__on__Inter,axiom,
! [B: $tType,A: $tType,S3: set @ ( set @ A ),F2: A > B] :
( ( S3
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [A8: set @ A] :
( ( member @ ( set @ A ) @ A8 @ S3 )
=> ( inj_on @ A @ B @ F2 @ A8 ) )
=> ( inj_on @ A @ B @ F2 @ ( complete_Inf_Inf @ ( set @ A ) @ S3 ) ) ) ) ).
% inj_on_Inter
thf(fact_6259_finite__inverse__image,axiom,
! [A: $tType,B: $tType,A3: set @ A,F2: B > A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] : ( member @ A @ ( F2 @ J3 ) @ A3 ) ) ) ) ) ).
% finite_inverse_image
thf(fact_6260_finite__inverse__image__gen,axiom,
! [A: $tType,B: $tType,A3: set @ A,F2: B > A,D6: set @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( inj_on @ B @ A @ F2 @ D6 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ D6 )
& ( member @ A @ ( F2 @ J3 ) @ A3 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_6261_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,A3: set @ A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A2 ) @ A3 ) ) ) ).
% inj_on_mult
thf(fact_6262_inj__on__add_H,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A3: set @ A] :
( inj_on @ A @ A
@ ^ [B4: A] : ( plus_plus @ A @ B4 @ A2 )
@ A3 ) ) ).
% inj_on_add'
thf(fact_6263_inj__on__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A3: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 ) ) ).
% inj_on_add
thf(fact_6264_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_6265_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F2: A > B] :
( ! [X5: A,Y4: A] :
( ( ord_less @ A @ X5 @ Y4 )
=> ( ( F2 @ X5 )
!= ( F2 @ Y4 ) ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_6266_sorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( inj_on @ A @ A
@ ^ [X: A] : X
@ ( top_top @ ( set @ A ) ) ) ) ).
% sorted_list_of_set.inj_on
thf(fact_6267_inj__fn,axiom,
! [A: $tType,F2: A > A,N2: nat] :
( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fn
thf(fact_6268_inj__on__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( inj_on @ A @ B @ F2 @ B3 ) ) ) ).
% inj_on_subset
thf(fact_6269_subset__inj__on,axiom,
! [B: $tType,A: $tType,F2: A > B,B3: set @ A,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( inj_on @ A @ B @ F2 @ A3 ) ) ) ).
% subset_inj_on
thf(fact_6270_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,F2: A > B] :
( ! [X5: A,Y4: A] :
( ( ord_less @ A @ X5 @ Y4 )
=> ( ( member @ A @ X5 @ A3 )
=> ( ( member @ A @ Y4 @ A3 )
=> ( ( F2 @ X5 )
!= ( F2 @ Y4 ) ) ) ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( member @ A @ Y4 @ A3 )
=> ( ( ord_less_eq @ A @ X5 @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X5 ) ) ) )
=> ( inj_on @ A @ B @ F2 @ A3 ) ) ) ) ).
% linorder_inj_onI
thf(fact_6271_subset__image__inj,axiom,
! [A: $tType,B: $tType,S3: set @ A,F2: B > A,T4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ ( image @ B @ A @ F2 @ T4 ) )
= ( ? [U5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U5 @ T4 )
& ( inj_on @ B @ A @ F2 @ U5 )
& ( S3
= ( image @ B @ A @ F2 @ U5 ) ) ) ) ) ).
% subset_image_inj
thf(fact_6272_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,C5: set @ A,A3: set @ A,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 )
=> ( ( ( image @ A @ B @ F2 @ A3 )
= ( image @ A @ B @ F2 @ B3 ) )
= ( A3 = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_6273_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,B3: set @ A,A2: A,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ B3 )
=> ( ( member @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ B @ ( F2 @ A2 ) @ ( image @ A @ B @ F2 @ A3 ) )
= ( member @ A @ A2 @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_6274_finite__image__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F2 @ A3 ) )
= ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_image_iff
thf(fact_6275_finite__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( inj_on @ B @ A @ F2 @ A3 )
=> ( finite_finite2 @ B @ A3 ) ) ) ).
% finite_imageD
thf(fact_6276_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,B3: set @ A,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ B3 )
=> ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B3 ) ) ) ) ).
% inj_on_strict_subset
thf(fact_6277_finite__UNIV__inj__surj,axiom,
! [A: $tType,F2: A > A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_6278_finite__UNIV__surj__inj,axiom,
! [A: $tType,F2: A > A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_6279_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_6280_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A3: set @ A,A16: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F3: A > B] :
( ( inj_on @ A @ B @ F3 @ A3 )
& ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A3 ) @ A16 ) ) )
= ( ? [G2: B > A] :
( ( image @ B @ A @ G2 @ A16 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_6281_finite__surj__inj,axiom,
! [A: $tType,A3: set @ A,F2: A > A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( image @ A @ A @ F2 @ A3 ) )
=> ( inj_on @ A @ A @ F2 @ A3 ) ) ) ).
% finite_surj_inj
thf(fact_6282_inj__on__finite,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B3: set @ B] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( finite_finite2 @ A @ A3 ) ) ) ) ).
% inj_on_finite
thf(fact_6283_endo__inj__surj,axiom,
! [A: $tType,A3: set @ A,F2: A > A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ F2 @ A3 ) @ A3 )
=> ( ( inj_on @ A @ A @ F2 @ A3 )
=> ( ( image @ A @ A @ F2 @ A3 )
= A3 ) ) ) ) ).
% endo_inj_surj
thf(fact_6284_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F2: A > B,C5: set @ A,A3: set @ A,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 )
=> ( ( image @ A @ B @ F2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
= ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B3 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_6285_inj__on__iff__eq__card,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( inj_on @ A @ B @ F2 @ A3 )
= ( ( finite_card @ B @ ( image @ A @ B @ F2 @ A3 ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_6286_eq__card__imp__inj__on,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( finite_card @ B @ ( image @ A @ B @ F2 @ A3 ) )
= ( finite_card @ A @ A3 ) )
=> ( inj_on @ A @ B @ F2 @ A3 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_6287_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F2: A > B,C5: set @ A,A3: set @ A,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 )
=> ( ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B3 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_6288_pigeonhole,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( ord_less @ nat @ ( finite_card @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( finite_card @ B @ A3 ) )
=> ~ ( inj_on @ B @ A @ F2 @ A3 ) ) ).
% pigeonhole
thf(fact_6289_continuous__inj__imp__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo8458572112393995274pology @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A2: A,X3: A,B2: A,F2: A > B] :
( ( ord_less @ A @ A2 @ X3 )
=> ( ( ord_less @ A @ X3 @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
=> ( ( ( ord_less @ B @ ( F2 @ A2 ) @ ( F2 @ X3 ) )
& ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ B2 ) ) )
| ( ( ord_less @ B @ ( F2 @ B2 ) @ ( F2 @ X3 ) )
& ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ A2 ) ) ) ) ) ) ) ) ) ).
% continuous_inj_imp_mono
thf(fact_6290_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,X3: B,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( member @ B @ X3 @ ( image @ A @ B @ F2 @ A3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A3 @ F2 @ X3 ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_6291_inj__on__UNION__chain,axiom,
! [C: $tType,B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B ),F2: B > C] :
( ! [I4: A,J2: A] :
( ( member @ A @ I4 @ I6 )
=> ( ( member @ A @ J2 @ I6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A3 @ I4 ) @ ( A3 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A3 @ J2 ) @ ( A3 @ I4 ) ) ) ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( inj_on @ B @ C @ F2 @ ( A3 @ I4 ) ) )
=> ( inj_on @ B @ C @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_6292_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I6: set @ A,F2: B > C,A3: A > ( set @ B )] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( inj_on @ B @ C @ F2 @ ( A3 @ I4 ) ) )
=> ( inj_on @ B @ C @ F2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_6293_surjective__iff__injective__gen,axiom,
! [B: $tType,A: $tType,S3: set @ A,T4: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ( ( finite_card @ A @ S3 )
= ( finite_card @ B @ T4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ S3 ) @ T4 )
=> ( ( ! [X: B] :
( ( member @ B @ X @ T4 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ S3 )
& ( ( F2 @ Y2 )
= X ) ) ) )
= ( inj_on @ A @ B @ F2 @ S3 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_6294_card__bij__eq,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B3: set @ B,G: B > A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B3 )
=> ( ( inj_on @ B @ A @ G @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ B3 ) @ A3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( finite_card @ A @ A3 )
= ( finite_card @ B @ B3 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_6295_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_6296_image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F2: A > B,C5: set @ A,A3: set @ C,B3: C > ( set @ A ),J: C] :
( ( inj_on @ A @ B @ F2 @ C5 )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( B3 @ X5 ) @ C5 ) )
=> ( ( member @ C @ J @ A3 )
=> ( ( image @ A @ B @ F2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ C @ ( set @ A ) @ B3 @ A3 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image @ C @ ( set @ B )
@ ^ [X: C] : ( image @ A @ B @ F2 @ ( B3 @ X ) )
@ A3 ) ) ) ) ) ) ).
% image_INT
thf(fact_6297_card__le__inj,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ B @ B3 ) )
=> ? [F4: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A3 ) @ B3 )
& ( inj_on @ A @ B @ F4 @ A3 ) ) ) ) ) ).
% card_le_inj
thf(fact_6298_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B3: set @ B] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ B @ B3 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_6299_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A3: set @ A,B3: set @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( ? [F3: A > B] :
( ( inj_on @ A @ B @ F3 @ A3 )
& ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A3 ) @ B3 ) ) )
= ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ B @ B3 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_6300_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_6301_funpow__inj__finite,axiom,
! [A: $tType,P4: A > A,X3: A] :
( ( inj_on @ A @ A @ P4 @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y2: A] :
? [N: nat] :
( Y2
= ( compow @ ( A > A ) @ N @ P4 @ X3 ) ) ) )
=> ~ ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( compow @ ( A > A ) @ N3 @ P4 @ X3 )
!= X3 ) ) ) ) ).
% funpow_inj_finite
thf(fact_6302_all__subset__image__inj,axiom,
! [A: $tType,B: $tType,F2: B > A,S3: set @ B,P: ( set @ A ) > $o] :
( ( ! [T8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T8 @ ( image @ B @ A @ F2 @ S3 ) )
=> ( P @ T8 ) ) )
= ( ! [T8: set @ B] :
( ( ( ord_less_eq @ ( set @ B ) @ T8 @ S3 )
& ( inj_on @ B @ A @ F2 @ T8 ) )
=> ( P @ ( image @ B @ A @ F2 @ T8 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_6303_ex__subset__image__inj,axiom,
! [A: $tType,B: $tType,F2: B > A,S3: set @ B,P: ( set @ A ) > $o] :
( ( ? [T8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T8 @ ( image @ B @ A @ F2 @ S3 ) )
& ( P @ T8 ) ) )
= ( ? [T8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ T8 @ S3 )
& ( inj_on @ B @ A @ F2 @ T8 )
& ( P @ ( image @ B @ A @ F2 @ T8 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_6304_inj__singleton,axiom,
! [A: $tType,A3: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X: A] : ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
@ A3 ) ).
% inj_singleton
thf(fact_6305_inj__Some,axiom,
! [A: $tType,A3: set @ A] : ( inj_on @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ).
% inj_Some
thf(fact_6306_inj__Suc,axiom,
! [N7: set @ nat] : ( inj_on @ nat @ nat @ suc @ N7 ) ).
% inj_Suc
thf(fact_6307_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_6308_inj__on__diff__nat,axiom,
! [N7: set @ nat,K: nat] :
( ! [N3: nat] :
( ( member @ nat @ N3 @ N7 )
=> ( ord_less_eq @ nat @ K @ N3 ) )
=> ( inj_on @ nat @ nat
@ ^ [N: nat] : ( minus_minus @ nat @ N @ K )
@ N7 ) ) ).
% inj_on_diff_nat
thf(fact_6309_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_6310_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F2: A > B,X8: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X: A] : ( product_Pair @ A @ B @ X @ ( F2 @ X ) )
@ X8 ) ).
% inj_on_convol_ident
thf(fact_6311_swap__inj__on,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] :
( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [I3: A,J3: B] : ( product_Pair @ B @ A @ J3 @ I3 ) )
@ A3 ) ).
% swap_inj_on
thf(fact_6312_range__inj__infinite,axiom,
! [A: $tType,F2: nat > A] :
( ( inj_on @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) )
=> ~ ( finite_finite2 @ A @ ( image @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% range_inj_infinite
thf(fact_6313_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [N3: nat,F4: nat > A] :
( ( A3
= ( image @ nat @ A @ F4
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) ) )
& ( inj_on @ nat @ A @ F4
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_6314_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [F4: A > nat,N3: nat] :
( ( ( image @ A @ nat @ F4 @ A3 )
= ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) )
& ( inj_on @ A @ nat @ F4 @ A3 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_6315_inj__on__nth,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ! [X5: nat] :
( ( member @ nat @ X5 @ I6 )
=> ( ord_less @ nat @ X5 @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( inj_on @ nat @ A @ ( nth @ A @ Xs ) @ I6 ) ) ) ).
% inj_on_nth
thf(fact_6316_infinite__iff__countable__subset,axiom,
! [A: $tType,S3: set @ A] :
( ( ~ ( finite_finite2 @ A @ S3 ) )
= ( ? [F3: nat > A] :
( ( inj_on @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_6317_infinite__countable__subset,axiom,
! [A: $tType,S3: set @ A] :
( ~ ( finite_finite2 @ A @ S3 )
=> ? [F4: nat > A] :
( ( inj_on @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ).
% infinite_countable_subset
thf(fact_6318_inj__on__funpow__least,axiom,
! [A: $tType,N2: nat,F2: A > A,S: A] :
( ( ( compow @ ( A > A ) @ N2 @ F2 @ S )
= S )
=> ( ! [M3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ( ord_less @ nat @ M3 @ N2 )
=> ( ( compow @ ( A > A ) @ M3 @ F2 @ S )
!= S ) ) )
=> ( inj_on @ nat @ A
@ ^ [K3: nat] : ( compow @ ( A > A ) @ K3 @ F2 @ S )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% inj_on_funpow_least
thf(fact_6319_flat__lub__def,axiom,
! [A: $tType] :
( ( partial_flat_lub @ A )
= ( ^ [B4: A,A6: set @ A] :
( if @ A @ ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) @ B4
@ ( the @ A
@ ^ [X: A] : ( member @ A @ X @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_6320_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 ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R )
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ).
% rtrancl_finite_eq_relpow
thf(fact_6321_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_6322_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_6323_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_6324_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 ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A5: A,B5: 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 @ A5 @ B5 ) ) @ ( 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 @ A5 @ B5 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 )
=> ( ( P @ A5 @ B5 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_6325_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa2: A,Xb: 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 @ Xb ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( ( product_Pair @ A @ B @ Xa2 @ Xb )
!= ( product_Pair @ A @ B @ Za @ Zb ) )
=> ~ ! [A5: A,B5: 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 @ Xb ) @ ( product_Pair @ A @ B @ A5 @ B5 ) ) @ 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 @ A5 @ B5 ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_6326_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 ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( P @ Bx @ By )
=> ( ! [A5: A,B5: 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 @ A5 @ B5 ) @ ( 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 ) )
=> ( ( P @ Aa2 @ Ba )
=> ( P @ A5 @ B5 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_6327_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_6328_converse__rtrancl__induct,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( P @ B2 )
=> ( ! [Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( P @ Z3 )
=> ( P @ Y4 ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_6329_converse__rtranclE,axiom,
! [A: $tType,X3: A,Z2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( X3 != Z2 )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_6330_rtrancl__induct,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( P @ A2 )
=> ( ! [Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ Y4 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 )
=> ( ( P @ Y4 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_6331_rtrancl__trans,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_6332_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 )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ Y4 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ B2 ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_6333_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C2: 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 @ C2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_6334_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_6335_rtrancl_Osimps,axiom,
! [A: $tType,A12: A,A23: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ ( transitive_rtrancl @ A @ R2 ) )
= ( ? [A4: A] :
( ( A12 = A4 )
& ( A23 = A4 ) )
| ? [A4: A,B4: A,C4: A] :
( ( A12 = A4 )
& ( A23 = C4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C4 ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_6336_rtrancl_Ocases,axiom,
! [A: $tType,A12: A,A23: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( A23 != A12 )
=> ~ ! [B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B5 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A23 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_6337_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Zs2: 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 @ Zs2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_6338_tranclD,axiom,
! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R ) )
=> ? [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z3 ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% tranclD
thf(fact_6339_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_6340_tranclD2,axiom,
! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R ) )
=> ? [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z3 ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ R ) ) ) ).
% tranclD2
thf(fact_6341_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_6342_rtrancl__eq__or__trancl,axiom,
! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R ) )
= ( ( X3 = Y )
| ( ( X3 != Y )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_6343_rtrancl__into__trancl1,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C2: 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 @ C2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_6344_rtrancl__into__trancl2,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_6345_rtrancl__trancl__trancl,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_6346_trancl__rtrancl__trancl,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C2: 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 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_6347_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) )
=> ( ( size_size @ ( list @ A ) @ X3 )
= ( size_size @ ( list @ A ) @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_6348_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_6349_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_6350_pred__nat__trancl__eq__le,axiom,
! [M: nat,N2: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N2 ) @ ( transitive_rtrancl @ nat @ pred_nat ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% pred_nat_trancl_eq_le
thf(fact_6351_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_6352_uniformly__continuous__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo7287701948861334536_space @ A )
& ( topolo7287701948861334536_space @ B ) )
=> ! [S: set @ A,F2: A > B,E5: ( product_prod @ B @ B ) > $o] :
( ( topolo6026614971017936543ous_on @ A @ B @ S @ F2 )
=> ( ( eventually @ ( product_prod @ B @ B ) @ E5 @ ( topolo7806501430040627800ormity @ B ) )
=> ( eventually @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X: A,Y2: A] :
( ( member @ A @ X @ S )
=> ( ( member @ A @ Y2 @ S )
=> ( E5 @ ( product_Pair @ B @ B @ ( F2 @ X ) @ ( F2 @ Y2 ) ) ) ) ) )
@ ( topolo7806501430040627800ormity @ A ) ) ) ) ) ).
% uniformly_continuous_onD
thf(fact_6353_measure__function__int,axiom,
fun_is_measure @ int @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ).
% measure_function_int
thf(fact_6354_measure__size,axiom,
! [A: $tType] :
( ( size @ A )
=> ( fun_is_measure @ A @ ( size_size @ A ) ) ) ).
% measure_size
thf(fact_6355_uniformly__continuous__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ( ( topolo6026614971017936543ous_on @ A @ B )
= ( ^ [S8: set @ A,F3: A > B] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X: A] :
( ( member @ A @ X @ S8 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ S8 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y2 @ X ) @ D5 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) ) @ E4 ) ) ) ) ) ) ) ) ) ).
% uniformly_continuous_on_def
thf(fact_6356_isUCont__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F2: A > B] :
( ( topolo6026614971017936543ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S8 )
& ! [X: A,Y2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ S8 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X ) @ ( F2 @ Y2 ) ) @ R5 ) ) ) ) ) ) ) ).
% isUCont_def
thf(fact_6357_set__list__bind,axiom,
! [A: $tType,B: $tType,Xs: list @ B,F2: B > ( list @ A )] :
( ( set2 @ A @ ( bind @ B @ A @ Xs @ F2 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X: B] : ( set2 @ A @ ( F2 @ X ) )
@ ( set2 @ B @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_6358_has__derivative__power__int,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,X3: C,F11: C > A,S3: set @ C,N2: int] :
( ( ( F2 @ X3 )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F2 @ F11 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) )
=> ( has_derivative @ C @ A
@ ^ [X: C] : ( power_int @ A @ ( F2 @ X ) @ N2 )
@ ^ [H: C] : ( times_times @ A @ ( F11 @ H ) @ ( times_times @ A @ ( ring_1_of_int @ A @ N2 ) @ ( power_int @ A @ ( F2 @ X3 ) @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) ) ) )
@ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) ) ) ) ).
% has_derivative_power_int
thf(fact_6359_power__int__1__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N2: int] :
( ( power_int @ A @ ( one_one @ A ) @ N2 )
= ( one_one @ A ) ) ) ).
% power_int_1_left
thf(fact_6360_power__int__mult__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: num,N2: num] :
( ( power_int @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
= ( power_int @ A @ X3 @ ( numeral_numeral @ int @ ( times_times @ num @ M @ N2 ) ) ) ) ) ).
% power_int_mult_numeral
thf(fact_6361_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_6362_power__int__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N2: int] :
( ( sgn_sgn @ A @ ( power_int @ A @ A2 @ N2 ) )
= ( power_int @ A @ ( sgn_sgn @ A @ A2 ) @ N2 ) ) ) ).
% power_int_sgn
thf(fact_6363_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F2: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F2 )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_6364_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [W2: num,Y: A,M: int] :
( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) @ M )
= ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M ) @ ( power_int @ A @ Y @ M ) ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_6365_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,W2: num,M: int] :
( ( power_int @ A @ ( times_times @ A @ X3 @ ( numeral_numeral @ A @ W2 ) ) @ M )
= ( times_times @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_6366_power__int__0__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M )
= ( zero_zero @ A ) ) ) ) ).
% power_int_0_left
thf(fact_6367_power__int__eq__0__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,N2: int] :
( ( ( power_int @ A @ X3 @ N2 )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( N2
!= ( zero_zero @ int ) ) ) ) ) ).
% power_int_eq_0_iff
thf(fact_6368_power__int__0__right,axiom,
! [B: $tType] :
( ( ( inverse @ B )
& ( power @ B ) )
=> ! [X3: B] :
( ( power_int @ B @ X3 @ ( zero_zero @ int ) )
= ( one_one @ B ) ) ) ).
% power_int_0_right
thf(fact_6369_abs__power__int__minus,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N2: int] :
( ( abs_abs @ A @ ( power_int @ A @ ( uminus_uminus @ A @ A2 ) @ N2 ) )
= ( abs_abs @ A @ ( power_int @ A @ A2 @ N2 ) ) ) ) ).
% abs_power_int_minus
thf(fact_6370_power__int__of__nat,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ! [X3: A,N2: nat] :
( ( power_int @ A @ X3 @ ( semiring_1_of_nat @ int @ N2 ) )
= ( power_power @ A @ X3 @ N2 ) ) ) ).
% power_int_of_nat
thf(fact_6371_power__int__numeral,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ! [X3: A,N2: num] :
( ( power_int @ A @ X3 @ ( numeral_numeral @ int @ N2 ) )
= ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ N2 ) ) ) ) ).
% power_int_numeral
thf(fact_6372_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_6373_power__int__add__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: num,N2: num] :
( ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ M ) ) @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ N2 ) ) )
= ( power_int @ A @ X3 @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N2 ) ) ) ) ) ).
% power_int_add_numeral
thf(fact_6374_power__int__add__numeral2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: num,N2: num,B2: A] :
( ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ M ) ) @ ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ N2 ) ) @ B2 ) )
= ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N2 ) ) ) @ B2 ) ) ) ).
% power_int_add_numeral2
thf(fact_6375_power__int__mono__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N2: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ A2 @ N2 ) @ ( power_int @ A @ B2 @ N2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ).
% power_int_mono_iff
thf(fact_6376_list__bind__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F2: A > ( list @ B ),G: A > ( list @ B )] :
( ( Xs = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( F2 @ X5 )
= ( G @ X5 ) ) )
=> ( ( bind @ A @ B @ Xs @ F2 )
= ( bind @ A @ B @ Ys @ G ) ) ) ) ).
% list_bind_cong
thf(fact_6377_continuous__on__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S: set @ A,F2: A > B,N2: int] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( F2 @ X5 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X: A] : ( power_int @ B @ ( F2 @ X ) @ N2 ) ) ) ) ) ).
% continuous_on_power_int
thf(fact_6378_power__int__abs,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N2: int] :
( ( abs_abs @ A @ ( power_int @ A @ A2 @ N2 ) )
= ( power_int @ A @ ( abs_abs @ A @ A2 ) @ N2 ) ) ) ).
% power_int_abs
thf(fact_6379_power__int__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,Y: A,M: int] :
( ( power_int @ A @ ( times_times @ A @ X3 @ Y ) @ M )
= ( times_times @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ Y @ M ) ) ) ) ).
% power_int_mult_distrib
thf(fact_6380_power__int__commutes,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,N2: int] :
( ( times_times @ A @ ( power_int @ A @ X3 @ N2 ) @ X3 )
= ( times_times @ A @ X3 @ ( power_int @ A @ X3 @ N2 ) ) ) ) ).
% power_int_commutes
thf(fact_6381_power__int__mult,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: int,N2: int] :
( ( power_int @ A @ X3 @ ( times_times @ int @ M @ N2 ) )
= ( power_int @ A @ ( power_int @ A @ X3 @ M ) @ N2 ) ) ) ).
% power_int_mult
thf(fact_6382_power__int__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,N2: int] :
( ( power_int @ A @ ( inverse_inverse @ A @ X3 ) @ N2 )
= ( inverse_inverse @ A @ ( power_int @ A @ X3 @ N2 ) ) ) ) ).
% power_int_inverse
thf(fact_6383_power__int__divide__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,Y: A,M: int] :
( ( power_int @ A @ ( divide_divide @ A @ X3 @ Y ) @ M )
= ( divide_divide @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ Y @ M ) ) ) ) ).
% power_int_divide_distrib
thf(fact_6384_power__int__not__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,N2: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( N2
= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X3 @ N2 )
!= ( zero_zero @ A ) ) ) ) ).
% power_int_not_zero
thf(fact_6385_power__int__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,N2: int] :
( ( power_int @ A @ ( divide_divide @ A @ ( one_one @ A ) @ X3 ) @ N2 )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_int @ A @ X3 @ N2 ) ) ) ) ).
% power_int_one_over
thf(fact_6386_zero__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,N2: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X3 @ N2 ) ) ) ) ).
% zero_less_power_int
thf(fact_6387_power__int__minus,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,N2: int] :
( ( power_int @ A @ X3 @ ( uminus_uminus @ int @ N2 ) )
= ( inverse_inverse @ A @ ( power_int @ A @ X3 @ N2 ) ) ) ) ).
% power_int_minus
thf(fact_6388_zero__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,N2: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X3 @ N2 ) ) ) ) ).
% zero_le_power_int
thf(fact_6389_power__int__0__left__If,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M: int] :
( ( ( M
= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M )
= ( one_one @ A ) ) )
& ( ( M
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% power_int_0_left_If
thf(fact_6390_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N7: int,A2: A] :
( ( ord_less_eq @ int @ N2 @ N7 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A2 @ N2 ) @ ( power_int @ A @ A2 @ N7 ) ) ) ) ) ).
% power_int_increasing
thf(fact_6391_power__int__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N7: int,A2: A] :
( ( ord_less @ int @ N2 @ N7 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_int @ A @ A2 @ N2 ) @ ( power_int @ A @ A2 @ N7 ) ) ) ) ) ).
% power_int_strict_increasing
thf(fact_6392_power__int__diff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,M: int,N2: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( M != N2 ) )
=> ( ( power_int @ A @ X3 @ ( minus_minus @ int @ M @ N2 ) )
= ( divide_divide @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ X3 @ N2 ) ) ) ) ) ).
% power_int_diff
thf(fact_6393_tendsto__power__int,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A2: A,F5: filter @ B,N2: int] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( power_int @ A @ ( F2 @ X ) @ N2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( power_int @ A @ A2 @ N2 ) )
@ F5 ) ) ) ) ).
% tendsto_power_int
thf(fact_6394_continuous__at__within__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A2: A,S: set @ A,F2: A > B,N2: int] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X: A] : ( power_int @ B @ ( F2 @ X ) @ N2 ) ) ) ) ) ).
% continuous_at_within_power_int
thf(fact_6395_differentiable__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X3: A,S: set @ A,N2: int] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( F2 @ X3 )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X: A] : ( power_int @ B @ ( F2 @ X ) @ N2 )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% differentiable_power_int
thf(fact_6396_continuous__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F5: filter @ A,F2: A > B,N2: int] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X: A] : X ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X: A] : ( power_int @ B @ ( F2 @ X ) @ N2 ) ) ) ) ) ).
% continuous_power_int
thf(fact_6397_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N7: int,A2: A] :
( ( ord_less @ int @ N2 @ N7 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A2 @ N7 ) @ ( power_int @ A @ A2 @ N2 ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_6398_power__int__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A,N2: int] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( power_int @ A @ X3 @ N2 ) @ ( power_int @ A @ Y @ N2 ) ) ) ) ) ) ).
% power_int_mono
thf(fact_6399_power__int__strict__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N2: int] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ int @ N2 @ ( zero_zero @ int ) )
=> ( ord_less @ A @ ( power_int @ A @ B2 @ N2 ) @ ( power_int @ A @ A2 @ N2 ) ) ) ) ) ) ).
% power_int_strict_antimono
thf(fact_6400_one__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,N2: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ X3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X3 @ N2 ) ) ) ) ) ).
% one_le_power_int
thf(fact_6401_one__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A2 @ N2 ) ) ) ) ) ).
% one_less_power_int
thf(fact_6402_power__int__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: int,N2: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( ( plus_plus @ int @ M @ N2 )
!= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X3 @ ( plus_plus @ int @ M @ N2 ) )
= ( times_times @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ X3 @ N2 ) ) ) ) ) ).
% power_int_add
thf(fact_6403_power__int__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N2: int] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ int @ N2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ B2 @ N2 ) @ ( power_int @ A @ A2 @ N2 ) ) ) ) ) ) ).
% power_int_antimono
thf(fact_6404_power__int__strict__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N2: int] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less @ A @ ( power_int @ A @ A2 @ N2 ) @ ( power_int @ A @ B2 @ N2 ) ) ) ) ) ) ).
% power_int_strict_mono
thf(fact_6405_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N7: int,A2: A] :
( ( ord_less_eq @ int @ N2 @ N7 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( A2
!= ( zero_zero @ A ) )
| ( N7
!= ( zero_zero @ int ) )
| ( N2
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A2 @ N7 ) @ ( power_int @ A @ A2 @ N2 ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_6406_power__int__le__one,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,N2: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( ord_less_eq @ A @ X3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ X3 @ N2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% power_int_le_one
thf(fact_6407_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,M: int,N2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ X3 @ N2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less_eq @ int @ M @ N2 ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_6408_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,M: int,N2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X3 )
=> ( ( ord_less @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ X3 @ N2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less @ int @ M @ N2 ) ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_6409_power__int__minus__mult,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,N2: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( N2
!= ( zero_zero @ int ) ) )
=> ( ( times_times @ A @ ( power_int @ A @ X3 @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) ) @ X3 )
= ( power_int @ A @ X3 @ N2 ) ) ) ) ).
% power_int_minus_mult
thf(fact_6410_power__int__add__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X3 @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ ( power_int @ A @ X3 @ M ) @ X3 ) ) ) ) ).
% power_int_add_1
thf(fact_6411_power__int__add__1_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X3 @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ X3 @ ( power_int @ A @ X3 @ M ) ) ) ) ) ).
% power_int_add_1'
thf(fact_6412_power__int__def,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ( ( power_int @ A )
= ( ^ [X: A,N: int] : ( if @ A @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N ) @ ( power_power @ A @ X @ ( nat2 @ N ) ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ ( nat2 @ ( uminus_uminus @ int @ N ) ) ) ) ) ) ) ).
% power_int_def
thf(fact_6413_powr__real__of__int_H,axiom,
! [X3: real,N2: int] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( X3
!= ( zero_zero @ real ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ N2 ) )
=> ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ N2 ) )
= ( power_int @ real @ X3 @ N2 ) ) ) ) ).
% powr_real_of_int'
thf(fact_6414_DERIV__power__int,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D2: A,X3: A,S: set @ A,N2: int] :
( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( F2 @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( power_int @ A @ ( F2 @ X ) @ N2 )
@ ( times_times @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ N2 ) @ ( power_int @ A @ ( F2 @ X3 ) @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) ) ) @ D2 )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% DERIV_power_int
thf(fact_6415_has__derivative__power__int_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X3: A,N2: int,S3: set @ A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A
@ ^ [X: A] : ( power_int @ A @ X @ N2 )
@ ^ [Y2: A] : ( times_times @ A @ Y2 @ ( times_times @ A @ ( ring_1_of_int @ A @ N2 ) @ ( power_int @ A @ X3 @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ).
% has_derivative_power_int'
thf(fact_6416_power__int__numeral__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [M: num,N2: num] :
( ( power_int @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( inverse_inverse @ A @ ( numeral_numeral @ A @ ( pow @ M @ N2 ) ) ) ) ) ).
% power_int_numeral_neg_numeral
thf(fact_6417_eventually__less__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ ( ring_1_of_int @ B @ ( archimedean_ceiling @ B @ L ) ) )
@ F5 ) ) ) ) ).
% eventually_less_ceiling
thf(fact_6418_Ints__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ B )
=> ! [A3: set @ A,F2: A > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( member @ B @ ( F2 @ X5 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_sum
thf(fact_6419_Ints__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ B )
& ( ring_1 @ B ) )
=> ! [A3: set @ A,F2: A > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( member @ B @ ( F2 @ X5 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_prod
thf(fact_6420_frac__eq__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ( archimedean_frac @ A @ X3 )
= ( zero_zero @ A ) )
= ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_eq_0_iff
thf(fact_6421_floor__add2,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
| ( member @ A @ Y @ ( ring_1_Ints @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ) ).
% floor_add2
thf(fact_6422_frac__gt__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X3 ) )
= ( ~ ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) ) ) ) ).
% frac_gt_0_iff
thf(fact_6423_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_6424_Ints__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_0
thf(fact_6425_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_6426_minus__in__Ints__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: A] :
( ( member @ A @ ( uminus_uminus @ A @ X3 ) @ ( ring_1_Ints @ A ) )
= ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) ) ) ).
% minus_in_Ints_iff
thf(fact_6427_Ints__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N2: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_nat
thf(fact_6428_Ints__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N2: nat] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( power_power @ A @ A2 @ N2 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_power
thf(fact_6429_Ints__of__int,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: int] : ( member @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_int
thf(fact_6430_Ints__induct,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q4: A,P: A > $o] :
( ( member @ A @ Q4 @ ( ring_1_Ints @ A ) )
=> ( ! [Z3: int] : ( P @ ( ring_1_of_int @ A @ Z3 ) )
=> ( P @ Q4 ) ) ) ) ).
% Ints_induct
thf(fact_6431_Ints__cases,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q4: A] :
( ( member @ A @ Q4 @ ( ring_1_Ints @ A ) )
=> ~ ! [Z3: int] :
( Q4
!= ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% Ints_cases
thf(fact_6432_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_6433_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_6434_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_6435_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_6436_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N2: num] : ( member @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_6437_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_6438_Ints__subset__Rats,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ord_less_eq @ ( set @ A ) @ ( ring_1_Ints @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Ints_subset_Rats
thf(fact_6439_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_6440_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A2 @ X )
& ( ord_less_eq @ A @ X @ B2 ) ) ) ) ) ).
% finite_int_segment
thf(fact_6441_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_6442_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_6443_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_6444_Nats__altdef2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [N: A] :
( ( member @ A @ N @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ N ) ) ) ) ) ).
% Nats_altdef2
thf(fact_6445_Ints__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_Ints @ A )
= ( image @ int @ A @ ( ring_1_of_int @ A ) @ ( top_top @ ( set @ int ) ) ) ) ) ).
% Ints_def
thf(fact_6446_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_6447_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
=> ( ( X3
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X3 ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_6448_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( abs_abs @ A @ X3 ) @ ( one_one @ A ) )
=> ( X3
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_6449_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y @ ( ring_1_Ints @ A ) )
=> ( ( X3 = Y )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_6450_frac__neg,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X3 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X3 ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X3 ) ) ) ) ) ) ).
% frac_neg
thf(fact_6451_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_6452_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,A2: A] :
( ( ( archimedean_frac @ A @ X3 )
= A2 )
= ( ( member @ A @ ( minus_minus @ A @ X3 @ 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_6453_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_6454_eventually__floor__less,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less @ B @ ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ B @ L ) ) @ ( F2 @ X ) )
@ F5 ) ) ) ) ).
% eventually_floor_less
thf(fact_6455_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X: A,Y2: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( ord_max @ A @ X ) @ Y2 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_6456_UN__le__eq__Un0,axiom,
! [A: $tType,M7: nat > ( set @ A ),N2: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_ord_atMost @ nat @ N2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ) @ ( M7 @ ( zero_zero @ nat ) ) ) ) ).
% UN_le_eq_Un0
thf(fact_6457_UnCI,axiom,
! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
( ( ~ ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ A3 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_6458_Un__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( ( member @ A @ C2 @ A3 )
| ( member @ A @ C2 @ B3 ) ) ) ).
% Un_iff
thf(fact_6459_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_6460_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X3 @ Z2 )
& ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% le_sup_iff
thf(fact_6461_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_6462_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_6463_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_6464_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_6465_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X3: A,Y: A] :
( ( ( sup_sup @ A @ X3 @ Y )
= ( bot_bot @ A ) )
= ( ( X3
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_6466_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X3: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X3 @ Y ) )
= ( ( X3
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_6467_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ ( bot_bot @ A ) )
= X3 ) ) ).
% sup_bot_right
thf(fact_6468_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X3 )
= X3 ) ) ).
% sup_bot_left
thf(fact_6469_Un__empty,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_6470_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_6471_Un__subset__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C5 )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
& ( ord_less_eq @ ( set @ A ) @ B3 @ C5 ) ) ) ).
% Un_subset_iff
thf(fact_6472_Un__insert__right,axiom,
! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% Un_insert_right
thf(fact_6473_Un__insert__left,axiom,
! [A: $tType,A2: A,B3: set @ A,C5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C5 )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) ) ) ).
% Un_insert_left
thf(fact_6474_Int__Un__eq_I4_J,axiom,
! [A: $tType,T4: set @ A,S3: set @ A] :
( ( sup_sup @ ( set @ A ) @ T4 @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) )
= T4 ) ).
% Int_Un_eq(4)
thf(fact_6475_Int__Un__eq_I3_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ S3 @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_6476_Int__Un__eq_I2_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) @ T4 )
= T4 ) ).
% Int_Un_eq(2)
thf(fact_6477_Int__Un__eq_I1_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_6478_Un__Int__eq_I4_J,axiom,
! [A: $tType,T4: set @ A,S3: set @ A] :
( ( inf_inf @ ( set @ A ) @ T4 @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) )
= T4 ) ).
% Un_Int_eq(4)
thf(fact_6479_Un__Int__eq_I3_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ S3 @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_6480_Un__Int__eq_I2_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) @ T4 )
= T4 ) ).
% Un_Int_eq(2)
thf(fact_6481_Un__Int__eq_I1_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_6482_Un__Diff__cancel2,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) @ A3 )
= ( sup_sup @ ( set @ A ) @ B3 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_6483_Un__Diff__cancel,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_Diff_cancel
thf(fact_6484_fold__empty,axiom,
! [B: $tType,A: $tType,F2: B > A > A,Z2: A] :
( ( finite_fold @ B @ A @ F2 @ Z2 @ ( bot_bot @ ( set @ B ) ) )
= Z2 ) ).
% fold_empty
thf(fact_6485_fold__infinite,axiom,
! [A: $tType,B: $tType,A3: set @ A,F2: A > B > B,Z2: B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ A3 )
= Z2 ) ) ).
% fold_infinite
thf(fact_6486_Compl__Diff__eq,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ B3 ) ) ).
% Compl_Diff_eq
thf(fact_6487_if__image__distrib,axiom,
! [A: $tType,B: $tType,P: B > $o,F2: B > A,G: B > A,S3: set @ B] :
( ( image @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( F2 @ X ) @ ( G @ X ) )
@ S3 )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ S3 @ ( collect @ B @ P ) ) )
@ ( image @ B @ A @ G
@ ( inf_inf @ ( set @ B ) @ S3
@ ( collect @ B
@ ^ [X: B] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_6488_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_6489_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set @ C,A3: C > ( set @ D ),B3: set @ D] :
( ( ( C5
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image @ C @ ( set @ D )
@ ^ [X: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X ) @ B3 )
@ C5 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image @ C @ ( set @ D )
@ ^ [X: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X ) @ B3 )
@ C5 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ A3 @ C5 ) ) @ B3 ) ) ) ) ).
% UN_simps(2)
thf(fact_6490_UN__simps_I3_J,axiom,
! [E2: $tType,F: $tType,C5: set @ F,A3: set @ E2,B3: F > ( set @ E2 )] :
( ( ( C5
= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E2 )
@ ( image @ F @ ( set @ E2 )
@ ^ [X: F] : ( sup_sup @ ( set @ E2 ) @ A3 @ ( B3 @ X ) )
@ C5 ) )
= ( bot_bot @ ( set @ E2 ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E2 )
@ ( image @ F @ ( set @ E2 )
@ ^ [X: F] : ( sup_sup @ ( set @ E2 ) @ A3 @ ( B3 @ X ) )
@ C5 ) )
= ( sup_sup @ ( set @ E2 ) @ A3 @ ( complete_Sup_Sup @ ( set @ E2 ) @ ( image @ F @ ( set @ E2 ) @ B3 @ C5 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_6491_rtrancl__Un__separator__converseE,axiom,
! [A: $tType,A2: A,B2: A,P: 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 ) ) @ P @ Q ) ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ B2 ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X5 ) @ Q )
=> ( Y4 = X5 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_6492_rtrancl__Un__separatorE,axiom,
! [A: $tType,A2: A,B2: A,P: 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 ) ) @ P @ Q ) ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X5 ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ Q )
=> ( X5 = Y4 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_6493_mono__Un,axiom,
! [B: $tType,A: $tType,F2: ( set @ A ) > ( set @ B ),A3: set @ A,B3: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F2 )
=> ( ord_less_eq @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) @ ( F2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% mono_Un
thf(fact_6494_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F2: A > B,A3: A,B3: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) @ ( F2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% mono_sup
thf(fact_6495_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A4: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_6496_Un__singleton__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X3: A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_6497_singleton__Un__iff,axiom,
! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
( ( ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_6498_boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ ( bot_bot @ A ) )
= X3 ) ) ).
% boolean_algebra.disj_zero_right
thf(fact_6499_Un__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Un_empty_right
thf(fact_6500_Un__empty__left,axiom,
! [A: $tType,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 )
= B3 ) ).
% Un_empty_left
thf(fact_6501_Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( complete_Sup_Sup @ A @ A3 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) @ A3 ) ) ) ) ).
% Sup_fold_sup
thf(fact_6502_cSup__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B3 )
=> ( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_6503_Compl__partition2,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ A3 )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition2
thf(fact_6504_Compl__partition,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition
thf(fact_6505_sup__Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ B3 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ B3 @ A3 ) ) ) ) ).
% sup_Sup_fold_sup
thf(fact_6506_infinite__Un,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) ) )
= ( ~ ( finite_finite2 @ A @ S3 )
| ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% infinite_Un
thf(fact_6507_Un__infinite,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ~ ( finite_finite2 @ A @ S3 )
=> ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) ) ) ).
% Un_infinite
thf(fact_6508_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_6509_insert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A4: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] : X = A4 ) ) ) ) ).
% insert_def
thf(fact_6510_Un__Int__crazy,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) @ ( inf_inf @ ( set @ A ) @ C5 @ A3 ) )
= ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) ) @ ( sup_sup @ ( set @ A ) @ C5 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_6511_Int__Un__distrib,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( inf_inf @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% Int_Un_distrib
thf(fact_6512_Un__Int__distrib,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ ( sup_sup @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% Un_Int_distrib
thf(fact_6513_Int__Un__distrib2,axiom,
! [A: $tType,B3: set @ A,C5: set @ A,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) @ A3 )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B3 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C5 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_6514_Un__Int__distrib2,axiom,
! [A: $tType,B3: set @ A,C5: set @ A,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) @ A3 )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B3 @ A3 ) @ ( sup_sup @ ( set @ A ) @ C5 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_6515_Un__Diff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C5 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ C5 ) @ ( minus_minus @ ( set @ A ) @ B3 @ C5 ) ) ) ).
% Un_Diff
thf(fact_6516_UnE,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ( ~ ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% UnE
thf(fact_6517_UnI1,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_6518_UnI2,axiom,
! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
( ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_6519_bex__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
& ( P @ X ) ) )
= ( ? [X: A] :
( ( member @ A @ X @ A3 )
& ( P @ X ) )
| ? [X: A] :
( ( member @ A @ X @ B3 )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_6520_ball__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ( P @ X ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( P @ X ) )
& ! [X: A] :
( ( member @ A @ X @ B3 )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_6521_Un__assoc,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C5 )
= ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) ) ) ).
% Un_assoc
thf(fact_6522_Un__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_6523_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_6524_Un__left__absorb,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_left_absorb
thf(fact_6525_Un__left__commute,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) )
= ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% Un_left_commute
thf(fact_6526_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A6 )
| ( member @ A @ X @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_6527_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_6528_Collect__imp__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( P @ X )
=> ( Q @ X ) ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_6529_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,A2: A,B2: A] :
( ( ord_less @ A @ X3 @ A2 )
=> ( ord_less @ A @ X3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI1
thf(fact_6530_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,B2: A,A2: A] :
( ( ord_less @ A @ X3 @ B2 )
=> ( ord_less @ A @ X3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI2
thf(fact_6531_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_6532_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_6533_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_6534_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A4: A] :
( ( A4
= ( sup_sup @ A @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_6535_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ A2 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_6536_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_6537_Un__UNIV__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_6538_Un__UNIV__left,axiom,
! [A: $tType,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_6539_Diff__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) )
= ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% Diff_Un
thf(fact_6540_Diff__Int,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% Diff_Int
thf(fact_6541_Int__Diff__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_6542_Un__Diff__Int,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_6543_Compl__Int,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) ) ).
% Compl_Int
thf(fact_6544_Compl__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) ) ).
% Compl_Un
thf(fact_6545_set__shuffles,axiom,
! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( set2 @ A @ Zs2 )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_6546_Diff__subset__conv,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ C5 )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) ) ) ).
% Diff_subset_conv
thf(fact_6547_Diff__partition,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) )
= B3 ) ) ).
% Diff_partition
thf(fact_6548_Un__Int__assoc__eq,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C5 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) ) )
= ( ord_less_eq @ ( set @ A ) @ C5 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_6549_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y ) @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ).
% distrib_sup_le
thf(fact_6550_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y ) @ ( inf_inf @ A @ X3 @ Z2 ) ) @ ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% distrib_inf_le
thf(fact_6551_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_6552_subset__UnE,axiom,
! [A: $tType,C5: set @ A,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ! [A15: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A15 @ A3 )
=> ! [B14: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B14 @ B3 )
=> ( C5
!= ( sup_sup @ ( set @ A ) @ A15 @ B14 ) ) ) ) ) ).
% subset_UnE
thf(fact_6553_Un__absorb2,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
= A3 ) ) ).
% Un_absorb2
thf(fact_6554_Un__absorb1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
= B3 ) ) ).
% Un_absorb1
thf(fact_6555_Un__upper2,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_upper2
thf(fact_6556_Un__upper1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_upper1
thf(fact_6557_Un__least,axiom,
! [A: $tType,A3: set @ A,C5: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C5 ) ) ) ).
% Un_least
thf(fact_6558_Un__mono,axiom,
! [A: $tType,A3: set @ A,C5: set @ A,B3: set @ A,D6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ D6 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ ( sup_sup @ ( set @ A ) @ C5 @ D6 ) ) ) ) ).
% Un_mono
thf(fact_6559_Un__Pow__subset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B3 ) ) @ ( pow2 @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% Un_Pow_subset
thf(fact_6560_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X3: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X3 @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_6561_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_6562_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A,X3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq @ A @ A2 @ X3 )
=> ~ ( ord_less_eq @ A @ B2 @ X3 ) ) ) ) ).
% le_supE
thf(fact_6563_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,X3: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
=> ( ( ord_less_eq @ A @ B2 @ X3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X3 ) ) ) ) ).
% le_supI
thf(fact_6564_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y ) ) ) ).
% sup_ge1
thf(fact_6565_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X3: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X3 @ Y ) ) ) ).
% sup_ge2
thf(fact_6566_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_6567_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ X3 @ B2 )
=> ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_6568_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,D2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D2 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D2 ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_6569_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C2 @ D2 ) ) ) ) ) ).
% sup_mono
thf(fact_6570_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X3: A,Z2: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X3 ) ) ) ) ).
% sup_least
thf(fact_6571_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y2: A] :
( ( sup_sup @ A @ X @ Y2 )
= Y2 ) ) ) ) ).
% le_iff_sup
thf(fact_6572_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_6573_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_6574_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F2: A > A > A,X3: A,Y: A] :
( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ X5 @ ( F2 @ X5 @ Y4 ) )
=> ( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ ( F2 @ X5 @ Y4 ) )
=> ( ! [X5: A,Y4: A,Z3: A] :
( ( ord_less_eq @ A @ Y4 @ X5 )
=> ( ( ord_less_eq @ A @ Z3 @ X5 )
=> ( ord_less_eq @ A @ ( F2 @ Y4 @ Z3 ) @ X5 ) ) )
=> ( ( sup_sup @ A @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_6575_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_6576_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_6577_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( sup_sup @ A @ X3 @ Y )
= X3 ) ) ) ).
% sup_absorb1
thf(fact_6578_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( sup_sup @ A @ X3 @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_6579_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_6580_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_6581_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A4: A] :
( A4
= ( sup_sup @ A @ A4 @ B4 ) ) ) ) ) ).
% sup.order_iff
thf(fact_6582_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_6583_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_6584_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A4: A] :
( ( sup_sup @ A @ A4 @ B4 )
= A4 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_6585_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
( ( sup_sup @ A @ A4 @ B4 )
= B4 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_6586_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_6587_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_6588_ivl__disj__un__two_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_6589_ivl__disj__un__two__touch_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_6590_ivl__disj__un__two_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_6591_sup__SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B3: A,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( sup_sup @ A @ B3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F2 ) @ B3 @ A3 ) ) ) ) ).
% sup_SUP_fold_sup
thf(fact_6592_image__Un,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B3: set @ B] :
( ( image @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ ( image @ B @ A @ F2 @ B3 ) ) ) ).
% image_Un
thf(fact_6593_cSUP__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,B3: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ B3 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ B3 ) ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_6594_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F2 ) @ ( bot_bot @ A ) @ A3 ) ) ) ) ).
% SUP_fold_sup
thf(fact_6595_sup__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ( sup_sup @ A @ X3 @ Y )
= ( top_top @ A ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) ) ) ).
% sup_shunt
thf(fact_6596_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P4: A,Q4: A,R2: A] :
( ( ord_less_eq @ A @ P4 @ ( sup_sup @ A @ Q4 @ R2 ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P4 @ ( uminus_uminus @ A @ Q4 ) ) @ R2 ) ) ) ).
% sup_neg_inf
thf(fact_6597_shunt2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) @ Z2 )
= ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% shunt2
thf(fact_6598_shunt1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ Z2 )
= ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y ) @ Z2 ) ) ) ) ).
% shunt1
thf(fact_6599_boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [A2: A,X3: A,Y: A] :
( ( ( inf_inf @ A @ A2 @ X3 )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A2 @ X3 )
= ( top_top @ A ) )
=> ( ( ( inf_inf @ A @ A2 @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A2 @ Y )
= ( top_top @ A ) )
=> ( X3 = Y ) ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_6600_finite__Sup__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( member @ A @ Y4 @ A3 )
=> ( member @ A @ ( sup_sup @ A @ X5 @ Y4 ) @ A3 ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A3 ) @ A3 ) ) ) ) ) ).
% finite_Sup_in
thf(fact_6601_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B3: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_6602_ivl__disj__un__two_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_6603_inf__Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A3 ) @ B3 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ B3 @ A3 ) ) ) ) ).
% inf_Inf_fold_inf
thf(fact_6604_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_6605_Union__image__empty,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: B > ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= A3 ) ).
% Union_image_empty
thf(fact_6606_card__Un__le,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ).
% card_Un_le
thf(fact_6607_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( set_or7035219750837199246ssThan @ nat @ I @ ( plus_plus @ nat @ J @ K ) )
= ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) @ ( set_or7035219750837199246ssThan @ nat @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_6608_ivl__disj__un__two_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_6609_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_6610_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_6611_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_6612_Pow__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( pow2 @ A @ ( insert @ A @ A2 @ A3 ) )
= ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ A2 ) @ ( pow2 @ A @ A3 ) ) ) ) ).
% Pow_insert
thf(fact_6613_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C5: set @ C,A3: C > ( set @ D ),B3: set @ D] :
( ( ( C5
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ A3 @ C5 ) ) @ B3 )
= B3 ) )
& ( ( C5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ A3 @ C5 ) ) @ B3 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image @ C @ ( set @ D )
@ ^ [X: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X ) @ B3 )
@ C5 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_6614_UN__extend__simps_I3_J,axiom,
! [E2: $tType,F: $tType,C5: set @ F,A3: set @ E2,B3: F > ( set @ E2 )] :
( ( ( C5
= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E2 ) @ A3 @ ( complete_Sup_Sup @ ( set @ E2 ) @ ( image @ F @ ( set @ E2 ) @ B3 @ C5 ) ) )
= A3 ) )
& ( ( C5
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E2 ) @ A3 @ ( complete_Sup_Sup @ ( set @ E2 ) @ ( image @ F @ ( set @ E2 ) @ B3 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ E2 )
@ ( image @ F @ ( set @ E2 )
@ ^ [X: F] : ( sup_sup @ ( set @ E2 ) @ A3 @ ( B3 @ X ) )
@ C5 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_6615_Inter__Un__subset,axiom,
! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B3 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A3 @ B3 ) ) ) ).
% Inter_Un_subset
thf(fact_6616_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ( inf_inf @ A @ X3 @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ X3 @ Y )
= ( top_top @ A ) )
=> ( ( uminus_uminus @ A @ X3 )
= Y ) ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_6617_cSup__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( condit941137186595557371_above @ A @ X8 )
=> ( ( ( X8
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ X8 ) )
= A2 ) )
& ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ X8 ) )
= ( sup_sup @ A @ A2 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_6618_cSup__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ X8 ) )
= ( sup_sup @ A @ A2 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ).
% cSup_insert
thf(fact_6619_ivl__disj__un__two__touch_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_6620_sum_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B3 ) ) ) ) ) ) ).
% sum.union_inter
thf(fact_6621_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B3 ) ) ) ) ) ) ).
% prod.union_inter
thf(fact_6622_Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( complete_Inf_Inf @ A @ A3 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ A3 ) ) ) ) ).
% Inf_fold_inf
thf(fact_6623_card__Un__Int,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( plus_plus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_6624_ivl__disj__un__two__touch_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_6625_ivl__disj__un__two_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_6626_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_6627_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_6628_sum_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups7311177749621191930dd_sum @ B @ A )
= ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% sum.eq_fold
thf(fact_6629_cInf__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B3 )
=> ( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_6630_ivl__disj__un__two_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_6631_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ ( lattic643756798349783984er_Max @ A @ B3 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_6632_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_6633_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_6634_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_6635_Max_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ A3 ) )
= ( finite_fold @ A @ A @ ( ord_max @ A ) @ X3 @ A3 ) ) ) ) ).
% Max.eq_fold
thf(fact_6636_ivl__disj__un__two__touch_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_6637_SUP__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: A,B3: A] :
( ( sup_sup @ A @ A3
@ ( complete_Sup_Sup @ A
@ ( image @ nat @ A
@ ^ [X: nat] : B3
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% SUP_nat_binary
thf(fact_6638_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_6639_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ G @ A3 ) )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A3 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [A4: B] : ( sup_sup @ A @ ( F2 @ A4 ) @ ( G @ A4 ) )
@ A3 ) ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_6640_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,G: A > B,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( inj_on @ A @ B @ G @ B3 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ G @ B3 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ A3 ) @ ( F2 @ X ) @ ( G @ X ) )
@ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_6641_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X )
@ ^ [X: A,Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_6642_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
@ ^ [X: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% rtrancl_insert
thf(fact_6643_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
@ ^ [X: A,Y2: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ ( transitive_trancl @ A @ R2 ) )
| ( X = A2 ) )
& ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y2 ) @ ( transitive_trancl @ A @ R2 ) )
| ( Y2 = B2 ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_6644_cSup__inter__less__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( condit941137186595557371_above @ A @ A3 )
=> ( ( condit941137186595557371_above @ A @ B3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_6645_cSUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,A2: B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( insert @ B @ A2 @ A3 ) ) )
= ( sup_sup @ A @ ( F2 @ A2 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_6646_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B3 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_6647_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,B3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) ) ) ) ) ) ).
% sum_Un
thf(fact_6648_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( ( inf_inf @ ( set @ B ) @ A3 @ B3 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B3 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_6649_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B3 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_6650_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( ( inf_inf @ ( set @ B ) @ A3 @ B3 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B3 ) ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_6651_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_6652_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A3: set @ A,B3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ) ).
% sum_Un2
thf(fact_6653_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B3 @ A3 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_6654_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B3 @ A3 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_6655_card__Un__disjoint,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_6656_inj__on__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( ( inj_on @ A @ B @ F2 @ A3 )
& ( inj_on @ A @ B @ F2 @ B3 )
& ( ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) @ ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_6657_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_6658_ivl__disj__un__two_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_6659_cINF__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,B3: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ B3 ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ B3 ) ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_6660_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_6661_inf__INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B3: A,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( inf_inf @ A @ B3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F2 ) @ B3 @ A3 ) ) ) ) ).
% inf_INF_fold_inf
thf(fact_6662_sum__Un__nat,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_6663_ivl__disj__un__two_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_6664_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_6665_trancl__insert,axiom,
! [A: $tType,Y: A,X3: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ R2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A4: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B4 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% trancl_insert
thf(fact_6666_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A3: set @ B,B3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) )
=> ( ( F2 @ X5 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_6667_INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F2 ) @ ( top_top @ A ) @ A3 ) ) ) ) ).
% INF_fold_inf
thf(fact_6668_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G: A > B,C5: set @ A,B3: set @ A,X3: A] :
( ( inj_on @ A @ B @ G @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ C5 @ ( sup_sup @ ( set @ A ) @ B3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ ( B > A )
@ ^ [I3: B] : ( if @ A @ ( member @ B @ I3 @ ( image @ A @ B @ G @ C5 ) ) @ ( the_inv_into @ A @ B @ C5 @ G @ I3 ) @ X3 )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_6669_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_6670_Ball__fold,axiom,
! [A: $tType,A3: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( P @ X ) ) )
= ( finite_fold @ A @ $o
@ ^ [K3: A,S8: $o] :
( S8
& ( P @ K3 ) )
@ $true
@ A3 ) ) ) ).
% Ball_fold
thf(fact_6671_sup__int__def,axiom,
( ( sup_sup @ int )
= ( ord_max @ int ) ) ).
% sup_int_def
thf(fact_6672_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A6 )
@ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_6673_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_6674_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R )
@ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S3 ) )
= ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ) ) ).
% sup_Un_eq2
thf(fact_6675_union__fold__insert,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
= ( finite_fold @ A @ ( set @ A ) @ ( insert @ A ) @ B3 @ A3 ) ) ) ).
% union_fold_insert
thf(fact_6676_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu3: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_6677_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
@ ^ [X: A] : X )
@ ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.eq_fold
thf(fact_6678_image__fold__insert,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( image @ A @ B @ F2 @ A3 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K3: A] : ( insert @ B @ ( F2 @ K3 ) )
@ ( bot_bot @ ( set @ B ) )
@ A3 ) ) ) ).
% image_fold_insert
thf(fact_6679_Id__on__fold,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( id_on @ A @ A3 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A3 ) ) ) ).
% Id_on_fold
thf(fact_6680_fold__union__pair,axiom,
! [B: $tType,A: $tType,B3: set @ A,X3: B,A3: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B3 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y2: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y2 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B3 ) )
@ A3 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y2: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y2 ) )
@ A3
@ B3 ) ) ) ).
% fold_union_pair
thf(fact_6681_Pow__fold,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( pow2 @ A @ A3 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X: A,A6: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A6 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X ) @ A6 ) )
@ ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A3 ) ) ) ).
% Pow_fold
thf(fact_6682_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_6683_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: B > A,A17: set @ B,B1: set @ A,F22: C > D,B22: set @ C,A26: set @ D] :
( ( ( image @ B @ A @ F1 @ A17 )
= B1 )
=> ( ( inj_on @ C @ D @ F22 @ B22 )
=> ( ( ord_less_eq @ ( set @ D ) @ ( image @ C @ D @ F22 @ B22 ) @ A26 )
=> ( ( ( B22
= ( bot_bot @ ( set @ C ) ) )
=> ( A26
= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( bNF_Wellorder_Func @ C @ A @ B22 @ B1 )
= ( image @ ( D > B ) @ ( C > A ) @ ( bNF_We4925052301507509544nc_map @ C @ B @ A @ D @ B22 @ F1 @ F22 ) @ ( bNF_Wellorder_Func @ D @ B @ A26 @ A17 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_6684_max__ext__additive,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,R: set @ ( product_prod @ A @ A ),C5: set @ A,D6: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A3 @ B3 ) @ ( max_ext @ A @ R ) )
=> ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ C5 @ D6 ) @ ( max_ext @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ C5 ) @ ( sup_sup @ ( set @ A ) @ B3 @ D6 ) ) @ ( max_ext @ A @ R ) ) ) ) ).
% max_ext_additive
thf(fact_6685_Func__map,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,G: A > B,A26: set @ A,A17: set @ B,F1: B > C,B1: set @ C,F22: D > A,B22: set @ D] :
( ( member @ ( A > B ) @ G @ ( bNF_Wellorder_Func @ A @ B @ A26 @ A17 ) )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ F1 @ A17 ) @ B1 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ D @ A @ F22 @ B22 ) @ A26 )
=> ( member @ ( D > C ) @ ( bNF_We4925052301507509544nc_map @ D @ B @ C @ A @ B22 @ F1 @ F22 @ G ) @ ( bNF_Wellorder_Func @ D @ C @ B22 @ B1 ) ) ) ) ) ).
% Func_map
thf(fact_6686_max__ext_Ocases,axiom,
! [A: $tType,A12: set @ A,A23: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A12 @ A23 ) @ ( max_ext @ A @ R ) )
=> ~ ( ( finite_finite2 @ A @ A12 )
=> ( ( finite_finite2 @ A @ A23 )
=> ( ( A23
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ A12 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A23 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa3 ) @ R ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_6687_max__ext_Osimps,axiom,
! [A: $tType,A12: set @ A,A23: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A12 @ A23 ) @ ( max_ext @ A @ R ) )
= ( ( finite_finite2 @ A @ A12 )
& ( finite_finite2 @ A @ A23 )
& ( A23
!= ( bot_bot @ ( set @ A ) ) )
& ! [X: A] :
( ( member @ A @ X @ A12 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ A23 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R ) ) ) ) ) ).
% max_ext.simps
thf(fact_6688_max__ext_Omax__extI,axiom,
! [A: $tType,X8: set @ A,Y7: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X8 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( Y7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Y7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa ) @ R ) ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X8 @ Y7 ) @ ( max_ext @ A @ R ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_6689_Func__non__emp,axiom,
! [A: $tType,B: $tType,B3: set @ A,A3: set @ B] :
( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bNF_Wellorder_Func @ B @ A @ A3 @ B3 )
!= ( bot_bot @ ( set @ ( B > A ) ) ) ) ) ).
% Func_non_emp
thf(fact_6690_Func__is__emp,axiom,
! [A: $tType,B: $tType,A3: set @ A,B3: set @ B] :
( ( ( bNF_Wellorder_Func @ A @ B @ A3 @ B3 )
= ( bot_bot @ ( set @ ( A > B ) ) ) )
= ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Func_is_emp
thf(fact_6691_Set__filter__fold,axiom,
! [A: $tType,A3: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( filter3 @ A @ P @ A3 )
= ( finite_fold @ A @ ( set @ A )
@ ^ [X: A,A18: set @ A] : ( if @ ( set @ A ) @ ( P @ X ) @ ( insert @ A @ X @ A18 ) @ A18 )
@ ( bot_bot @ ( set @ A ) )
@ A3 ) ) ) ).
% Set_filter_fold
thf(fact_6692_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X: A,Y2: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( sup_sup @ A @ X ) @ Y2 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_6693_member__filter,axiom,
! [A: $tType,X3: A,P: A > $o,A3: set @ A] :
( ( member @ A @ X3 @ ( filter3 @ A @ P @ A3 ) )
= ( ( member @ A @ X3 @ A3 )
& ( P @ X3 ) ) ) ).
% member_filter
thf(fact_6694_inf__Sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ( ( inf_inf @ A @ A2 @ ( lattic5882676163264333800up_fin @ A @ A3 ) )
= A2 ) ) ) ) ).
% inf_Sup_absorb
thf(fact_6695_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A] :
( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Sup_fin.singleton
thf(fact_6696_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ A3 ) )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_6697_Sup__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A3 ) )
= ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_6698_Sup__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ A @ A2 @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_6699_Sup__fin__Max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% Sup_fin_Max
thf(fact_6700_Set_Ofilter__def,axiom,
! [A: $tType] :
( ( filter3 @ A )
= ( ^ [P3: A > $o,A6: set @ A] :
( collect @ A
@ ^ [A4: A] :
( ( member @ A @ A4 @ A6 )
& ( P3 @ A4 ) ) ) ) ) ).
% Set.filter_def
thf(fact_6701_finite__filter,axiom,
! [A: $tType,S3: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ S3 )
=> ( finite_finite2 @ A @ ( filter3 @ A @ P @ S3 ) ) ) ).
% finite_filter
thf(fact_6702_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X3 )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_6703_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A3 )
=> ( ord_less_eq @ A @ A5 @ X3 ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X3 ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_6704_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X3 )
=> ! [A11: A] :
( ( member @ A @ A11 @ A3 )
=> ( ord_less_eq @ A @ A11 @ X3 ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_6705_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= ( lattic5882676163264333800up_fin @ A @ X8 ) ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_6706_Sup__fin__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A3 )
= ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% Sup_fin_Sup
thf(fact_6707_Sup__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( lattic5882676163264333800up_fin @ A @ A3 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Sup_fin.infinite
thf(fact_6708_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B3 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_6709_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H2: A > A,N7: set @ A] :
( ! [X5: A,Y4: A] :
( ( H2 @ ( sup_sup @ A @ X5 @ Y4 ) )
= ( sup_sup @ A @ ( H2 @ X5 ) @ ( H2 @ Y4 ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic5882676163264333800up_fin @ A @ N7 ) )
= ( lattic5882676163264333800up_fin @ A @ ( image @ A @ A @ H2 @ N7 ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_6710_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B3 ) @ ( lattic5882676163264333800up_fin @ A @ A3 ) )
= ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_6711_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ A3 ) )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_6712_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] : ( member @ A @ ( sup_sup @ A @ X5 @ Y4 ) @ ( insert @ A @ X5 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_6713_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B3 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_6714_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ A3 ) )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X3 @ A3 ) ) ) ) ).
% Sup_fin.eq_fold
thf(fact_6715_inter__Set__filter,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( filter3 @ A
@ ^ [X: A] : ( member @ A @ X @ A3 )
@ B3 ) ) ) ).
% inter_Set_filter
thf(fact_6716_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A3 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A4: A] :
( ( Uu3
= ( inf_inf @ A @ X3 @ A4 ) )
& ( member @ A @ A4 @ A3 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_6717_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B3 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A4: A,B4: A] :
( ( Uu3
= ( inf_inf @ A @ A4 @ B4 ) )
& ( member @ A @ A4 @ A3 )
& ( member @ A @ B4 @ B3 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_6718_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ A3 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ A3 ) )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_6719_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A3 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A3 )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_6720_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X: A,Y2: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( inf_inf @ A @ X ) @ Y2 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_6721_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
@ ^ [X: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R6 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_6722_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A] :
( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Inf_fin.singleton
thf(fact_6723_sup__Inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ A2 )
= A2 ) ) ) ) ).
% sup_Inf_absorb
thf(fact_6724_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ A3 ) )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_6725_Inf__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ A2 ) ) ) ) ).
% Inf_fin.coboundedI
thf(fact_6726_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
= ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_6727_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
=> ! [A11: A] :
( ( member @ A @ A11 @ A3 )
=> ( ord_less_eq @ A @ X3 @ A11 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_6728_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A3 )
=> ( ord_less_eq @ A @ X3 @ A5 ) )
=> ( ord_less_eq @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_6729_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_6730_Inf__fin__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A3 )
= ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ).
% Inf_fin_Inf
thf(fact_6731_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= ( lattic7752659483105999362nf_fin @ A @ X8 ) ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_6732_Inf__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( lattic7752659483105999362nf_fin @ A @ A3 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Inf_fin.infinite
thf(fact_6733_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B3 ) @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_6734_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H2: A > A,N7: set @ A] :
( ! [X5: A,Y4: A] :
( ( H2 @ ( inf_inf @ A @ X5 @ Y4 ) )
= ( inf_inf @ A @ ( H2 @ X5 ) @ ( H2 @ Y4 ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic7752659483105999362nf_fin @ A @ N7 ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image @ A @ A @ H2 @ N7 ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_6735_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B3 ) @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
= ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_6736_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] : ( member @ A @ ( inf_inf @ A @ X5 @ Y4 ) @ ( insert @ A @ X5 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_6737_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ A3 ) )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_6738_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic7752659483105999362nf_fin @ A @ B3 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_6739_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_6740_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ A3 ) )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X3 @ A3 ) ) ) ) ).
% Inf_fin.eq_fold
thf(fact_6741_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic7752659483105999362nf_fin @ A @ B3 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A4: A,B4: A] :
( ( Uu3
= ( sup_sup @ A @ A4 @ B4 ) )
& ( member @ A @ A4 @ A3 )
& ( member @ A @ B4 @ B3 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_6742_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A4: A] :
( ( Uu3
= ( sup_sup @ A @ X3 @ A4 ) )
& ( member @ A @ A4 @ A3 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_6743_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A3 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A3 )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_6744_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ A3 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ A3 ) )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_6745_max__extp__max__ext__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R ) )
= ( ^ [X: set @ A,Y2: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X @ Y2 ) @ ( max_ext @ A @ R ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_6746_max__extp__eq,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R5: A > A > $o,X: set @ A,Y2: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X @ Y2 ) @ ( max_ext @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% max_extp_eq
thf(fact_6747_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X: A,Y2: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( ord_min @ A @ X ) @ Y2 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Min.eq_fold'
thf(fact_6748_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B ),X3: product_prod @ C @ A,X8: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X3 @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S3 ) @ X8 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z6: B,A18: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X3 )
= W3 )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X3 ) @ Z6 ) @ A18 )
@ A18 ) )
@ X8
@ S3 ) ) ) ).
% insert_relcomp_union_fold
thf(fact_6749_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_6750_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_6751_finite__relcomp,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S3 )
=> ( finite_finite2 @ ( product_prod @ A @ C ) @ ( relcomp @ A @ B @ C @ R @ S3 ) ) ) ) ).
% finite_relcomp
thf(fact_6752_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Min_singleton
thf(fact_6753_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_6754_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_6755_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ B,C2: A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image @ B @ A
@ ^ [Uu3: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% Min_const
thf(fact_6756_Min__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ A3 ) )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min_insert
thf(fact_6757_minus__Max__eq__Min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S3 ) )
= ( lattic643756798350308766er_Min @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_6758_minus__Min__eq__Max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S3 ) )
= ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_6759_Inf__fin__Min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% Inf_fin_Min
thf(fact_6760_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A12: A,A23: C,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A12 @ A23 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) )
=> ~ ! [B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A12 @ B5 ) @ R2 )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ A23 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_6761_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A12: A,A23: C,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A12 @ A23 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) )
= ( ? [A4: A,B4: B,C4: C] :
( ( A12 = A4 )
& ( A23 = C4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R2 )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ C4 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_6762_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),C2: 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 @ C2 ) @ S )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A2 @ C2 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_6763_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 ) )
=> ~ ! [X5: A,Y4: C,Z3: B] :
( ( Xz
= ( product_Pair @ A @ B @ X5 @ Z3 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X5 @ Y4 ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y4 @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_6764_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A2: A,C2: B,R2: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ C2 ) @ ( relcomp @ A @ C @ B @ R2 @ S ) )
=> ~ ! [B5: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A2 @ B5 ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B5 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_6765_Min_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ A2 ) ) ) ) ).
% Min.coboundedI
thf(fact_6766_Min__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ A3 )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( lattic643756798350308766er_Min @ A @ A3 )
= X3 ) ) ) ) ) ).
% Min_eqI
thf(fact_6767_Min__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ X3 ) ) ) ) ).
% Min_le
thf(fact_6768_relpow__add,axiom,
! [A: $tType,M: nat,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( plus_plus @ nat @ M @ N2 ) @ R )
= ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M @ R ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ).
% relpow_add
thf(fact_6769_relpow_Osimps_I2_J,axiom,
! [A: $tType,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R )
= ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) @ R ) ) ).
% relpow.simps(2)
thf(fact_6770_Min_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ).
% Min.in_idem
thf(fact_6771_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ A3 ) ) ) ) ).
% Min_in
thf(fact_6772_union__comp__emptyL,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),C5: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A3 @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ B3 @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B3 ) @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyL
thf(fact_6773_union__comp__emptyR,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A ),C5: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ A3 @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ A3 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ B3 @ C5 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyR
thf(fact_6774_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A3 )
=> ( ord_less_eq @ A @ X3 @ A5 ) )
=> ( ord_less_eq @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_6775_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) )
=> ! [A11: A] :
( ( member @ A @ A11 @ A3 )
=> ( ord_less_eq @ A @ X3 @ A11 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_6776_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( ( member @ A @ M @ A3 )
& ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ M @ X ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_6777_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ X3 )
= ( ? [X: A] :
( ( member @ A @ X @ A3 )
& ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_6778_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798350308766er_Min @ A @ A3 )
= M )
= ( ( member @ A @ M @ A3 )
& ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ M @ X ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_6779_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ X3 )
= ( ? [X: A] :
( ( member @ A @ X @ A3 )
& ( ord_less @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_6780_Min__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ A3 )
=> ( ord_less_eq @ A @ A2 @ B5 ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ A2 @ A3 ) )
= A2 ) ) ) ) ).
% Min_insert2
thf(fact_6781_cInf__eq__Min,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= ( lattic643756798350308766er_Min @ A @ X8 ) ) ) ) ) ).
% cInf_eq_Min
thf(fact_6782_Min__Inf,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A3 )
= ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ).
% Min_Inf
thf(fact_6783_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
@ ^ [X: A,Z6: B] :
? [Y2: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ Y2 ) @ R5 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y2 @ Z6 ) @ S8 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_6784_Min_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( lattic643756798350308766er_Min @ A @ A3 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Min.infinite
thf(fact_6785_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B3 ) @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_6786_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M7: set @ A,N7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M7 @ N7 )
=> ( ( M7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N7 ) @ ( lattic643756798350308766er_Min @ A @ M7 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_6787_max__ext__compat,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S3 ) @ 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 @ S3 ) @ ( insert @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( max_ext @ A @ R ) ) ) ).
% max_ext_compat
thf(fact_6788_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N7: set @ A] :
( ! [X5: A,Y4: A] :
( ( H2 @ ( ord_min @ A @ X5 @ Y4 ) )
= ( ord_min @ A @ ( H2 @ X5 ) @ ( H2 @ Y4 ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798350308766er_Min @ A @ N7 ) )
= ( lattic643756798350308766er_Min @ A @ ( image @ A @ A @ H2 @ N7 ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_6789_min__ext__compat,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S3 ) @ 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 @ S3 ) @ ( insert @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( min_ext @ A @ R ) ) ) ).
% min_ext_compat
thf(fact_6790_Min_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B3 ) @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min.subset
thf(fact_6791_Min_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] : ( member @ A @ ( ord_min @ A @ X5 @ Y4 ) @ ( insert @ A @ X5 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Min.closed
thf(fact_6792_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ A3 ) )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_6793_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F2 @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( lattic643756798350308766er_Min @ B @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_6794_Min_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ ( lattic643756798350308766er_Min @ A @ B3 ) ) ) ) ) ) ) ) ).
% Min.union
thf(fact_6795_Min_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ A3 ) )
= ( finite_fold @ A @ A @ ( ord_min @ A ) @ X3 @ A3 ) ) ) ) ).
% Min.eq_fold
thf(fact_6796_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S3: set @ B,F2: B > A,K: A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( F2 @ X ) @ K )
@ S3 ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image @ B @ A @ F2 @ S3 ) ) @ K ) ) ) ) ) ).
% Min_add_commute
thf(fact_6797_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S3 )
=> ( ( relcomp @ A @ B @ C @ R @ S3 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [X: A,Y2: B,A6: set @ ( product_prod @ A @ C )] :
( finite_fold @ ( product_prod @ B @ C ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ B @ C @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [W3: B,Z6: C,A18: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y2 = W3 ) @ ( insert @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ Z6 ) @ A18 ) @ A18 ) )
@ A6
@ S3 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R ) ) ) ) ).
% relcomp_fold
thf(fact_6798_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ A3 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ A3 ) )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_6799_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A3 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A3 )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_6800_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B ),X3: product_prod @ C @ A,R: set @ ( product_prod @ C @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X3 @ R ) @ S3 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z6: B,A18: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X3 )
= W3 )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X3 ) @ Z6 ) @ A18 )
@ A18 ) )
@ ( relcomp @ C @ A @ B @ R @ S3 )
@ S3 ) ) ) ).
% insert_relcomp_fold
thf(fact_6801_card__Min__le__sum,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A3 ) @ ( lattic643756798350308766er_Min @ nat @ ( image @ A @ nat @ F2 @ A3 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) ) ) ).
% card_Min_le_sum
thf(fact_6802_comp__fun__commute__relcomp__fold,axiom,
! [A: $tType,B: $tType,C: $tType,S3: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( finite6289374366891150609ommute @ ( product_prod @ C @ A ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ C @ A @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [X: C,Y2: A,A6: set @ ( product_prod @ C @ B )] :
( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z6: B,A18: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y2 = W3 ) @ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X @ Z6 ) @ A18 ) @ A18 ) )
@ A6
@ S3 ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_6803_dual__Max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Max @ A
@ ^ [X: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X ) )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% dual_Max
thf(fact_6804_linorder_OMax_Ocong,axiom,
! [A: $tType] :
( ( lattices_Max @ A )
= ( lattices_Max @ A ) ) ).
% linorder.Max.cong
thf(fact_6805_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X: B,Z6: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y2: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y2 ) )
@ Z6
@ B3 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_6806_f__arg__min__list__f,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [Xs: list @ A,F2: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( F2 @ ( arg_min_list @ A @ B @ F2 @ Xs ) )
= ( lattic643756798350308766er_Min @ B @ ( image @ A @ B @ F2 @ ( set2 @ A @ Xs ) ) ) ) ) ) ).
% f_arg_min_list_f
thf(fact_6807_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A3: B > ( set @ A ),I: B,B3: set @ A,J4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ ( fun_upd @ B @ ( set @ A ) @ A3 @ I @ B3 ) @ J4 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ B ) @ J4 @ ( insert @ B @ I @ ( bot_bot @ ( set @ B ) ) ) ) ) ) @ ( if @ ( set @ A ) @ ( member @ B @ I @ J4 ) @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% UNION_fun_upd
thf(fact_6808_finite__update__induct,axiom,
! [B: $tType,A: $tType,F2: A > B,C2: B,P: ( A > B ) > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A4: A] :
( ( F2 @ A4 )
!= C2 ) ) )
=> ( ( P
@ ^ [A4: A] : C2 )
=> ( ! [A5: A,B5: B,F4: A > B] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [C4: A] :
( ( F4 @ C4 )
!= C2 ) ) )
=> ( ( ( F4 @ A5 )
= C2 )
=> ( ( B5 != C2 )
=> ( ( P @ F4 )
=> ( P @ ( fun_upd @ A @ B @ F4 @ A5 @ B5 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_update_induct
thf(fact_6809_arg__min__list__in,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [Xs: list @ A,F2: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( member @ A @ ( arg_min_list @ A @ B @ F2 @ Xs ) @ ( set2 @ A @ Xs ) ) ) ) ).
% arg_min_list_in
thf(fact_6810_fun__upd__image,axiom,
! [A: $tType,B: $tType,X3: B,A3: set @ B,F2: B > A,Y: A] :
( ( ( member @ B @ X3 @ A3 )
=> ( ( image @ B @ A @ ( fun_upd @ B @ A @ F2 @ X3 @ Y ) @ A3 )
= ( insert @ A @ Y @ ( image @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ X3 @ A3 )
=> ( ( image @ B @ A @ ( fun_upd @ B @ A @ F2 @ X3 @ Y ) @ A3 )
= ( image @ B @ A @ F2 @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_6811_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,A3: set @ A,B3: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ S3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( finite_fold @ A @ B @ F2 @ ( finite_fold @ A @ B @ F2 @ Z2 @ A3 ) @ B3 ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
thf(fact_6812_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) )
= ( F2 @ X3 @ ( finite_fold @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert_remove
thf(fact_6813_Finite__Set_Ofold__cong,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,G: A > B > B,A3: set @ A,S: B,T2: B,B3: set @ A] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 )
=> ( ( finite4664212375090638736ute_on @ A @ B @ S3 @ G )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ S3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( F2 @ X5 )
= ( G @ X5 ) ) )
=> ( ( S = T2 )
=> ( ( A3 = B3 )
=> ( ( finite_fold @ A @ B @ F2 @ S @ A3 )
= ( finite_fold @ A @ B @ G @ T2 @ B3 ) ) ) ) ) ) ) ) ) ).
% Finite_Set.fold_cong
thf(fact_6814_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
! [B: $tType,A: $tType,C: $tType,S3: set @ A,F2: A > B > B,G: C > A,R: set @ C] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ G @ ( top_top @ ( set @ C ) ) ) @ S3 )
=> ( finite4664212375090638736ute_on @ C @ B @ R @ ( comp @ A @ ( B > B ) @ C @ F2 @ G ) ) ) ) ).
% comp_fun_commute_on.comp_comp_fun_commute_on
thf(fact_6815_comp__fun__commute__on_Ofold__insert,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) )
= ( F2 @ X3 @ ( finite_fold @ A @ B @ F2 @ Z2 @ A3 ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert
thf(fact_6816_comp__fun__commute__on_Ofold__insert2,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) )
= ( finite_fold @ A @ B @ F2 @ ( F2 @ X3 @ Z2 ) @ A3 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert2
thf(fact_6817_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( F2 @ X3 @ ( finite_fold @ A @ B @ F2 @ Z2 @ A3 ) )
= ( finite_fold @ A @ B @ F2 @ ( F2 @ X3 @ Z2 ) @ A3 ) ) ) ) ) ).
% comp_fun_commute_on.fold_fun_left_comm
thf(fact_6818_comp__fun__commute__on_Ofold__rec,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,A3: set @ A,X3: A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ S3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ A3 )
= ( F2 @ X3 @ ( finite_fold @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_rec
thf(fact_6819_image__map__upd,axiom,
! [B: $tType,A: $tType,X3: A,A3: set @ A,M: A > ( option @ B ),Y: B] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ( image @ A @ ( option @ B ) @ ( fun_upd @ A @ ( option @ B ) @ M @ X3 @ ( some @ B @ Y ) ) @ A3 )
= ( image @ A @ ( option @ B ) @ M @ A3 ) ) ) ).
% image_map_upd
thf(fact_6820_finite__range__updI,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),A2: B,B2: A] :
( ( finite_finite2 @ ( option @ A ) @ ( image @ B @ ( option @ A ) @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( option @ A ) @ ( image @ B @ ( option @ A ) @ ( fun_upd @ B @ ( option @ A ) @ F2 @ A2 @ ( some @ A @ B2 ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_updI
thf(fact_6821_empty__upd__none,axiom,
! [B: $tType,A: $tType,X3: A] :
( ( fun_upd @ A @ ( option @ B )
@ ^ [X: A] : ( none @ B )
@ X3
@ ( none @ B ) )
= ( ^ [X: A] : ( none @ B ) ) ) ).
% empty_upd_none
thf(fact_6822_map__upd__Some__unfold,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),A2: B,B2: A,X3: B,Y: A] :
( ( ( fun_upd @ B @ ( option @ A ) @ M @ A2 @ ( some @ A @ B2 ) @ X3 )
= ( some @ A @ Y ) )
= ( ( ( X3 = A2 )
& ( B2 = Y ) )
| ( ( X3 != A2 )
& ( ( M @ X3 )
= ( some @ A @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_6823_map__upd__triv,axiom,
! [A: $tType,B: $tType,T2: B > ( option @ A ),K: B,X3: A] :
( ( ( T2 @ K )
= ( some @ A @ X3 ) )
=> ( ( fun_upd @ B @ ( option @ A ) @ T2 @ K @ ( some @ A @ X3 ) )
= T2 ) ) ).
% map_upd_triv
thf(fact_6824_map__upd__eqD1,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),A2: A,X3: B,N2: A > ( option @ B ),Y: B] :
( ( ( fun_upd @ A @ ( option @ B ) @ M @ A2 @ ( some @ B @ X3 ) )
= ( fun_upd @ A @ ( option @ B ) @ N2 @ A2 @ ( some @ B @ Y ) ) )
=> ( X3 = Y ) ) ).
% map_upd_eqD1
thf(fact_6825_map__upd__nonempty,axiom,
! [B: $tType,A: $tType,T2: A > ( option @ B ),K: A,X3: B] :
( ( fun_upd @ A @ ( option @ B ) @ T2 @ K @ ( some @ B @ X3 ) )
!= ( ^ [X: A] : ( none @ B ) ) ) ).
% map_upd_nonempty
thf(fact_6826_graph__map__upd,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),K: A,V2: B] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( some @ B @ V2 ) ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( none @ B ) ) ) ) ) ).
% graph_map_upd
thf(fact_6827_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),X3: A,Y: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X3 @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_6828_restrict__out,axiom,
! [A: $tType,B: $tType,X3: A,A3: set @ A,M: A > ( option @ B )] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ( restrict_map @ A @ B @ M @ A3 @ X3 )
= ( none @ B ) ) ) ).
% restrict_out
thf(fact_6829_restrict__map__empty,axiom,
! [B: $tType,A: $tType,D6: set @ A] :
( ( restrict_map @ A @ B
@ ^ [X: A] : ( none @ B )
@ D6 )
= ( ^ [X: A] : ( none @ B ) ) ) ).
% restrict_map_empty
thf(fact_6830_restrict__map__to__empty,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M @ ( bot_bot @ ( set @ A ) ) )
= ( ^ [X: A] : ( none @ B ) ) ) ).
% restrict_map_to_empty
thf(fact_6831_graph__empty,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B
@ ^ [X: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% graph_empty
thf(fact_6832_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X3: A,D6: set @ A,M: A > ( option @ B ),Y: option @ B] :
( ( member @ A @ X3 @ D6 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D6 ) @ X3 @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X3 @ Y ) ) ) ).
% fun_upd_restrict_conv
thf(fact_6833_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,X3: A,D6: set @ A,M: A > ( option @ B ),Y: option @ B] :
( ( ( member @ A @ X3 @ D6 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X3 @ Y ) @ D6 )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X3 @ Y ) ) )
& ( ~ ( member @ A @ X3 @ D6 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X3 @ Y ) @ D6 )
= ( restrict_map @ A @ B @ M @ D6 ) ) ) ) ).
% restrict_fun_upd
thf(fact_6834_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X3: A,D6: set @ A,M: A > ( option @ B )] :
( ( ( member @ A @ X3 @ D6 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D6 ) @ X3 @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
& ( ~ ( member @ A @ X3 @ D6 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D6 ) @ X3 @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ D6 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_6835_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V2: B,M: A > ( option @ B ),A3: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M @ A3 ) ) )
=> ( member @ A @ K @ A3 ) ) ).
% graph_restrictD(1)
thf(fact_6836_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K: A,V2: B,M: A > ( option @ B ),A3: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M @ A3 ) ) )
=> ( ( M @ K )
= ( some @ B @ V2 ) ) ) ).
% graph_restrictD(2)
thf(fact_6837_in__graphD,axiom,
! [A: $tType,B: $tType,K: A,V2: B,M: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ M ) )
=> ( ( M @ K )
= ( some @ B @ V2 ) ) ) ).
% in_graphD
thf(fact_6838_in__graphI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),K: B,V2: A] :
( ( ( M @ K )
= ( some @ A @ V2 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V2 ) @ ( graph @ B @ A @ M ) ) ) ).
% in_graphI
thf(fact_6839_restrict__map__def,axiom,
! [B: $tType,A: $tType] :
( ( restrict_map @ A @ B )
= ( ^ [M2: A > ( option @ B ),A6: set @ A,X: A] : ( if @ ( option @ B ) @ ( member @ A @ X @ A6 ) @ ( M2 @ X ) @ ( none @ B ) ) ) ) ).
% restrict_map_def
thf(fact_6840_graph__def,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M2: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A4: A,B4: B] :
( ( Uu3
= ( product_Pair @ A @ B @ A4 @ B4 ) )
& ( ( M2 @ A4 )
= ( some @ B @ B4 ) ) ) ) ) ) ).
% graph_def
thf(fact_6841_graph__fun__upd__None,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K: A] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( none @ B ) ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [E4: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ E4 @ ( graph @ A @ B @ M ) )
& ( ( product_fst @ A @ B @ E4 )
!= K ) ) ) ) ).
% graph_fun_upd_None
thf(fact_6842_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),D6: set @ A,X3: A,Y: option @ B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D6 ) @ X3 @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X3 @ Y ) ) ).
% fun_upd_restrict
thf(fact_6843_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B ),X3: A] :
( ( restrict_map @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( fun_upd @ A @ ( option @ B ) @ F2 @ X3 @ ( none @ B ) ) ) ).
% restrict_complement_singleton_eq
thf(fact_6844_restrict__map__upds,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,D6: set @ A,M: 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 @ M @ Xs @ Ys ) @ D6 )
= ( map_upds @ A @ B @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D6 @ ( set2 @ A @ Xs ) ) ) @ Xs @ Ys ) ) ) ) ).
% restrict_map_upds
thf(fact_6845_ran__map__upd,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A2: B,B2: A] :
( ( ( M @ A2 )
= ( none @ A ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M @ A2 @ ( some @ A @ B2 ) ) )
= ( insert @ A @ B2 @ ( ran @ B @ A @ M ) ) ) ) ).
% ran_map_upd
thf(fact_6846_map__upds__apply__nontin,axiom,
! [B: $tType,A: $tType,X3: A,Xs: list @ A,F2: A > ( option @ B ),Ys: list @ B] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( map_upds @ A @ B @ F2 @ Xs @ Ys @ X3 )
= ( F2 @ X3 ) ) ) ).
% map_upds_apply_nontin
thf(fact_6847_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_6848_map__upds__list__update2__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,M: A > ( option @ B ),Ys: list @ B,Y: B] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( map_upds @ A @ B @ M @ Xs @ ( list_update @ B @ Ys @ I @ Y ) )
= ( map_upds @ A @ B @ M @ Xs @ Ys ) ) ) ).
% map_upds_list_update2_drop
thf(fact_6849_map__upds__twist,axiom,
! [A: $tType,B: $tType,A2: A,As: list @ A,M: A > ( option @ B ),B2: B,Bs: list @ B] :
( ~ ( member @ A @ A2 @ ( set2 @ A @ As ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ A2 @ ( some @ B @ B2 ) ) @ As @ Bs )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ As @ Bs ) @ A2 @ ( some @ B @ B2 ) ) ) ) ).
% map_upds_twist
thf(fact_6850_ranI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A2: B,B2: A] :
( ( ( M @ A2 )
= ( some @ A @ B2 ) )
=> ( member @ A @ B2 @ ( ran @ B @ A @ M ) ) ) ).
% ranI
thf(fact_6851_ran__restrictD,axiom,
! [B: $tType,A: $tType,Y: A,M: B > ( option @ A ),A3: set @ B] :
( ( member @ A @ Y @ ( ran @ B @ A @ ( restrict_map @ B @ A @ M @ A3 ) ) )
=> ? [X5: B] :
( ( member @ B @ X5 @ A3 )
& ( ( M @ X5 )
= ( some @ A @ Y ) ) ) ) ).
% ran_restrictD
thf(fact_6852_ran__def,axiom,
! [B: $tType,A: $tType] :
( ( ran @ A @ B )
= ( ^ [M2: A > ( option @ B )] :
( collect @ B
@ ^ [B4: B] :
? [A4: A] :
( ( M2 @ A4 )
= ( some @ B @ B4 ) ) ) ) ) ).
% ran_def
thf(fact_6853_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),X3: B,Y: A,Z2: A] :
( ( ( M @ X3 )
= ( some @ A @ Y ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M @ ( dom @ B @ A @ M ) )
=> ( ~ ( member @ A @ Z2 @ ( ran @ B @ A @ M ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M @ X3 @ ( some @ A @ Z2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_6854_map__upds__fold__map__upd,axiom,
! [B: $tType,A: $tType] :
( ( map_upds @ A @ B )
= ( ^ [M2: A > ( option @ B ),Ks: list @ A,Vs: list @ B] :
( foldl @ ( A > ( option @ B ) ) @ ( product_prod @ A @ B )
@ ^ [N: A > ( option @ B )] :
( product_case_prod @ A @ B @ ( A > ( option @ B ) )
@ ^ [K3: A,V5: B] : ( fun_upd @ A @ ( option @ B ) @ N @ K3 @ ( some @ B @ V5 ) ) )
@ M2
@ ( zip @ A @ B @ Ks @ Vs ) ) ) ) ).
% map_upds_fold_map_upd
thf(fact_6855_dom__eq__empty__conv,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B )] :
( ( ( dom @ A @ B @ F2 )
= ( bot_bot @ ( set @ A ) ) )
= ( F2
= ( ^ [X: A] : ( none @ B ) ) ) ) ).
% dom_eq_empty_conv
thf(fact_6856_fun__upd__None__if__notin__dom,axiom,
! [B: $tType,A: $tType,K: A,M: A > ( option @ B )] :
( ~ ( member @ A @ K @ ( dom @ A @ B @ M ) )
=> ( ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( none @ B ) )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_6857_dom__const,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( dom @ A @ B
@ ^ [X: A] : ( some @ B @ ( F2 @ X ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% dom_const
thf(fact_6858_dom__empty,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B
@ ^ [X: A] : ( none @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% dom_empty
thf(fact_6859_finite__graph__iff__finite__dom,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( graph @ A @ B @ M ) )
= ( finite_finite2 @ A @ ( dom @ A @ B @ M ) ) ) ).
% finite_graph_iff_finite_dom
thf(fact_6860_dom__fun__upd,axiom,
! [B: $tType,A: $tType,Y: option @ B,F2: A > ( option @ B ),X3: A] :
( ( ( Y
= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X3 @ Y ) )
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F2 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( ( Y
!= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X3 @ Y ) )
= ( insert @ A @ X3 @ ( dom @ A @ B @ F2 ) ) ) ) ) ).
% dom_fun_upd
thf(fact_6861_finite__ran,axiom,
! [B: $tType,A: $tType,P4: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ P4 ) )
=> ( finite_finite2 @ B @ ( ran @ A @ B @ P4 ) ) ) ).
% finite_ran
thf(fact_6862_insert__dom,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),X3: B,Y: A] :
( ( ( F2 @ X3 )
= ( some @ A @ Y ) )
=> ( ( insert @ B @ X3 @ ( dom @ B @ A @ F2 ) )
= ( dom @ B @ A @ F2 ) ) ) ).
% insert_dom
thf(fact_6863_domD,axiom,
! [A: $tType,B: $tType,A2: A,M: A > ( option @ B )] :
( ( member @ A @ A2 @ ( dom @ A @ B @ M ) )
=> ? [B5: B] :
( ( M @ A2 )
= ( some @ B @ B5 ) ) ) ).
% domD
thf(fact_6864_domI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A2: B,B2: A] :
( ( ( M @ A2 )
= ( some @ A @ B2 ) )
=> ( member @ B @ A2 @ ( dom @ B @ A @ M ) ) ) ).
% domI
thf(fact_6865_domIff,axiom,
! [A: $tType,B: $tType,A2: A,M: A > ( option @ B )] :
( ( member @ A @ A2 @ ( dom @ A @ B @ M ) )
= ( ( M @ A2 )
!= ( none @ B ) ) ) ).
% domIff
thf(fact_6866_dom__def,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B )
= ( ^ [M2: A > ( option @ B )] :
( collect @ A
@ ^ [A4: A] :
( ( M2 @ A4 )
!= ( none @ B ) ) ) ) ) ).
% dom_def
thf(fact_6867_foldl__cong,axiom,
! [A: $tType,B: $tType,A2: A,B2: A,L: list @ B,K: list @ B,F2: A > B > A,G: A > B > A] :
( ( A2 = B2 )
=> ( ( L = K )
=> ( ! [A5: A,X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ L ) )
=> ( ( F2 @ A5 @ X5 )
= ( G @ A5 @ X5 ) ) )
=> ( ( foldl @ A @ B @ F2 @ A2 @ L )
= ( foldl @ A @ B @ G @ B2 @ K ) ) ) ) ) ).
% foldl_cong
thf(fact_6868_foldl__Nil,axiom,
! [A: $tType,B: $tType,F2: B > A > B,A2: B] :
( ( foldl @ B @ A @ F2 @ A2 @ ( nil @ A ) )
= A2 ) ).
% foldl_Nil
thf(fact_6869_finite__map__freshness,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ F2 ) )
=> ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ? [X5: A] :
( ( F2 @ X5 )
= ( none @ B ) ) ) ) ).
% finite_map_freshness
thf(fact_6870_dom__minus,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),X3: B,A3: set @ B] :
( ( ( F2 @ X3 )
= ( none @ A ) )
=> ( ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F2 ) @ ( insert @ B @ X3 @ A3 ) )
= ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F2 ) @ A3 ) ) ) ).
% dom_minus
thf(fact_6871_finite__set__of__finite__maps,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( finite_finite2 @ ( A > ( option @ B ) )
@ ( collect @ ( A > ( option @ B ) )
@ ^ [M2: A > ( option @ B )] :
( ( ( dom @ A @ B @ M2 )
= A3 )
& ( ord_less_eq @ ( set @ B ) @ ( ran @ A @ B @ M2 ) @ B3 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_6872_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M2: A > ( option @ B )] :
( image @ A @ ( product_prod @ A @ B )
@ ^ [X: A] : ( product_Pair @ A @ B @ X @ ( the2 @ B @ ( M2 @ X ) ) )
@ ( dom @ A @ B @ M2 ) ) ) ) ).
% graph_eq_to_snd_dom
thf(fact_6873_finite__Map__induct,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),P: ( A > ( option @ B ) ) > $o] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M ) )
=> ( ( P
@ ^ [X: A] : ( none @ B ) )
=> ( ! [K2: A,V3: B,M3: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M3 ) )
=> ( ~ ( member @ A @ K2 @ ( dom @ A @ B @ M3 ) )
=> ( ( P @ M3 )
=> ( P @ ( fun_upd @ A @ ( option @ B ) @ M3 @ K2 @ ( some @ B @ V3 ) ) ) ) ) )
=> ( P @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_6874_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B ),X3: A] :
( ( ( dom @ A @ B @ F2 )
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ? [V5: B] :
( F2
= ( fun_upd @ A @ ( option @ B )
@ ^ [X: A] : ( none @ B )
@ X3
@ ( some @ B @ V5 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_6875_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_6876_minus__fold__remove,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ B3 @ A3 )
= ( finite_fold @ A @ ( set @ A ) @ ( remove @ A ) @ B3 @ A3 ) ) ) ).
% minus_fold_remove
thf(fact_6877_member__remove,axiom,
! [A: $tType,X3: A,Y: A,A3: set @ A] :
( ( member @ A @ X3 @ ( remove @ A @ Y @ A3 ) )
= ( ( member @ A @ X3 @ A3 )
& ( X3 != Y ) ) ) ).
% member_remove
thf(fact_6878_remove__code_I1_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( remove @ A @ X3 @ ( set2 @ A @ Xs ) )
= ( set2 @ A @ ( removeAll @ A @ X3 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_6879_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X: A,A6: set @ A] : ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_6880_finite__subsets__at__top__finite,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite5375528669736107172at_top @ A @ A3 )
= ( principal @ ( set @ A ) @ ( insert @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ).
% finite_subsets_at_top_finite
thf(fact_6881_pos__deriv__imp__strict__mono,axiom,
! [F2: real > real,F11: real > real] :
( ! [X5: real] : ( has_field_derivative @ real @ F2 @ ( F11 @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
=> ( ! [X5: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F11 @ X5 ) )
=> ( order_strict_mono @ real @ real @ F2 ) ) ) ).
% pos_deriv_imp_strict_mono
thf(fact_6882_eventually__finite__subsets__at__top__weakI,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ! [X10: set @ A] :
( ( finite_finite2 @ A @ X10 )
=> ( ( ord_less_eq @ ( set @ A ) @ X10 @ A3 )
=> ( P @ X10 ) ) )
=> ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A3 ) ) ) ).
% eventually_finite_subsets_at_top_weakI
thf(fact_6883_eventually__finite__subsets__at__top__finite,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A3 ) )
= ( P @ A3 ) ) ) ).
% eventually_finite_subsets_at_top_finite
thf(fact_6884_strict__mono__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( order_mono @ A @ B @ F2 ) ) ) ).
% strict_mono_mono
thf(fact_6885_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y ) )
= ( X3 = Y ) ) ) ) ).
% strict_mono_eq
thf(fact_6886_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) )
= ( ord_less @ A @ X3 @ Y ) ) ) ) ).
% strict_mono_less
thf(fact_6887_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F3: A > B] :
! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_6888_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X5: A,Y4: A] :
( ( ord_less @ A @ X5 @ Y4 )
=> ( ord_less @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( order_strict_mono @ A @ B @ F2 ) ) ) ).
% strict_monoI
thf(fact_6889_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ) ).
% strict_monoD
thf(fact_6890_strict__mono__add,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A] :
( order_strict_mono @ A @ A
@ ^ [N: A] : ( plus_plus @ A @ N @ K ) ) ) ).
% strict_mono_add
thf(fact_6891_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) )
= ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ).
% strict_mono_less_eq
thf(fact_6892_strict__mono__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [R2: A > B,M: A,N2: A] :
( ( order_strict_mono @ A @ B @ R2 )
=> ( ( ord_less_eq @ A @ M @ N2 )
=> ( ord_less_eq @ B @ ( R2 @ M ) @ ( R2 @ N2 ) ) ) ) ) ).
% strict_mono_leD
thf(fact_6893_finite__subsets__at__top__neq__bot,axiom,
! [A: $tType,A3: set @ A] :
( ( finite5375528669736107172at_top @ A @ A3 )
!= ( bot_bot @ ( filter @ ( set @ A ) ) ) ) ).
% finite_subsets_at_top_neq_bot
thf(fact_6894_eventually__finite__subsets__at__top,axiom,
! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] :
( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A3 ) )
= ( ? [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
& ( ord_less_eq @ ( set @ A ) @ X6 @ A3 )
& ! [Y9: set @ A] :
( ( ( finite_finite2 @ A @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ X6 @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ Y9 @ A3 ) )
=> ( P @ Y9 ) ) ) ) ) ).
% eventually_finite_subsets_at_top
thf(fact_6895_finite__subsets__at__top__def,axiom,
! [A: $tType] :
( ( finite5375528669736107172at_top @ A )
= ( ^ [A6: set @ A] :
( complete_Inf_Inf @ ( filter @ ( set @ A ) )
@ ( image @ ( set @ A ) @ ( filter @ ( set @ A ) )
@ ^ [X6: set @ A] :
( principal @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Y9: set @ A] :
( ( finite_finite2 @ A @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ X6 @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ Y9 @ A6 ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
& ( ord_less_eq @ ( set @ A ) @ X6 @ A6 ) ) ) ) ) ) ) ).
% finite_subsets_at_top_def
thf(fact_6896_filterlim__finite__subsets__at__top,axiom,
! [A: $tType,B: $tType,F2: A > ( set @ B ),A3: set @ B,F5: filter @ A] :
( ( filterlim @ A @ ( set @ B ) @ F2 @ ( finite5375528669736107172at_top @ B @ A3 ) @ F5 )
= ( ! [X6: set @ B] :
( ( ( finite_finite2 @ B @ X6 )
& ( ord_less_eq @ ( set @ B ) @ X6 @ A3 ) )
=> ( eventually @ A
@ ^ [Y2: A] :
( ( finite_finite2 @ B @ ( F2 @ Y2 ) )
& ( ord_less_eq @ ( set @ B ) @ X6 @ ( F2 @ Y2 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F2 @ Y2 ) @ A3 ) )
@ F5 ) ) ) ) ).
% filterlim_finite_subsets_at_top
thf(fact_6897_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
@ ^ [I3: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ) ).
% ring_1_class.of_int_def
thf(fact_6898_sorted__key__list__of__set__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord144544945434240204of_set @ B @ A )
= ( ^ [F3: B > A] : ( finite_folding_F @ B @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F3 ) @ ( nil @ B ) ) ) ) ) ).
% sorted_key_list_of_set_def
thf(fact_6899_of__nat__eq__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( id @ nat ) ) ).
% of_nat_eq_id
thf(fact_6900_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_6901_id__funpow,axiom,
! [A: $tType,N2: nat] :
( ( compow @ ( A > A ) @ N2 @ ( id @ A ) )
= ( id @ A ) ) ).
% id_funpow
thf(fact_6902_apfst__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apfst @ A @ A @ B @ ( id @ A ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apfst_id
thf(fact_6903_apsnd__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apsnd_id
thf(fact_6904_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_6905_drop__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A @ ( zero_zero @ nat ) )
= ( id @ A ) ) ) ).
% drop_bit_0
thf(fact_6906_comp__the__Some,axiom,
! [A: $tType] :
( ( comp @ ( option @ A ) @ A @ A @ ( the2 @ A ) @ ( some @ A ) )
= ( id @ A ) ) ).
% comp_the_Some
thf(fact_6907_infinite__enumerate,axiom,
! [S3: set @ nat] :
( ~ ( finite_finite2 @ nat @ S3 )
=> ? [R4: nat > nat] :
( ( order_strict_mono @ nat @ nat @ R4 )
& ! [N6: nat] : ( member @ nat @ ( R4 @ N6 ) @ S3 ) ) ) ).
% infinite_enumerate
thf(fact_6908_strict__mono__imp__increasing,axiom,
! [F2: nat > nat,N2: nat] :
( ( order_strict_mono @ nat @ nat @ F2 )
=> ( ord_less_eq @ nat @ N2 @ ( F2 @ N2 ) ) ) ).
% strict_mono_imp_increasing
thf(fact_6909_strict__mono__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_strict_mono @ nat @ A )
= ( ^ [F3: nat > A] :
! [N: nat] : ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ ( suc @ N ) ) ) ) ) ) ).
% strict_mono_Suc_iff
thf(fact_6910_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 )
@ ^ [X: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y2 ) ) ) ) ) ) ).
% less_int_def
thf(fact_6911_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_6912_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 )
@ ^ [X: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y2 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_6913_funpow__simps__right_I1_J,axiom,
! [A: $tType,F2: A > A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F2 )
= ( id @ A ) ) ).
% funpow_simps_right(1)
thf(fact_6914_fst__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% fst_diag_id
thf(fact_6915_snd__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% snd_diag_id
thf(fact_6916_summable__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G: nat > nat,F2: nat > A] :
( ( order_strict_mono @ nat @ nat @ G )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ ( image @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F2 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( summable @ A
@ ^ [N: nat] : ( F2 @ ( G @ N ) ) )
= ( summable @ A @ F2 ) ) ) ) ) ).
% summable_mono_reindex
thf(fact_6917_sums__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G: nat > nat,F2: nat > A,C2: A] :
( ( order_strict_mono @ nat @ nat @ G )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ ( image @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F2 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [N: nat] : ( F2 @ ( G @ N ) )
@ C2 )
= ( sums @ A @ F2 @ C2 ) ) ) ) ) ).
% sums_mono_reindex
thf(fact_6918_suminf__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G: nat > nat,F2: nat > A] :
( ( order_strict_mono @ nat @ nat @ G )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ ( image @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F2 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A
@ ^ [N: nat] : ( F2 @ ( G @ N ) ) )
= ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_mono_reindex
thf(fact_6919_increasing__Bseq__subseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > nat] :
( ! [X5: nat,Y4: nat] :
( ( ord_less_eq @ nat @ X5 @ Y4 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X5 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ Y4 ) ) ) )
=> ( ( order_strict_mono @ nat @ nat @ G )
=> ( ( bfun @ nat @ A
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ) ).
% increasing_Bseq_subseq_iff
thf(fact_6920_folding__on_Oinsert__remove,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite_folding_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) )
= ( F2 @ X3 @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% folding_on.insert_remove
thf(fact_6921_folding__on_Oremove,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,A3: set @ A,X3: A,Z2: B] :
( ( finite_folding_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ S3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ A3 )
= ( F2 @ X3 @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% folding_on.remove
thf(fact_6922_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_6923_folding__on_Oempty,axiom,
! [A: $tType,B: $tType,S3: set @ A,F2: A > B > B,Z2: B] :
( ( finite_folding_on @ A @ B @ S3 @ F2 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( bot_bot @ ( set @ A ) ) )
= Z2 ) ) ).
% folding_on.empty
thf(fact_6924_folding__on_Oinfinite,axiom,
! [A: $tType,B: $tType,S3: set @ A,F2: A > B > B,A3: set @ A,Z2: B] :
( ( finite_folding_on @ A @ B @ S3 @ F2 )
=> ( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ A3 )
= Z2 ) ) ) ).
% folding_on.infinite
thf(fact_6925_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
@ ^ [X: A] : X ) ) ) ).
% sorted_list_of_set.fold_insort_key.folding_on_axioms
thf(fact_6926_card__def,axiom,
! [B: $tType] :
( ( finite_card @ B )
= ( finite_folding_F @ B @ nat
@ ^ [Uu3: B] : suc
@ ( zero_zero @ nat ) ) ) ).
% card_def
thf(fact_6927_folding__on_Oinsert,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite_folding_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) )
= ( F2 @ X3 @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ A3 ) ) ) ) ) ) ) ).
% folding_on.insert
thf(fact_6928_folding__idem__on_Oinsert__idem,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite1890593828518410140dem_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) )
= ( F2 @ X3 @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ A3 ) ) ) ) ) ) ).
% folding_idem_on.insert_idem
thf(fact_6929_comp__fun__idem__on_Ofold__insert__idem,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) )
= ( F2 @ X3 @ ( finite_fold @ A @ B @ F2 @ Z2 @ A3 ) ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem
thf(fact_6930_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
! [B: $tType,A: $tType,C: $tType,S3: set @ A,F2: A > B > B,G: C > A,R: set @ C] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ G @ ( top_top @ ( set @ C ) ) ) @ S3 )
=> ( finite673082921795544331dem_on @ C @ B @ R @ ( comp @ A @ ( B > B ) @ C @ F2 @ G ) ) ) ) ).
% comp_fun_idem_on.comp_comp_fun_idem_on
thf(fact_6931_comp__fun__idem__on_Ofold__insert__idem2,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) )
= ( finite_fold @ A @ B @ F2 @ ( F2 @ X3 @ Z2 ) @ A3 ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem2
thf(fact_6932_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_6933_nth__image,axiom,
! [A: $tType,L: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( image @ nat @ A @ ( nth @ A @ Xs ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ L ) )
= ( set2 @ A @ ( take @ A @ L @ Xs ) ) ) ) ).
% nth_image
thf(fact_6934_take__take,axiom,
! [A: $tType,N2: nat,M: nat,Xs: list @ A] :
( ( take @ A @ N2 @ ( take @ A @ M @ Xs ) )
= ( take @ A @ ( ord_min @ nat @ N2 @ M ) @ Xs ) ) ).
% take_take
thf(fact_6935_take__eq__Nil2,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( nil @ A )
= ( take @ A @ N2 @ Xs ) )
= ( ( N2
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil2
thf(fact_6936_take__eq__Nil,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( take @ A @ N2 @ Xs )
= ( nil @ A ) )
= ( ( N2
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil
thf(fact_6937_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs2: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_6938_take__all__iff,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( take @ A @ N2 @ Xs )
= Xs )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ).
% take_all_iff
thf(fact_6939_take__all,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 )
=> ( ( take @ A @ N2 @ Xs )
= Xs ) ) ).
% take_all
thf(fact_6940_nth__take,axiom,
! [A: $tType,I: nat,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ N2 )
=> ( ( nth @ A @ ( take @ A @ N2 @ Xs ) @ I )
= ( nth @ A @ Xs @ I ) ) ) ).
% nth_take
thf(fact_6941_take__update__cancel,axiom,
! [A: $tType,N2: nat,M: nat,Xs: list @ A,Y: A] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( take @ A @ N2 @ ( list_update @ A @ Xs @ M @ Y ) )
= ( take @ A @ N2 @ Xs ) ) ) ).
% take_update_cancel
thf(fact_6942_length__take,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( take @ A @ N2 @ Xs ) )
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ).
% length_take
thf(fact_6943_nths__upt__eq__take,axiom,
! [A: $tType,L: list @ A,N2: nat] :
( ( nths @ A @ L @ ( set_ord_lessThan @ nat @ N2 ) )
= ( take @ A @ N2 @ L ) ) ).
% nths_upt_eq_take
thf(fact_6944_take__replicate,axiom,
! [A: $tType,I: nat,K: nat,X3: A] :
( ( take @ A @ I @ ( replicate @ A @ K @ X3 ) )
= ( replicate @ A @ ( ord_min @ nat @ I @ K ) @ X3 ) ) ).
% take_replicate
thf(fact_6945_dom__map__upds,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),Xs: list @ A,Ys: list @ B] :
( ( dom @ A @ B @ ( map_upds @ A @ B @ M @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) @ ( dom @ A @ B @ M ) ) ) ).
% dom_map_upds
thf(fact_6946_in__set__takeD,axiom,
! [A: $tType,X3: A,N2: nat,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( take @ A @ N2 @ Xs ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% in_set_takeD
thf(fact_6947_take__zip,axiom,
! [A: $tType,B: $tType,N2: nat,Xs: list @ A,Ys: list @ B] :
( ( take @ ( product_prod @ A @ B ) @ N2 @ ( zip @ A @ B @ Xs @ Ys ) )
= ( zip @ A @ B @ ( take @ A @ N2 @ Xs ) @ ( take @ B @ N2 @ Ys ) ) ) ).
% take_zip
thf(fact_6948_take__Nil,axiom,
! [A: $tType,N2: nat] :
( ( take @ A @ N2 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% take_Nil
thf(fact_6949_take__update__swap,axiom,
! [A: $tType,M: nat,Xs: list @ A,N2: nat,X3: A] :
( ( take @ A @ M @ ( list_update @ A @ Xs @ N2 @ X3 ) )
= ( list_update @ A @ ( take @ A @ M @ Xs ) @ N2 @ X3 ) ) ).
% take_update_swap
thf(fact_6950_take__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ! [I4: nat] :
( ( take @ A @ I4 @ Xs )
= ( take @ A @ I4 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_6951_distinct__take,axiom,
! [A: $tType,Xs: list @ A,I: nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( take @ A @ I @ Xs ) ) ) ).
% distinct_take
thf(fact_6952_take__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( take @ A @ ( zero_zero @ nat ) @ Xs )
= ( nil @ A ) ) ).
% take_0
thf(fact_6953_set__take__subset,axiom,
! [A: $tType,N2: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ N2 @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_take_subset
thf(fact_6954_set__take__subset__set__take,axiom,
! [A: $tType,M: nat,N2: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ M @ Xs ) ) @ ( set2 @ A @ ( take @ A @ N2 @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_6955_zip__obtain__same__length,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ ( product_prod @ A @ B ) ) > $o] :
( ! [Zs: list @ A,Ws: list @ B,N3: nat] :
( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ B ) @ Ws ) )
=> ( ( N3
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( Zs
= ( take @ A @ N3 @ Xs ) )
=> ( ( Ws
= ( take @ B @ N3 @ Ys ) )
=> ( P @ ( zip @ A @ B @ Zs @ Ws ) ) ) ) ) )
=> ( P @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_obtain_same_length
thf(fact_6956_nth__take__lemma,axiom,
! [A: $tType,K: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ( ( nth @ A @ Xs @ I4 )
= ( nth @ A @ Ys @ I4 ) ) )
=> ( ( take @ A @ K @ Xs )
= ( take @ A @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_6957_less__rat__def,axiom,
( ( ord_less @ rat )
= ( ^ [X: rat,Y2: rat] : ( positive @ ( minus_minus @ rat @ Y2 @ X ) ) ) ) ).
% less_rat_def
thf(fact_6958_map__upd__upds__conv__if,axiom,
! [A: $tType,B: $tType,X3: A,Ys: list @ B,Xs: list @ A,F2: A > ( option @ B ),Y: B] :
( ( ( member @ A @ X3 @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X3 @ ( some @ B @ Y ) ) @ Xs @ Ys )
= ( map_upds @ A @ B @ F2 @ Xs @ Ys ) ) )
& ( ~ ( member @ A @ X3 @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X3 @ ( some @ B @ Y ) ) @ Xs @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ F2 @ Xs @ Ys ) @ X3 @ ( some @ B @ Y ) ) ) ) ) ).
% map_upd_upds_conv_if
thf(fact_6959_Rat_Opositive_Orep__eq,axiom,
( positive
= ( ^ [X: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ ( rep_Rat @ X ) ) @ ( product_snd @ int @ int @ ( rep_Rat @ X ) ) ) ) ) ) ).
% Rat.positive.rep_eq
thf(fact_6960_lexord__take__index__conv,axiom,
! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X3 ) @ ( size_size @ ( list @ A ) @ Y ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X3 ) @ Y )
= X3 ) )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X3 ) @ ( size_size @ ( list @ A ) @ Y ) ) )
& ( ( take @ A @ I3 @ X3 )
= ( take @ A @ I3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X3 @ I3 ) @ ( nth @ A @ Y @ I3 ) ) @ R2 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_6961_lexord__linear,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: list @ A,Y: list @ A] :
( ! [A5: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ R2 )
| ( A5 = B5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A5 ) @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) )
| ( X3 = Y )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X3 ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_linear
thf(fact_6962_lexord__irreflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X5: A] :
~ ( 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 ) @ Xs @ Xs ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_irreflexive
thf(fact_6963_lexord__Nil__right,axiom,
! [A: $tType,X3: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ ( nil @ A ) ) @ ( lexord @ A @ R2 ) ) ).
% lexord_Nil_right
thf(fact_6964_lexord__partial__trans,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A ),Ys: list @ A,Zs2: list @ A] :
( ! [X5: A,Y4: A,Z3: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ 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 @ Zs2 ) @ ( lexord @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_6965_List_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [R5: A > A > $o,Xs2: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% List.lexordp_def
thf(fact_6966_Rat_Opositive__def,axiom,
( positive
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ $o @ $o @ rep_Rat @ ( id @ $o )
@ ^ [X: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% Rat.positive_def
thf(fact_6967_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 ) )
=> ~ ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( take @ A @ I4 @ Xs )
= ( take @ A @ I4 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Ys @ I4 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_6968_ntrancl__Suc,axiom,
! [A: $tType,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( suc @ N2 ) @ R )
= ( relcomp @ A @ A @ A @ ( transitive_ntrancl @ A @ N2 @ R ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ R ) ) ) ).
% ntrancl_Suc
thf(fact_6969_IdI,axiom,
! [A: $tType,A2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id2 @ A ) ) ).
% IdI
thf(fact_6970_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_6971_rtrancl__empty,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( id2 @ A ) ) ).
% rtrancl_empty
thf(fact_6972_Id__def,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [P6: product_prod @ A @ A] :
? [X: A] :
( P6
= ( product_Pair @ A @ A @ X @ X ) ) ) ) ).
% Id_def
thf(fact_6973_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_6974_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_6975_IdE,axiom,
! [A: $tType,P4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P4 @ ( id2 @ A ) )
=> ~ ! [X5: A] :
( P4
!= ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ).
% IdE
thf(fact_6976_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_6977_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_6978_reflcl__set__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( A > A > $o )
@ ^ [X: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 )
@ ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [X: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ).
% reflcl_set_eq
thf(fact_6979_lexord__lex,axiom,
! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lex @ A @ R2 ) )
= ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) )
& ( ( size_size @ ( list @ A ) @ X3 )
= ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% lexord_lex
thf(fact_6980_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys3: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys3 ) @ ( lex @ A @ R5 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_6981_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,I: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ ( nth @ A @ Xs @ I ) )
= ( some @ B @ ( nth @ B @ Ys @ I ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_6982_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_6983_map__of__eq__empty__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
( ( ( map_of @ A @ B @ Xys )
= ( ^ [X: A] : ( none @ B ) ) )
= ( Xys
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% map_of_eq_empty_iff
thf(fact_6984_empty__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
( ( ( ^ [X: A] : ( none @ B ) )
= ( map_of @ A @ B @ Xys ) )
= ( Xys
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% empty_eq_map_of_iff
thf(fact_6985_map__of__zip__is__None,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,X3: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ X3 )
= ( none @ B ) )
= ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_6986_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_6987_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_6988_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_6989_map__of__Cons__code_I1_J,axiom,
! [B: $tType,A: $tType,K: B] :
( ( map_of @ B @ A @ ( nil @ ( product_prod @ B @ A ) ) @ K )
= ( none @ A ) ) ).
% map_of_Cons_code(1)
thf(fact_6990_map__of_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( map_of @ A @ B @ ( nil @ ( product_prod @ A @ B ) ) )
= ( ^ [X: A] : ( none @ B ) ) ) ).
% map_of.simps(1)
thf(fact_6991_lenlex__irreflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X5: A] :
~ ( 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 ) @ Xs @ Xs ) @ ( lenlex @ A @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_6992_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_6993_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K: A,X3: B,L: list @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ X3 ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) )
=> ? [X5: B] :
( ( map_of @ A @ B @ L @ K )
= ( some @ B @ X5 ) ) ) ).
% weak_map_of_SomeI
thf(fact_6994_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Y: A] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ Y ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ Y ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% map_of_SomeD
thf(fact_6995_map__of__zip__inject,axiom,
! [B: $tType,A: $tType,Ys: list @ A,Xs: list @ B,Zs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys )
= ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs2 )
= ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( distinct @ B @ Xs )
=> ( ( ( map_of @ B @ A @ ( zip @ B @ A @ Xs @ Ys ) )
= ( map_of @ B @ A @ ( zip @ B @ A @ Xs @ Zs2 ) ) )
=> ( Ys = Zs2 ) ) ) ) ) ).
% map_of_zip_inject
thf(fact_6996_finite__range__map__of,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A )] : ( finite_finite2 @ ( option @ A ) @ ( image @ B @ ( option @ A ) @ ( map_of @ B @ A @ Xys ) @ ( top_top @ ( set @ B ) ) ) ) ).
% finite_range_map_of
thf(fact_6997_map__of__zip__is__Some,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X3: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
= ( ? [Y2: B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ X3 )
= ( some @ B @ Y2 ) ) ) ) ) ).
% map_of_zip_is_Some
thf(fact_6998_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_6999_map__of__eq__None__iff,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A ),X3: B] :
( ( ( map_of @ B @ A @ Xys @ X3 )
= ( none @ A ) )
= ( ~ ( member @ B @ X3 @ ( image @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ ( set2 @ ( product_prod @ B @ A ) @ Xys ) ) ) ) ) ).
% map_of_eq_None_iff
thf(fact_7000_map__of__zip__upd,axiom,
! [A: $tType,B: $tType,Ys: list @ B,Xs: list @ A,Zs2: list @ B,X3: A,Y: B,Z2: B] :
( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( size_size @ ( list @ B ) @ Zs2 )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) @ X3 @ ( some @ B @ Y ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs2 ) ) @ X3 @ ( some @ B @ Z2 ) ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) )
= ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs2 ) ) ) ) ) ) ) ).
% map_of_zip_upd
thf(fact_7001_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_7002_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ A @ B ),X3: 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 @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) )
=> ( ( map_of @ A @ B @ Xys @ X3 )
= ( some @ B @ Y ) ) ) ) ).
% map_of_is_SomeI
thf(fact_7003_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),Y: B,X3: A] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( some @ B @ Y )
= ( map_of @ A @ B @ Xys @ X3 ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% Some_eq_map_of_iff
thf(fact_7004_map__ident,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X: A] : X )
= ( ^ [Xs2: list @ A] : Xs2 ) ) ).
% map_ident
thf(fact_7005_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: list @ A] :
( ( ( map @ A @ B @ F2 @ A2 )
= ( nil @ B ) )
= ( A2
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_7006_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( ( nil @ A )
= ( map @ B @ A @ F2 @ Xs ) )
= ( Xs
= ( nil @ B ) ) ) ).
% Nil_is_map_conv
thf(fact_7007_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ B ) ) ) ).
% map_is_Nil_conv
thf(fact_7008_map__map,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > A,G: C > B,Xs: list @ C] :
( ( map @ B @ A @ F2 @ ( map @ C @ B @ G @ Xs ) )
= ( map @ C @ A @ ( comp @ B @ A @ C @ F2 @ G ) @ Xs ) ) ).
% map_map
thf(fact_7009_List_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F2: B > C,G: A > B,List: list @ A] :
( ( map @ B @ C @ F2 @ ( map @ A @ B @ G @ List ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_7010_list_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F2: A > B,V2: list @ A] :
( ( map @ B @ C @ G @ ( map @ A @ B @ F2 @ V2 ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ G @ F2 ) @ V2 ) ) ).
% list.map_comp
thf(fact_7011_map__eq__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,G: B > A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ B @ A @ G @ Xs ) )
= ( ! [X: B] :
( ( member @ B @ X @ ( set2 @ B @ Xs ) )
=> ( ( F2 @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_7012_length__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ).
% length_map
thf(fact_7013_list_Omap__id0,axiom,
! [A: $tType] :
( ( map @ A @ A @ ( id @ A ) )
= ( id @ ( list @ A ) ) ) ).
% list.map_id0
thf(fact_7014_map__replicate,axiom,
! [A: $tType,B: $tType,F2: B > A,N2: nat,X3: B] :
( ( map @ B @ A @ F2 @ ( replicate @ B @ N2 @ X3 ) )
= ( replicate @ A @ N2 @ ( F2 @ X3 ) ) ) ).
% map_replicate
thf(fact_7015_list_Oset__map,axiom,
! [B: $tType,A: $tType,F2: A > B,V2: list @ A] :
( ( set2 @ B @ ( map @ A @ B @ F2 @ V2 ) )
= ( image @ A @ B @ F2 @ ( set2 @ A @ V2 ) ) ) ).
% list.set_map
thf(fact_7016_map__snd__enumerate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( map @ ( product_prod @ nat @ A ) @ A @ ( product_snd @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) )
= Xs ) ).
% map_snd_enumerate
thf(fact_7017_inj__map__eq__map,axiom,
! [B: $tType,A: $tType,F2: A > B,Xs: list @ A,Ys: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map @ A @ B @ F2 @ Xs )
= ( map @ A @ B @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_7018_map__fun__upd,axiom,
! [B: $tType,A: $tType,Y: A,Xs: list @ A,F2: A > B,V2: B] :
( ~ ( member @ A @ Y @ ( set2 @ A @ Xs ) )
=> ( ( map @ A @ B @ ( fun_upd @ A @ B @ F2 @ Y @ V2 ) @ Xs )
= ( map @ A @ B @ F2 @ Xs ) ) ) ).
% map_fun_upd
thf(fact_7019_map__comp__map,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,G: A > C] :
( ( comp @ ( list @ C ) @ ( list @ B ) @ ( list @ A ) @ ( map @ C @ B @ F2 ) @ ( map @ A @ C @ G ) )
= ( map @ A @ B @ ( comp @ C @ B @ A @ F2 @ G ) ) ) ).
% map_comp_map
thf(fact_7020_List_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F2: B > C,G: A > B] :
( ( comp @ ( list @ B ) @ ( list @ C ) @ ( list @ A ) @ ( map @ B @ C @ F2 ) @ ( map @ A @ B @ G ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) ) ) ).
% List.map.comp
thf(fact_7021_size__list__map,axiom,
! [A: $tType,B: $tType,F2: A > nat,G: B > A,Xs: list @ B] :
( ( size_list @ A @ F2 @ ( map @ B @ A @ G @ Xs ) )
= ( size_list @ B @ ( comp @ A @ nat @ B @ F2 @ G ) @ Xs ) ) ).
% size_list_map
thf(fact_7022_nth__map,axiom,
! [B: $tType,A: $tType,N2: nat,Xs: list @ A,F2: A > B] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ B @ ( map @ A @ B @ F2 @ Xs ) @ N2 )
= ( F2 @ ( nth @ A @ Xs @ N2 ) ) ) ) ).
% nth_map
thf(fact_7023_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_7024_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_7025_inj__map,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
= ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_map
thf(fact_7026_inj__mapI,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ) ).
% inj_mapI
thf(fact_7027_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),X3: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( map_of @ A @ B @ Xys @ X3 )
= ( some @ B @ Y ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% map_of_eq_Some_iff
thf(fact_7028_List_Omap_Oidentity,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X: A] : X )
= ( id @ ( list @ A ) ) ) ).
% List.map.identity
thf(fact_7029_list_Omap__id,axiom,
! [A: $tType,T2: list @ A] :
( ( map @ A @ A @ ( id @ A ) @ T2 )
= T2 ) ).
% list.map_id
thf(fact_7030_list_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F2: B > nat,G: A > B] :
( ( comp @ ( list @ B ) @ nat @ ( list @ A ) @ ( size_list @ B @ F2 ) @ ( map @ A @ B @ G ) )
= ( size_list @ A @ ( comp @ B @ nat @ A @ F2 @ G ) ) ) ).
% list.size_gen_o_map
thf(fact_7031_map2__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,H2: B > C > A,F2: D > B,Xs: list @ D,G: D > C] :
( ( map @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ H2 ) @ ( zip @ B @ C @ ( map @ D @ B @ F2 @ Xs ) @ ( map @ D @ C @ G @ Xs ) ) )
= ( map @ D @ A
@ ^ [X: D] : ( H2 @ ( F2 @ X ) @ ( G @ X ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_7032_zip__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,F2: C > A,Xs: list @ C,G: D > B,Ys: list @ D] :
( ( zip @ A @ B @ ( map @ C @ A @ F2 @ Xs ) @ ( map @ D @ B @ G @ Ys ) )
= ( map @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X: C,Y2: D] : ( product_Pair @ A @ B @ ( F2 @ X ) @ ( G @ Y2 ) ) )
@ ( zip @ C @ D @ Xs @ Ys ) ) ) ).
% zip_map_map
thf(fact_7033_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,F2: C > B,Ys: list @ C] :
( ( zip @ A @ B @ Xs @ ( map @ C @ B @ F2 @ Ys ) )
= ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [X: A,Y2: C] : ( product_Pair @ A @ B @ X @ ( F2 @ Y2 ) ) )
@ ( zip @ A @ C @ Xs @ Ys ) ) ) ).
% zip_map2
thf(fact_7034_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F2: C > A,Xs: list @ C,Ys: list @ B] :
( ( zip @ A @ B @ ( map @ C @ A @ F2 @ Xs ) @ Ys )
= ( map @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ B @ ( product_prod @ A @ B )
@ ^ [X: C] : ( product_Pair @ A @ B @ ( F2 @ X ) ) )
@ ( zip @ C @ B @ Xs @ Ys ) ) ) ).
% zip_map1
thf(fact_7035_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: ( product_prod @ B @ C ) > A,Xs: list @ B,G: D > C,Ys: list @ D] :
( ( map @ ( product_prod @ B @ C ) @ A @ F2 @ ( zip @ B @ C @ Xs @ ( map @ D @ C @ G @ Ys ) ) )
= ( map @ ( product_prod @ B @ D ) @ A
@ ( product_case_prod @ B @ D @ A
@ ^ [X: B,Y2: D] : ( F2 @ ( product_Pair @ B @ C @ X @ ( G @ Y2 ) ) ) )
@ ( zip @ B @ D @ Xs @ Ys ) ) ) ).
% map_zip_map2
thf(fact_7036_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F2: ( product_prod @ B @ C ) > A,G: D > B,Xs: list @ D,Ys: list @ C] :
( ( map @ ( product_prod @ B @ C ) @ A @ F2 @ ( zip @ B @ C @ ( map @ D @ B @ G @ Xs ) @ Ys ) )
= ( map @ ( product_prod @ D @ C ) @ A
@ ( product_case_prod @ D @ C @ A
@ ^ [X: D,Y2: C] : ( F2 @ ( product_Pair @ B @ C @ ( G @ X ) @ Y2 ) ) )
@ ( zip @ D @ C @ Xs @ Ys ) ) ) ).
% map_zip_map
thf(fact_7037_map__injective,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ B] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_7038_remdups__map__remdups,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( remdups @ A @ ( map @ B @ A @ F2 @ ( remdups @ B @ Xs ) ) )
= ( remdups @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_7039_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,Xs: list @ B,G: C > A,Ys: list @ C] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ C @ A @ G @ Ys ) )
=> ( ( size_size @ ( list @ B ) @ Xs )
= ( size_size @ ( list @ C ) @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_7040_rotate1__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( rotate1 @ A @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( rotate1 @ B @ Xs ) ) ) ).
% rotate1_map
thf(fact_7041_list_Omap__ident,axiom,
! [A: $tType,T2: list @ A] :
( ( map @ A @ A
@ ^ [X: A] : X
@ T2 )
= T2 ) ).
% list.map_ident
thf(fact_7042_map__concat,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ ( list @ B )] :
( ( map @ B @ A @ F2 @ ( concat @ B @ Xs ) )
= ( concat @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ Xs ) ) ) ).
% map_concat
thf(fact_7043_map__update,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,K: nat,Y: B] :
( ( map @ B @ A @ F2 @ ( list_update @ B @ Xs @ K @ Y ) )
= ( list_update @ A @ ( map @ B @ A @ F2 @ Xs ) @ K @ ( F2 @ Y ) ) ) ).
% map_update
thf(fact_7044_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( map @ A @ B @ F2 @ ( nil @ A ) )
= ( nil @ B ) ) ).
% list.simps(8)
thf(fact_7045_list_Omap__cong,axiom,
! [B: $tType,A: $tType,X3: list @ A,Ya: list @ A,F2: A > B,G: A > B] :
( ( X3 = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ Ya ) )
=> ( ( F2 @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map @ A @ B @ F2 @ X3 )
= ( map @ A @ B @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_7046_list_Omap__cong0,axiom,
! [B: $tType,A: $tType,X3: list @ A,F2: A > B,G: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ X3 ) )
=> ( ( F2 @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map @ A @ B @ F2 @ X3 )
= ( map @ A @ B @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_7047_list_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X3: list @ A,Xa2: list @ A,F2: A > B,Fa: A > B] :
( ! [Z3: A,Za2: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ X3 ) )
=> ( ( member @ A @ Za2 @ ( set2 @ A @ Xa2 ) )
=> ( ( ( F2 @ Z3 )
= ( Fa @ Za2 ) )
=> ( Z3 = Za2 ) ) ) )
=> ( ( ( map @ A @ B @ F2 @ X3 )
= ( map @ A @ B @ Fa @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_7048_map__ext,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B,G: A > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( F2 @ X5 )
= ( G @ X5 ) ) )
=> ( ( map @ A @ B @ F2 @ Xs )
= ( map @ A @ B @ G @ Xs ) ) ) ).
% map_ext
thf(fact_7049_map__idI,axiom,
! [A: $tType,Xs: list @ A,F2: A > A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( F2 @ X5 )
= X5 ) )
=> ( ( map @ A @ A @ F2 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_7050_map__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F2: A > B,G: A > B] :
( ( Xs = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( ( F2 @ X5 )
= ( G @ X5 ) ) )
=> ( ( map @ A @ B @ F2 @ Xs )
= ( map @ A @ B @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_7051_ex__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ B,F2: A > B] :
( ( ? [Xs2: list @ A] :
( Ys
= ( map @ A @ B @ F2 @ Xs2 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ ( set2 @ B @ Ys ) )
=> ? [Y2: A] :
( X
= ( F2 @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_7052_nths__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,I6: set @ nat] :
( ( nths @ A @ ( map @ B @ A @ F2 @ Xs ) @ I6 )
= ( map @ B @ A @ F2 @ ( nths @ B @ Xs @ I6 ) ) ) ).
% nths_map
thf(fact_7053_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_7054_enumerate__Suc__eq,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( enumerate @ A @ ( suc @ N2 ) @ Xs )
= ( map @ ( product_prod @ nat @ A ) @ ( product_prod @ nat @ A ) @ ( product_apfst @ nat @ nat @ A @ suc ) @ ( enumerate @ A @ N2 @ Xs ) ) ) ).
% enumerate_Suc_eq
thf(fact_7055_map__replicate__const,axiom,
! [B: $tType,A: $tType,K: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X: B] : K
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_7056_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,G: A > B > A,A2: A,F2: C > B,Xs: list @ C] :
( ( foldl @ A @ B @ G @ A2 @ ( map @ C @ B @ F2 @ Xs ) )
= ( foldl @ A @ C
@ ^ [A4: A,X: C] : ( G @ A4 @ ( F2 @ X ) )
@ A2
@ Xs ) ) ).
% foldl_map
thf(fact_7057_image__set,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( image @ B @ A @ F2 @ ( set2 @ B @ Xs ) )
= ( set2 @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ).
% image_set
thf(fact_7058_take__map,axiom,
! [A: $tType,B: $tType,N2: nat,F2: B > A,Xs: list @ B] :
( ( take @ A @ N2 @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( take @ B @ N2 @ Xs ) ) ) ).
% take_map
thf(fact_7059_inj__on__map__eq__map,axiom,
! [B: $tType,A: $tType,F2: A > B,Xs: list @ A,Ys: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) )
=> ( ( ( map @ A @ B @ F2 @ Xs )
= ( map @ A @ B @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_7060_map__inj__on,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ B] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
=> ( ( inj_on @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_7061_distinct__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) )
= ( ( distinct @ B @ Xs )
& ( inj_on @ B @ A @ F2 @ ( set2 @ B @ Xs ) ) ) ) ).
% distinct_map
thf(fact_7062_remdups__adj__map__injective,axiom,
! [B: $tType,A: $tType,F2: A > B,Xs: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( remdups_adj @ B @ ( map @ A @ B @ F2 @ Xs ) )
= ( map @ A @ B @ F2 @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% remdups_adj_map_injective
thf(fact_7063_zip__map__fst__snd,axiom,
! [B: $tType,A: $tType,Zs2: list @ ( product_prod @ A @ B )] :
( ( zip @ A @ B @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs2 ) @ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs2 ) )
= Zs2 ) ).
% zip_map_fst_snd
thf(fact_7064_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 ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ( map_of @ A @ B @ Xs @ X5 )
= ( map_of @ A @ B @ Ys @ X5 ) ) )
=> ( ( map_of @ A @ B @ Xs )
= ( map_of @ A @ B @ Ys ) ) ) ) ).
% map_of_eqI
thf(fact_7065_map__removeAll__inj,axiom,
! [B: $tType,A: $tType,F2: A > B,X3: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( map @ A @ B @ F2 @ ( removeAll @ A @ X3 @ Xs ) )
= ( removeAll @ B @ ( F2 @ X3 ) @ ( map @ A @ B @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_7066_inj__mapD,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_mapD
thf(fact_7067_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs2 ) )
= ( 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 ) ) )
@ ^ [X: A,Y2: B,Z6: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C @ Y2 @ Z6 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys ) @ Zs2 ) ) ) ).
% zip_assoc
thf(fact_7068_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs2 ) )
= ( 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 ) )
@ ^ [Y2: B] :
( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [X: A,Z6: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C @ Y2 @ Z6 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys @ ( zip @ A @ C @ Xs @ Zs2 ) ) ) ) ).
% zip_left_commute
thf(fact_7069_zip__commute,axiom,
! [B: $tType,A: $tType] :
( ( zip @ A @ B )
= ( ^ [Xs2: list @ A,Ys3: list @ B] :
( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X: B,Y2: A] : ( product_Pair @ A @ B @ Y2 @ X ) )
@ ( zip @ B @ A @ Ys3 @ Xs2 ) ) ) ) ).
% zip_commute
thf(fact_7070_zip__eq__conv,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ ( product_prod @ A @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( zip @ A @ B @ Xs @ Ys )
= Zs2 )
= ( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs2 )
= Xs )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs2 )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_7071_distinct__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X3: B,Xs: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) )
= ( ~ ( member @ A @ ( F2 @ X3 ) @ ( image @ B @ A @ F2 @ ( set2 @ B @ Xs ) ) )
& ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ) ).
% distinct_insort_key
thf(fact_7072_map__removeAll__inj__on,axiom,
! [B: $tType,A: $tType,F2: A > B,X3: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert @ A @ X3 @ ( set2 @ A @ Xs ) ) )
=> ( ( map @ A @ B @ F2 @ ( removeAll @ A @ X3 @ Xs ) )
= ( removeAll @ B @ ( F2 @ X3 ) @ ( map @ A @ B @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_7073_eq__key__imp__eq__value,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),K: 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 @ K @ V1 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V22 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( V1 = V22 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_7074_inj__on__mapI,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ ( list @ A )] :
( ( inj_on @ A @ B @ F2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A3 ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ A3 ) ) ).
% inj_on_mapI
thf(fact_7075_map__of__zip__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ ( map @ A @ B @ F2 @ Xs ) ) )
= ( ^ [X: A] : ( if @ ( option @ B ) @ ( member @ A @ X @ ( set2 @ A @ Xs ) ) @ ( some @ B @ ( F2 @ X ) ) @ ( none @ B ) ) ) ) ).
% map_of_zip_map
thf(fact_7076_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_7077_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_7078_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,T2: list @ ( product_prod @ A @ C ),K: A,X3: C] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map_of @ A @ C @ T2 @ K )
= ( some @ C @ X3 ) )
=> ( ( 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 @ ( F2 @ K3 ) ) )
@ T2 )
@ ( F2 @ K ) )
= ( some @ C @ X3 ) ) ) ) ).
% map_of_mapk_SomeI
thf(fact_7079_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_7080_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_7081_dual__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( max @ A
@ ^ [X: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X ) )
= ( ord_min @ A ) ) ) ).
% dual_max
thf(fact_7082_list_Oinject,axiom,
! [A: $tType,X21: A,X222: list @ A,Y21: A,Y222: list @ A] :
( ( ( cons @ A @ X21 @ X222 )
= ( cons @ A @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_7083_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_7084_nth__Cons__Suc,axiom,
! [A: $tType,X3: A,Xs: list @ A,N2: nat] :
( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ ( suc @ N2 ) )
= ( nth @ A @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_7085_nth__Cons__0,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ ( zero_zero @ nat ) )
= X3 ) ).
% nth_Cons_0
thf(fact_7086_take__Suc__Cons,axiom,
! [A: $tType,N2: nat,X3: A,Xs: list @ A] :
( ( take @ A @ ( suc @ N2 ) @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( take @ A @ N2 @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_7087_nths__singleton,axiom,
! [A: $tType,A3: set @ nat,X3: A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( ( nths @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ A3 )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( ( nths @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ A3 )
= ( nil @ A ) ) ) ) ).
% nths_singleton
thf(fact_7088_Cons__listrel1__Cons,axiom,
! [A: $tType,X3: 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 @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X3 = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_7089_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F2: B > A,A2: A,X3: B,Xs: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ ( cons @ B @ X3 @ Xs ) )
= ( plus_plus @ A @ ( F2 @ X3 ) @ ( times_times @ A @ A2 @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ Xs ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_7090_lexord__cons__cons,axiom,
! [A: $tType,A2: A,X3: 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 @ X3 ) @ ( 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 ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_7091_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 ) )
= ( ? [A4: A,X: list @ A] :
( Y
= ( cons @ A @ A4 @ X ) ) ) ) ).
% lexord_Nil_left
thf(fact_7092_enumerate__simps_I2_J,axiom,
! [B: $tType,N2: nat,X3: B,Xs: list @ B] :
( ( enumerate @ B @ N2 @ ( cons @ B @ X3 @ Xs ) )
= ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N2 @ X3 ) @ ( enumerate @ B @ ( suc @ N2 ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_7093_map__upds__Cons,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),A2: A,As: list @ A,B2: B,Bs: list @ B] :
( ( map_upds @ A @ B @ M @ ( cons @ A @ A2 @ As ) @ ( cons @ B @ B2 @ Bs ) )
= ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ A2 @ ( some @ B @ B2 ) ) @ As @ Bs ) ) ).
% map_upds_Cons
thf(fact_7094_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_7095_nth__Cons__numeral,axiom,
! [A: $tType,X3: A,Xs: list @ A,V2: num] :
( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ ( numeral_numeral @ nat @ V2 ) )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_7096_take__Cons__numeral,axiom,
! [A: $tType,V2: num,X3: A,Xs: list @ A] :
( ( take @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_7097_Cons__in__lex,axiom,
! [A: $tType,X3: 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 @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( lex @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) )
| ( ( X3 = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_7098_concat__map__singleton,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( concat @ A
@ ( map @ B @ ( list @ A )
@ ^ [X: B] : ( cons @ A @ ( F2 @ X ) @ ( nil @ A ) )
@ Xs ) )
= ( map @ B @ A @ F2 @ Xs ) ) ).
% concat_map_singleton
thf(fact_7099_nth__Cons__pos,axiom,
! [A: $tType,N2: nat,X3: A,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_7100_zip__same__conv__map,axiom,
! [A: $tType,Xs: list @ A] :
( ( zip @ A @ A @ Xs @ Xs )
= ( map @ A @ ( product_prod @ A @ A )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ Xs ) ) ).
% zip_same_conv_map
thf(fact_7101_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs2: list @ A,Ys3: list @ B] :
( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys3 )
@ Xs2 ) ) ) ) ).
% product_concat_map
thf(fact_7102_n__lists_Osimps_I2_J,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( n_lists @ A @ ( suc @ N2 ) @ Xs )
= ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ^ [Ys3: list @ A] :
( map @ A @ ( list @ A )
@ ^ [Y2: A] : ( cons @ A @ Y2 @ Ys3 )
@ Xs )
@ ( n_lists @ A @ N2 @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_7103_map__eq__Cons__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( ? [Z6: B,Zs3: list @ B] :
( ( Xs
= ( cons @ B @ Z6 @ Zs3 ) )
& ( ( F2 @ Z6 )
= Y )
& ( ( map @ B @ A @ F2 @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_7104_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ A,F2: B > A,Ys: list @ B] :
( ( ( cons @ A @ X3 @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
= ( ? [Z6: B,Zs3: list @ B] :
( ( Ys
= ( cons @ B @ Z6 @ Zs3 ) )
& ( X3
= ( F2 @ Z6 ) )
& ( Xs
= ( map @ B @ A @ F2 @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_7105_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F2: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( cons @ A @ Y @ Ys ) )
=> ? [Z3: B,Zs: list @ B] :
( ( Xs
= ( cons @ B @ Z3 @ Zs ) )
& ( ( F2 @ Z3 )
= Y )
& ( ( map @ B @ A @ F2 @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_7106_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ A,F2: B > A,Ys: list @ B] :
( ( ( cons @ A @ X3 @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
=> ? [Z3: B,Zs: list @ B] :
( ( Ys
= ( cons @ B @ Z3 @ Zs ) )
& ( X3
= ( F2 @ Z3 ) )
& ( Xs
= ( map @ B @ A @ F2 @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_7107_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F2: A > B,X21: A,X222: list @ A] :
( ( map @ A @ B @ F2 @ ( cons @ A @ X21 @ X222 ) )
= ( cons @ B @ ( F2 @ X21 ) @ ( map @ A @ B @ F2 @ X222 ) ) ) ).
% list.simps(9)
thf(fact_7108_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 ) )
@ ^ [X: A] : ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( product_lists @ A @ Xss ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_7109_List_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( bind @ A @ B )
= ( ^ [Xs2: list @ A,F3: A > ( list @ B )] : ( concat @ B @ ( map @ A @ ( list @ B ) @ F3 @ Xs2 ) ) ) ) ).
% List.bind_def
thf(fact_7110_take__Cons,axiom,
! [A: $tType,N2: nat,X3: A,Xs: list @ A] :
( ( take @ A @ N2 @ ( cons @ A @ X3 @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( nil @ A )
@ ^ [M2: nat] : ( cons @ A @ X3 @ ( take @ A @ M2 @ Xs ) )
@ N2 ) ) ).
% take_Cons
thf(fact_7111_shuffles_Osimps_I3_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( shuffles @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X3 @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_7112_Cons__shuffles__subset1,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ ( cons @ A @ X3 @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_7113_Cons__shuffles__subset2,axiom,
! [A: $tType,Y: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ).
% Cons_shuffles_subset2
thf(fact_7114_nth__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A,N2: nat] :
( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N2 )
= ( case_nat @ A @ X3 @ ( nth @ A @ Xs ) @ N2 ) ) ).
% nth_Cons
thf(fact_7115_foldl__Cons,axiom,
! [B: $tType,A: $tType,F2: B > A > B,A2: B,X3: A,Xs: list @ A] :
( ( foldl @ B @ A @ F2 @ A2 @ ( cons @ A @ X3 @ Xs ) )
= ( foldl @ B @ A @ F2 @ ( F2 @ A2 @ X3 ) @ Xs ) ) ).
% foldl_Cons
thf(fact_7116_gen__length__code_I2_J,axiom,
! [B: $tType,N2: nat,X3: B,Xs: list @ B] :
( ( gen_length @ B @ N2 @ ( cons @ B @ X3 @ Xs ) )
= ( gen_length @ B @ ( suc @ N2 ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_7117_arg__min__list_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,X3: A] :
( ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= X3 ) ) ).
% arg_min_list.simps(1)
thf(fact_7118_find_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ( ( P @ X3 )
=> ( ( find @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( some @ A @ X3 ) ) )
& ( ~ ( P @ X3 )
=> ( ( find @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( find @ A @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_7119_shufflesE,axiom,
! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( ( Zs2 = Xs )
=> ( Ys
!= ( nil @ A ) ) )
=> ( ( ( Zs2 = Ys )
=> ( Xs
!= ( nil @ A ) ) )
=> ( ! [X5: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X5 @ Xs4 ) )
=> ! [Z3: A,Zs4: list @ A] :
( ( Zs2
= ( cons @ A @ Z3 @ Zs4 ) )
=> ( ( X5 = Z3 )
=> ~ ( member @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs4 @ Ys ) ) ) ) )
=> ~ ! [Y4: A,Ys5: list @ A] :
( ( Ys
= ( cons @ A @ Y4 @ Ys5 ) )
=> ! [Z3: A,Zs4: list @ A] :
( ( Zs2
= ( cons @ A @ Z3 @ Zs4 ) )
=> ( ( Y4 = Z3 )
=> ~ ( member @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs @ Ys5 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_7120_insort__key_Osimps_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X3: B] :
( ( linorder_insort_key @ B @ A @ F2 @ X3 @ ( nil @ B ) )
= ( cons @ B @ X3 @ ( nil @ B ) ) ) ) ).
% insort_key.simps(1)
thf(fact_7121_list__update__code_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: A] :
( ( list_update @ A @ ( cons @ A @ X3 @ Xs ) @ ( zero_zero @ nat ) @ Y )
= ( cons @ A @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_7122_distinct__singleton,axiom,
! [A: $tType,X3: A] : ( distinct @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% distinct_singleton
thf(fact_7123_distinct_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ X3 @ Xs ) )
= ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_7124_length__Suc__conv,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N2 ) )
= ( ? [Y2: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y2 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_7125_Suc__length__conv,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( suc @ N2 )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [Y2: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y2 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_7126_length__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_Cons
thf(fact_7127_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ C,Ws2: list @ D,P: ( 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 ) @ Zs2 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs2 )
= ( size_size @ ( list @ D ) @ Ws2 ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
=> ( ! [X5: A,Xs3: list @ A,Y4: B,Ys4: list @ B,Z3: C,Zs: list @ C,W: D,Ws: list @ D] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs )
= ( size_size @ ( list @ D ) @ Ws ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs @ Ws )
=> ( P @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) @ ( cons @ C @ Z3 @ Zs ) @ ( cons @ D @ W @ Ws ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs2 @ Ws2 ) ) ) ) ) ) ).
% list_induct4
thf(fact_7128_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ C,P: ( 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 ) @ Zs2 ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X5: A,Xs3: list @ A,Y4: B,Ys4: list @ B,Z3: C,Zs: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs )
=> ( P @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) @ ( cons @ C @ Z3 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_7129_list__induct2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X5: A,Xs3: list @ A,Y4: B,Ys4: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_7130_listrelp_OCons,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X3: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( R2 @ X3 @ Y )
=> ( ( listrelp @ A @ B @ R2 @ Xs @ Ys )
=> ( listrelp @ A @ B @ R2 @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_7131_list__update_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,I: nat,V2: A] :
( ( list_update @ A @ ( cons @ A @ X3 @ Xs ) @ I @ V2 )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ V2 @ Xs )
@ ^ [J3: nat] : ( cons @ A @ X3 @ ( list_update @ A @ Xs @ J3 @ V2 ) )
@ I ) ) ).
% list_update.simps(2)
thf(fact_7132_set__ConsD,axiom,
! [A: $tType,Y: A,X3: A,Xs: list @ A] :
( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_7133_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_7134_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_7135_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_7136_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X5: A] : ( P @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [X5: A,Xs3: list @ A] :
( ( Xs3
!= ( nil @ A ) )
=> ( ( P @ Xs3 )
=> ( P @ ( cons @ A @ X5 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_7137_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X5: A,Xs3: list @ A] : ( P @ ( cons @ A @ X5 @ Xs3 ) @ ( nil @ B ) )
=> ( ! [Y4: B,Ys4: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y4 @ Ys4 ) )
=> ( ! [X5: A,Xs3: list @ A,Y4: B,Ys4: list @ B] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_7138_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y2: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_7139_remdups__adj_Ocases,axiom,
! [A: $tType,X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ( ! [X5: A] :
( X3
!= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ~ ! [X5: A,Y4: A,Xs3: list @ A] :
( X3
!= ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_7140_transpose_Ocases,axiom,
! [A: $tType,X3: list @ ( list @ A )] :
( ( X3
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ~ ! [X5: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_7141_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: list @ A] :
( ! [X5: A,Xs3: list @ A] :
( X3
!= ( cons @ A @ X5 @ Xs3 ) )
=> ( X3
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_7142_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_7143_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_7144_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_7145_removeAll_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Y: A,Xs: list @ A] :
( ( ( X3 = Y )
=> ( ( removeAll @ A @ X3 @ ( cons @ A @ Y @ Xs ) )
= ( removeAll @ A @ X3 @ Xs ) ) )
& ( ( X3 != Y )
=> ( ( removeAll @ A @ X3 @ ( cons @ A @ Y @ Xs ) )
= ( cons @ A @ Y @ ( removeAll @ A @ X3 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_7146_remove1_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Y: A,Xs: list @ A] :
( ( ( X3 = Y )
=> ( ( remove1 @ A @ X3 @ ( cons @ A @ Y @ Xs ) )
= Xs ) )
& ( ( X3 != Y )
=> ( ( remove1 @ A @ X3 @ ( cons @ A @ Y @ Xs ) )
= ( cons @ A @ Y @ ( remove1 @ A @ X3 @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_7147_Cons__in__shuffles__rightI,axiom,
! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys: list @ A,Z2: A] :
( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ Z2 @ Zs2 ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Z2 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_7148_Cons__in__shuffles__leftI,axiom,
! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys: list @ A,Z2: A] :
( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ Z2 @ Zs2 ) @ ( shuffles @ A @ ( cons @ A @ Z2 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_7149_not__Cons__self2,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( cons @ A @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_7150_ord_Omax__def,axiom,
! [A: $tType] :
( ( max @ A )
= ( ^ [Less_eq: A > A > $o,A4: A,B4: A] : ( if @ A @ ( Less_eq @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% ord.max_def
thf(fact_7151_ord_Omax_Ocong,axiom,
! [A: $tType] :
( ( max @ A )
= ( max @ A ) ) ).
% ord.max.cong
thf(fact_7152_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_7153_list__update__code_I3_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,I: nat,Y: A] :
( ( list_update @ A @ ( cons @ A @ X3 @ Xs ) @ ( suc @ I ) @ Y )
= ( cons @ A @ X3 @ ( list_update @ A @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_7154_replicate__Suc,axiom,
! [A: $tType,N2: nat,X3: A] :
( ( replicate @ A @ ( suc @ N2 ) @ X3 )
= ( cons @ A @ X3 @ ( replicate @ A @ N2 @ X3 ) ) ) ).
% replicate_Suc
thf(fact_7155_remdups_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( remdups @ A @ ( cons @ A @ X3 @ Xs ) )
= ( remdups @ A @ Xs ) ) )
& ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( remdups @ A @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( remdups @ A @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_7156_remdups__adj_Osimps_I3_J,axiom,
! [A: $tType,X3: A,Y: A,Xs: list @ A] :
( ( ( X3 = Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) )
& ( ( X3 != Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
= ( cons @ A @ X3 @ ( remdups_adj @ A @ ( cons @ A @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_7157_remdups__adj_Osimps_I2_J,axiom,
! [A: $tType,X3: A] :
( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% remdups_adj.simps(2)
thf(fact_7158_remdups__adj_Oelims,axiom,
! [A: $tType,X3: list @ A,Y: list @ A] :
( ( ( remdups_adj @ A @ X3 )
= Y )
=> ( ( ( X3
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ( ! [X5: A] :
( ( X3
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( Y
!= ( cons @ A @ X5 @ ( nil @ A ) ) ) )
=> ~ ! [X5: A,Y4: A,Xs3: list @ A] :
( ( X3
= ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Xs3 ) ) )
=> ~ ( ( ( X5 = Y4 )
=> ( Y
= ( remdups_adj @ A @ ( cons @ A @ X5 @ Xs3 ) ) ) )
& ( ( X5 != Y4 )
=> ( Y
= ( cons @ A @ X5 @ ( remdups_adj @ A @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_7159_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 ) )
=> ~ ! [X5: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X5 @ Xs4 ) )
=> ! [Y4: B,Ys5: list @ B] :
( ( Ys
= ( cons @ B @ Y4 @ Ys5 ) )
=> ( ( Xy2
= ( product_Pair @ A @ B @ X5 @ Y4 ) )
=> ( Xys
!= ( zip @ A @ B @ Xs4 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_7160_successively_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P8: A > A > $o] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( nil @ A ) ) )
=> ( ! [P8: A > A > $o,X5: A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) )
=> ~ ! [P8: A > A > $o,X5: A,Y4: A,Xs3: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_7161_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X3: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F4: A > B,X5: A] :
( X3
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) )
=> ( ! [F4: A > B,X5: A,Y4: A,Zs: list @ A] :
( X3
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Zs ) ) ) )
=> ~ ! [A5: A > B] :
( X3
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A5 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_7162_sorted__wrt_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P8: A > A > $o] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( nil @ A ) ) )
=> ~ ! [P8: A > A > $o,X5: A,Ys4: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X5 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_7163_splice_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ~ ! [X5: A,Xs3: list @ A,Ys4: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_7164_shuffles_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( ! [Xs3: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ ( nil @ A ) ) )
=> ~ ! [X5: A,Xs3: list @ A,Y4: A,Ys4: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ A @ Y4 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_7165_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X3: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F4: A > B,Bs2: list @ B] :
( X3
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) )
=> ~ ! [F4: A > B,A5: A,As2: list @ A,Bs2: list @ B] :
( X3
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As2 ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_7166_listrel1I2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),X3: 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 @ X3 @ Xs ) @ ( cons @ A @ X3 @ Ys ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1I2
thf(fact_7167_inj__on__Cons1,axiom,
! [A: $tType,X3: A,A3: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ A3 ) ).
% inj_on_Cons1
thf(fact_7168_listrelp_Ocases,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A12: list @ A,A23: list @ B] :
( ( listrelp @ A @ B @ R2 @ A12 @ A23 )
=> ( ( ( A12
= ( nil @ A ) )
=> ( A23
!= ( nil @ B ) ) )
=> ~ ! [X5: A,Y4: B,Xs3: list @ A] :
( ( A12
= ( cons @ A @ X5 @ Xs3 ) )
=> ! [Ys4: list @ B] :
( ( A23
= ( cons @ B @ Y4 @ Ys4 ) )
=> ( ( R2 @ X5 @ Y4 )
=> ~ ( listrelp @ A @ B @ R2 @ Xs3 @ Ys4 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_7169_listrelp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( listrelp @ A @ B )
= ( ^ [R5: A > B > $o,A1: list @ A,A22: list @ B] :
( ( ( A1
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X: A,Y2: B,Xs2: list @ A,Ys3: list @ B] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
& ( A22
= ( cons @ B @ Y2 @ Ys3 ) )
& ( R5 @ X @ Y2 )
& ( listrelp @ A @ B @ R5 @ Xs2 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_7170_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X3: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_7171_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,X3: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs
!= ( cons @ A @ X3 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_7172_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X3: B,Y: B,Ys: list @ B] :
( ( ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F2 @ X3 @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ X3 @ ( cons @ B @ Y @ Ys ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F2 @ X3 @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ Y @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Ys ) ) ) ) ) ) ).
% insort_key.simps(2)
thf(fact_7173_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_7174_inj__split__Cons,axiom,
! [A: $tType,X8: set @ ( product_prod @ ( list @ A ) @ A )] :
( inj_on @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs2: list @ A,N: A] : ( cons @ A @ N @ Xs2 ) )
@ X8 ) ).
% inj_split_Cons
thf(fact_7175_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,X24: list @ A] :
( ? [Y2: A,Ys3: list @ A] :
( ( X15
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y2 @ Ys3 ) ) )
| ? [X: A,Y2: A,Xs2: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X @ Xs2 ) )
& ( X24
= ( cons @ A @ Y2 @ Ys3 ) )
& ( Less @ X @ Y2 ) )
| ? [X: A,Y2: A,Xs2: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X @ Xs2 ) )
& ( X24
= ( cons @ A @ Y2 @ Ys3 ) )
& ~ ( Less @ X @ Y2 )
& ~ ( Less @ Y2 @ X )
& ( P6 @ Xs2 @ Ys3 ) ) ) ) ).
% ord.lexordp.mono
thf(fact_7176_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,X3: A,Y: A,Zs2: list @ A] :
( ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Zs2 ) ) )
= ( if @ A @ ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ Y @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ Y @ Zs2 ) ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_7177_Suc__le__length__iff,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [X: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ X @ Ys3 ) )
& ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_7178_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_7179_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,F2: B > A,A2: B] :
( ! [X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ Xs ) )
=> ( ord_less_eq @ A @ ( F2 @ A2 ) @ ( F2 @ X5 ) ) )
=> ( ( linorder_insort_key @ B @ A @ F2 @ A2 @ Xs )
= ( cons @ B @ A2 @ Xs ) ) ) ) ).
% insort_is_Cons
thf(fact_7180_map__of__Cons__code_I2_J,axiom,
! [C: $tType,B: $tType,L: B,K: B,V2: C,Ps: list @ ( product_prod @ B @ C )] :
( ( ( L = K )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V2 ) @ Ps ) @ K )
= ( some @ C @ V2 ) ) )
& ( ( L != K )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V2 ) @ Ps ) @ K )
= ( map_of @ B @ C @ Ps @ K ) ) ) ) ).
% map_of_Cons_code(2)
thf(fact_7181_listrel1I1,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1I1
thf(fact_7182_Cons__listrel1E1,axiom,
! [A: $tType,X3: 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 @ X3 @ Xs ) @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( ! [Y4: A] :
( ( Ys
= ( cons @ A @ Y4 @ Xs ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 ) )
=> ~ ! [Zs: list @ A] :
( ( Ys
= ( cons @ A @ X3 @ Zs ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_7183_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 ) )
=> ( ! [X5: A] :
( ( Xs
= ( cons @ A @ X5 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y ) @ R2 ) )
=> ~ ! [Zs: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Zs ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_7184_listrel_OCons,axiom,
! [B: $tType,A: $tType,X3: A,Y: B,R2: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ 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 @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_7185_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 ) )
=> ~ ! [Y4: B,Ys4: list @ B] :
( ( Xs
= ( cons @ B @ Y4 @ Ys4 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y4 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys @ Ys4 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_7186_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 ) )
=> ~ ! [X5: A,Xs3: list @ A] :
( ( Xs
= ( cons @ A @ X5 @ Xs3 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_7187_count__list_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Y: A,Xs: list @ A] :
( ( ( X3 = Y )
=> ( ( count_list @ A @ ( cons @ A @ X3 @ Xs ) @ Y )
= ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y ) @ ( one_one @ nat ) ) ) )
& ( ( X3 != Y )
=> ( ( count_list @ A @ ( cons @ A @ X3 @ Xs ) @ Y )
= ( count_list @ A @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_7188_the__elem__set,axiom,
! [A: $tType,X3: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= X3 ) ).
% the_elem_set
thf(fact_7189_rtrancl__listrel1__ConsI1,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),X3: 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 @ X3 @ Xs ) @ ( cons @ A @ X3 @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ).
% rtrancl_listrel1_ConsI1
thf(fact_7190_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,X24: list @ A] :
( ? [Y2: A,Ys3: list @ A] :
( ( X15
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y2 @ Ys3 ) ) )
| ? [X: A,Y2: A,Xs2: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X @ Xs2 ) )
& ( X24
= ( cons @ A @ Y2 @ Ys3 ) )
& ( ord_less @ A @ X @ Y2 ) )
| ? [X: A,Y2: A,Xs2: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X @ Xs2 ) )
& ( X24
= ( cons @ A @ Y2 @ Ys3 ) )
& ~ ( ord_less @ A @ X @ Y2 )
& ~ ( ord_less @ A @ Y2 @ X )
& ( P6 @ Xs2 @ Ys3 ) ) ) ) ) ).
% lexordp.mono
thf(fact_7191_zip__replicate1,axiom,
! [A: $tType,B: $tType,N2: nat,X3: A,Ys: list @ B] :
( ( zip @ A @ B @ ( replicate @ A @ N2 @ X3 ) @ Ys )
= ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ ( take @ B @ N2 @ Ys ) ) ) ).
% zip_replicate1
thf(fact_7192_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_7193_nth__Cons_H,axiom,
! [A: $tType,N2: nat,X3: A,Xs: list @ A] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N2 )
= X3 ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_7194_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,N2: nat,Y: B] :
( ( zip @ A @ B @ Xs @ ( replicate @ B @ N2 @ Y ) )
= ( map @ A @ ( product_prod @ A @ B )
@ ^ [X: A] : ( product_Pair @ A @ B @ X @ Y )
@ ( take @ A @ N2 @ Xs ) ) ) ).
% zip_replicate2
thf(fact_7195_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A12: list @ A,A23: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A12 @ A23 ) @ ( listrel @ A @ B @ R2 ) )
=> ( ( ( A12
= ( nil @ A ) )
=> ( A23
!= ( nil @ B ) ) )
=> ~ ! [X5: A,Y4: B,Xs3: list @ A] :
( ( A12
= ( cons @ A @ X5 @ Xs3 ) )
=> ! [Ys4: list @ B] :
( ( A23
= ( cons @ B @ Y4 @ Ys4 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys4 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_7196_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A12: list @ A,A23: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A12 @ A23 ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( A12
= ( nil @ A ) )
& ( A23
= ( nil @ B ) ) )
| ? [X: A,Y2: B,Xs2: list @ A,Ys3: list @ B] :
( ( A12
= ( cons @ A @ X @ Xs2 ) )
& ( A23
= ( cons @ B @ Y2 @ Ys3 ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys3 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_7197_remdups__adj__replicate,axiom,
! [A: $tType,N2: nat,X3: A] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N2 @ X3 ) )
= ( nil @ A ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N2 @ X3 ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_replicate
thf(fact_7198_remdups__adj__singleton,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( ( remdups_adj @ A @ Xs )
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( Xs
= ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X3 ) ) ) ).
% remdups_adj_singleton
thf(fact_7199_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 )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ Xs ) ) ) ).
% Id_on_set
thf(fact_7200_list_Osize__gen_I2_J,axiom,
! [A: $tType,X3: A > nat,X21: A,X222: list @ A] :
( ( size_list @ A @ X3 @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X3 @ X21 ) @ ( size_list @ A @ X3 @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_7201_shuffles_Oelims,axiom,
! [A: $tType,X3: list @ A,Xa2: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X3 @ Xa2 )
= Y )
=> ( ( ( X3
= ( nil @ A ) )
=> ( Y
!= ( insert @ ( list @ A ) @ Xa2 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ( Y
!= ( insert @ ( list @ A ) @ X3 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ~ ! [X5: A,Xs3: list @ A] :
( ( X3
= ( cons @ A @ X5 @ Xs3 ) )
=> ! [Y4: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ Y4 @ Ys4 ) )
=> ( Y
!= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 ) @ ( shuffles @ A @ Xs3 @ ( cons @ A @ Y4 @ Ys4 ) ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y4 ) @ ( shuffles @ A @ ( cons @ A @ X5 @ Xs3 ) @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_7202_nth__equal__first__eq,axiom,
! [A: $tType,X3: A,Xs: list @ A,N2: nat] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N2 )
= X3 )
= ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_7203_nth__non__equal__first__eq,axiom,
! [A: $tType,X3: A,Y: A,Xs: list @ A,N2: nat] :
( ( X3 != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N2 )
= Y )
= ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
= Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_7204_take__Cons_H,axiom,
! [A: $tType,N2: nat,X3: A,Xs: list @ A] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( take @ A @ N2 @ ( cons @ A @ X3 @ Xs ) )
= ( nil @ A ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( take @ A @ N2 @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( take @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_7205_Cons__replicate__eq,axiom,
! [A: $tType,X3: A,Xs: list @ A,N2: nat,Y: A] :
( ( ( cons @ A @ X3 @ Xs )
= ( replicate @ A @ N2 @ Y ) )
= ( ( X3 = Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( Xs
= ( replicate @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ X3 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_7206_map__of__map__restrict,axiom,
! [B: $tType,A: $tType,F2: A > B,Ks2: list @ A] :
( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( F2 @ K3 ) )
@ Ks2 ) )
= ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F2 ) @ ( set2 @ A @ Ks2 ) ) ) ).
% map_of_map_restrict
thf(fact_7207_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs: list @ A,M: A > ( option @ B )] :
( ( ( set2 @ A @ Xs )
= ( dom @ A @ B @ M ) )
=> ( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( the2 @ B @ ( M @ K3 ) ) )
@ Xs ) )
= M ) ) ).
% map_of_map_keys
thf(fact_7208_Cons__lenlex__iff,axiom,
! [A: $tType,M: A,Ms: list @ A,N2: 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 @ M @ Ms ) @ ( cons @ A @ N2 @ 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 @ M @ N2 ) @ R2 ) )
| ( ( M = N2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_7209_map__of_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B,Ps: list @ ( product_prod @ A @ B )] :
( ( map_of @ A @ B @ ( cons @ ( product_prod @ A @ B ) @ P4 @ Ps ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Ps ) @ ( product_fst @ A @ B @ P4 ) @ ( some @ B @ ( product_snd @ A @ B @ P4 ) ) ) ) ).
% map_of.simps(2)
thf(fact_7210_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ 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 @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_7211_Pow__set_I2_J,axiom,
! [B: $tType,X3: B,Xs: list @ B] :
( ( pow2 @ B @ ( set2 @ B @ ( cons @ B @ X3 @ Xs ) ) )
= ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) @ ( image @ ( set @ B ) @ ( set @ B ) @ ( insert @ B @ X3 ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_7212_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( linord4507533701916653071of_set @ A @ A3 )
= ( cons @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_7213_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 ) )
@ ^ [X: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys )
@ Xs ) ) ) ) ).
% product_code
thf(fact_7214_set__Cons__sing__Nil,axiom,
! [A: $tType,A3: set @ A] :
( ( set_Cons @ A @ A3 @ ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
= ( image @ A @ ( list @ A )
@ ^ [X: A] : ( cons @ A @ X @ ( nil @ A ) )
@ A3 ) ) ).
% set_Cons_sing_Nil
thf(fact_7215_n__lists__Nil,axiom,
! [A: $tType,N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N2 @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N2 @ ( nil @ A ) )
= ( nil @ ( list @ A ) ) ) ) ) ).
% n_lists_Nil
thf(fact_7216_listset_Osimps_I2_J,axiom,
! [A: $tType,A3: set @ A,As3: list @ ( set @ A )] :
( ( listset @ A @ ( cons @ ( set @ A ) @ A3 @ As3 ) )
= ( set_Cons @ A @ A3 @ ( listset @ A @ As3 ) ) ) ).
% listset.simps(2)
thf(fact_7217_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
thf(fact_7218_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_7219_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_7220_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ ( suc @ I ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) )
= ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_7221_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) )
= ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_7222_set__Cons__def,axiom,
! [A: $tType] :
( ( set_Cons @ A )
= ( ^ [A6: set @ A,XS2: set @ ( list @ A )] :
( collect @ ( list @ A )
@ ^ [Z6: list @ A] :
? [X: A,Xs2: list @ A] :
( ( Z6
= ( cons @ A @ X @ Xs2 ) )
& ( member @ A @ X @ A6 )
& ( member @ ( list @ A ) @ Xs2 @ XS2 ) ) ) ) ) ).
% set_Cons_def
thf(fact_7223_concat__inth,axiom,
! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X3 ) ).
% concat_inth
thf(fact_7224_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,M: A > ( option @ B ),X3: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ Xs @ Ys ) @ X3 @ ( some @ B @ ( nth @ B @ Ys @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_7225_append_Oassoc,axiom,
! [A: $tType,A2: list @ A,B2: list @ A,C2: list @ A] :
( ( append @ A @ ( append @ A @ A2 @ B2 ) @ C2 )
= ( append @ A @ A2 @ ( append @ A @ B2 @ C2 ) ) ) ).
% append.assoc
thf(fact_7226_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs2 )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs2 ) ) ) ).
% append_assoc
thf(fact_7227_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs2 @ Xs ) )
= ( Ys = Zs2 ) ) ).
% append_same_eq
thf(fact_7228_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs2 ) )
= ( Ys = Zs2 ) ) ).
% same_append_eq
thf(fact_7229_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_7230_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_7231_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_7232_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_7233_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_7234_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_7235_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_7236_append_Oright__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ A2 @ ( nil @ A ) )
= A2 ) ).
% append.right_neutral
thf(fact_7237_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_7238_map__append,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ B] :
( ( map @ B @ A @ F2 @ ( append @ B @ Xs @ Ys ) )
= ( append @ A @ ( map @ B @ A @ F2 @ Xs ) @ ( map @ B @ A @ F2 @ Ys ) ) ) ).
% map_append
thf(fact_7239_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_7240_removeAll__append,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A] :
( ( removeAll @ A @ X3 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( removeAll @ A @ X3 @ Xs ) @ ( removeAll @ A @ X3 @ Ys ) ) ) ).
% removeAll_append
thf(fact_7241_foldl__append,axiom,
! [A: $tType,B: $tType,F2: A > B > A,A2: A,Xs: list @ B,Ys: list @ B] :
( ( foldl @ A @ B @ F2 @ A2 @ ( append @ B @ Xs @ Ys ) )
= ( foldl @ A @ B @ F2 @ ( foldl @ A @ B @ F2 @ A2 @ Xs ) @ Ys ) ) ).
% foldl_append
thf(fact_7242_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_7243_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_7244_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_7245_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_7246_fun__upds__append__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,M: A > ( option @ B ),Zs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M @ ( append @ A @ Xs @ Zs2 ) @ Ys )
= ( map_upds @ A @ B @ M @ Xs @ Ys ) ) ) ).
% fun_upds_append_drop
thf(fact_7247_fun__upds__append2__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,M: A > ( option @ B ),Zs2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M @ Xs @ ( append @ B @ Ys @ Zs2 ) )
= ( map_upds @ A @ B @ M @ Xs @ Ys ) ) ) ).
% fun_upds_append2_drop
thf(fact_7248_size__list__append,axiom,
! [A: $tType,F2: A > nat,Xs: list @ A,Ys: list @ A] :
( ( size_list @ A @ F2 @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_list @ A @ F2 @ Xs ) @ ( size_list @ A @ F2 @ Ys ) ) ) ).
% size_list_append
thf(fact_7249_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X3: B,Xs: list @ B,F2: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X3 @ Xs ) @ F2 )
= ( append @ A @ ( F2 @ X3 ) @ ( bind @ B @ A @ Xs @ F2 ) ) ) ).
% bind_simps(2)
thf(fact_7250_nth__append__length,axiom,
! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X3 ) ).
% nth_append_length
thf(fact_7251_nth__append__length__plus,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,N2: nat] :
( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) )
= ( nth @ A @ Ys @ N2 ) ) ).
% nth_append_length_plus
thf(fact_7252_take__append,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Ys: list @ A] :
( ( take @ A @ N2 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( take @ A @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_7253_list__update__length,axiom,
! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A,Y: A] :
( ( list_update @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) @ Y )
= ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_7254_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ ( suc @ K ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ K ) ) @ ( cons @ nat @ K @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_7255_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ ( suc @ K ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ K ) ) @ ( cons @ nat @ ( suc @ K ) @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_7256_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_7257_map__eq__append__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ A,Zs2: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( append @ A @ Ys @ Zs2 ) )
= ( ? [Us2: list @ B,Vs: list @ B] :
( ( Xs
= ( append @ B @ Us2 @ Vs ) )
& ( Ys
= ( map @ B @ A @ F2 @ Us2 ) )
& ( Zs2
= ( map @ B @ A @ F2 @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_7258_append__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ A,Zs2: list @ A,F2: B > A,Xs: list @ B] :
( ( ( append @ A @ Ys @ Zs2 )
= ( map @ B @ A @ F2 @ Xs ) )
= ( ? [Us2: list @ B,Vs: list @ B] :
( ( Xs
= ( append @ B @ Us2 @ Vs ) )
& ( Ys
= ( map @ B @ A @ F2 @ Us2 ) )
& ( Zs2
= ( map @ B @ A @ F2 @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_7259_split__list,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ? [Ys4: list @ A,Zs: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs ) ) ) ) ).
% split_list
thf(fact_7260_split__list__last,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ? [Ys4: list @ A,Zs: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs ) ) )
& ~ ( member @ A @ X3 @ ( set2 @ A @ Zs ) ) ) ) ).
% split_list_last
thf(fact_7261_split__list__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys4: list @ A,X5: A] :
( ? [Zs: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) )
& ( P @ X5 ) ) ) ).
% split_list_prop
thf(fact_7262_split__list__first,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ? [Ys4: list @ A,Zs: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs ) ) )
& ~ ( member @ A @ X3 @ ( set2 @ A @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_7263_split__list__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys4: list @ A,X5: A] :
( ? [Zs: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) )
=> ~ ( P @ X5 ) ) ) ).
% split_list_propE
thf(fact_7264_append__Cons__eq__iff,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A,Xs5: list @ A,Ys6: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ~ ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
=> ( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ Ys ) )
= ( append @ A @ Xs5 @ ( cons @ A @ X3 @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_7265_in__set__conv__decomp,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_7266_split__list__last__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys4: list @ A,X5: A,Zs: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) )
& ( P @ X5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_7267_split__list__first__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys4: list @ A,X5: A] :
( ? [Zs: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) )
& ( P @ X5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_7268_split__list__last__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys4: list @ A,X5: A,Zs: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) )
=> ( ( P @ X5 )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_7269_split__list__first__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys4: list @ A,X5: A] :
( ? [Zs: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) )
=> ( ( P @ X5 )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_7270_in__set__conv__decomp__last,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs3 ) ) )
& ~ ( member @ A @ X3 @ ( set2 @ A @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_7271_in__set__conv__decomp__first,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs3 ) ) )
& ~ ( member @ A @ X3 @ ( set2 @ A @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_7272_split__list__last__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys3: list @ A,X: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_7273_split__list__first__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys3: list @ A,X: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_7274_concat__eq__appendD,axiom,
! [A: $tType,Xss: list @ ( list @ A ),Ys: list @ A,Zs2: list @ A] :
( ( ( concat @ A @ Xss )
= ( append @ A @ Ys @ Zs2 ) )
=> ( ( Xss
!= ( nil @ ( list @ A ) ) )
=> ? [Xss1: list @ ( list @ A ),Xs3: list @ A,Xs4: list @ A,Xss22: list @ ( list @ A )] :
( ( Xss
= ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss1 ) @ Xs3 ) )
& ( Zs2
= ( append @ A @ Xs4 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_7275_concat__eq__append__conv,axiom,
! [A: $tType,Xss: list @ ( list @ A ),Ys: list @ A,Zs2: list @ A] :
( ( ( concat @ A @ Xss )
= ( append @ A @ Ys @ Zs2 ) )
= ( ( ( Xss
= ( nil @ ( list @ A ) ) )
=> ( ( Ys
= ( nil @ A ) )
& ( Zs2
= ( nil @ A ) ) ) )
& ( ( Xss
!= ( nil @ ( list @ A ) ) )
=> ? [Xss12: list @ ( list @ A ),Xs2: list @ A,Xs6: list @ A,Xss23: list @ ( list @ A )] :
( ( Xss
= ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs2 @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss12 ) @ Xs2 ) )
& ( Zs2
= ( append @ A @ Xs6 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_7276_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X5: A] : ( P @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [X5: A,Xs3: list @ A] :
( ( Xs3
!= ( nil @ A ) )
=> ( ( P @ Xs3 )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_7277_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs2: list @ A,X3: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs2 )
= ( cons @ A @ X3 @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs2
= ( cons @ A @ X3 @ Xs ) ) )
| ? [Ys7: list @ A] :
( ( Ys
= ( cons @ A @ X3 @ Ys7 ) )
& ( ( append @ A @ Ys7 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_7278_Cons__eq__append__conv,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( cons @ A @ X3 @ Xs )
= ( append @ A @ Ys @ Zs2 ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X3 @ Xs )
= Zs2 ) )
| ? [Ys7: list @ A] :
( ( ( cons @ A @ X3 @ Ys7 )
= Ys )
& ( Xs
= ( append @ A @ Ys7 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_7279_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys4: list @ A,Y4: A] :
( Xs
!= ( append @ A @ Ys4 @ ( cons @ A @ Y4 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_7280_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X5: A,Xs3: list @ A] :
( ( P @ Xs3 )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_7281_append__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X3 @ Xs ) @ Ys )
= ( cons @ A @ X3 @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_7282_Cons__eq__appendI,axiom,
! [A: $tType,X3: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs2: list @ A] :
( ( ( cons @ A @ X3 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs2 ) )
=> ( ( cons @ A @ X3 @ Xs )
= ( append @ A @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_7283_concat_Osimps_I2_J,axiom,
! [A: $tType,X3: list @ A,Xs: list @ ( list @ A )] :
( ( concat @ A @ ( cons @ ( list @ A ) @ X3 @ Xs ) )
= ( append @ A @ X3 @ ( concat @ A @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_7284_replicate__app__Cons__same,axiom,
! [A: $tType,N2: nat,X3: A,Xs: list @ A] :
( ( append @ A @ ( replicate @ A @ N2 @ X3 ) @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( append @ A @ ( replicate @ A @ N2 @ X3 ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_7285_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_7286_lexord__append__leftI,axiom,
! [A: $tType,U: list @ A,V2: list @ A,R2: set @ ( product_prod @ A @ A ),X3: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X3 @ U ) @ ( append @ A @ X3 @ V2 ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_leftI
thf(fact_7287_lex__append__leftI,axiom,
! [A: $tType,Ys: list @ A,Zs2: 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 @ Zs2 ) @ ( lex @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R2 ) ) ) ).
% lex_append_leftI
thf(fact_7288_comm__append__are__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Ys @ Xs ) )
=> ? [M3: nat,N3: nat,Zs: list @ A] :
( ( ( concat @ A @ ( replicate @ ( list @ A ) @ M3 @ Zs ) )
= Xs )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N3 @ Zs ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_7289_replicate__add,axiom,
! [A: $tType,N2: nat,M: nat,X3: A] :
( ( replicate @ A @ ( plus_plus @ nat @ N2 @ M ) @ X3 )
= ( append @ A @ ( replicate @ A @ N2 @ X3 ) @ ( replicate @ A @ M @ X3 ) ) ) ).
% replicate_add
thf(fact_7290_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_7291_append_Oleft__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ ( nil @ A ) @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_7292_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_7293_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs2: list @ A,Ys: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs2 @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_7294_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs2 @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs
= ( append @ A @ Zs2 @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us2 )
= Zs2 )
& ( Ys
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_7295_append__replicate__commute,axiom,
! [A: $tType,N2: nat,X3: A,K: nat] :
( ( append @ A @ ( replicate @ A @ N2 @ X3 ) @ ( replicate @ A @ K @ X3 ) )
= ( append @ A @ ( replicate @ A @ K @ X3 ) @ ( replicate @ A @ N2 @ X3 ) ) ) ).
% append_replicate_commute
thf(fact_7296_enumerate__append__eq,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Ys: list @ A] :
( ( enumerate @ A @ N2 @ ( append @ A @ Xs @ Ys ) )
= ( append @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) @ ( enumerate @ A @ ( plus_plus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_7297_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_7298_remove1__append,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A] :
( ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X3 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( remove1 @ A @ X3 @ Xs ) @ Ys ) ) )
& ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X3 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( remove1 @ A @ X3 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_7299_same__length__different,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ? [Pre: list @ A,X5: A,Xs4: list @ A,Y4: A,Ys5: list @ A] :
( ( X5 != Y4 )
& ( Xs
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X5 @ ( nil @ A ) ) @ Xs4 ) ) )
& ( Ys
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_7300_not__distinct__decomp,axiom,
! [A: $tType,Ws2: list @ A] :
( ~ ( distinct @ A @ Ws2 )
=> ? [Xs3: list @ A,Ys4: list @ A,Zs: list @ A,Y4: A] :
( Ws2
= ( append @ A @ Xs3 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ ( append @ A @ Ys4 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Zs ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_7301_not__distinct__conv__prefix,axiom,
! [A: $tType,As: list @ A] :
( ( ~ ( distinct @ A @ As ) )
= ( ? [Xs2: list @ A,Y2: A,Ys3: list @ A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Xs2 ) )
& ( distinct @ A @ Xs2 )
& ( As
= ( append @ A @ Xs2 @ ( cons @ A @ Y2 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_7302_replicate__append__same,axiom,
! [A: $tType,I: nat,X3: A] :
( ( append @ A @ ( replicate @ A @ I @ X3 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( cons @ A @ X3 @ ( replicate @ A @ I @ X3 ) ) ) ).
% replicate_append_same
thf(fact_7303_list__update__append1,axiom,
! [A: $tType,I: nat,Xs: list @ A,Ys: list @ A,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ I @ X3 )
= ( append @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_7304_remdups__adj__append__two,axiom,
! [A: $tType,Xs: list @ A,X3: A,Y: A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) @ ( if @ ( list @ A ) @ ( X3 = Y ) @ ( nil @ A ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_append_two
thf(fact_7305_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_7306_lexord__append__leftD,axiom,
! [A: $tType,X3: list @ A,U: list @ A,V2: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X3 @ U ) @ ( append @ A @ X3 @ V2 ) ) @ ( lexord @ A @ R2 ) )
=> ( ! [A5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_append_leftD
thf(fact_7307_lexord__append__rightI,axiom,
! [A: $tType,Y: list @ A,X3: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ? [B11: A,Z4: list @ A] :
( Y
= ( cons @ A @ B11 @ Z4 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ ( append @ A @ X3 @ Y ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_rightI
thf(fact_7308_lexord__sufE,axiom,
! [A: $tType,Xs: list @ A,Zs2: 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 @ Zs2 ) @ ( append @ A @ Ys @ Qs ) ) @ ( lexord @ A @ R2 ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs2 )
= ( 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_7309_lex__append__leftD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ! [X5: A] :
~ ( 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 ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_7310_lex__append__left__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ! [X5: A] :
~ ( 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 ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R2 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_7311_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X3 @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_7312_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_7313_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_7314_nths__append,axiom,
! [A: $tType,L: list @ A,L3: list @ A,A3: set @ nat] :
( ( nths @ A @ ( append @ A @ L @ L3 ) @ A3 )
= ( append @ A @ ( nths @ A @ L @ A3 )
@ ( nths @ A @ L3
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( plus_plus @ nat @ J3 @ ( size_size @ ( list @ A ) @ L ) ) @ A3 ) ) ) ) ) ).
% nths_append
thf(fact_7315_length__Suc__conv__rev,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N2 ) )
= ( ? [Y2: A,Ys3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N2 ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_7316_length__append__singleton,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_append_singleton
thf(fact_7317_subseqs_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( subseqs @ A @ ( cons @ A @ X3 @ Xs ) )
= ( append @ ( list @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( subseqs @ A @ Xs ) ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_7318_nth__append,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Ys: list @ A] :
( ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N2 )
= ( nth @ A @ Xs @ N2 ) ) )
& ( ~ ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N2 )
= ( nth @ A @ Ys @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_7319_list__update__append,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Ys: list @ A,X3: A] :
( ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N2 @ X3 )
= ( append @ A @ ( list_update @ A @ Xs @ N2 @ X3 ) @ Ys ) ) )
& ( ~ ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N2 @ X3 )
= ( append @ A @ Xs @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) @ X3 ) ) ) ) ) ).
% list_update_append
thf(fact_7320_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 ) )
=> ~ ! [X5: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ R2 )
=> ! [Us3: list @ A,Vs3: list @ A] :
( ( Xs
= ( append @ A @ Us3 @ ( cons @ A @ X5 @ Vs3 ) ) )
=> ( Ys
!= ( append @ A @ Us3 @ ( cons @ A @ Y4 @ Vs3 ) ) ) ) ) ) ).
% listrel1E
thf(fact_7321_listrel1I,axiom,
! [A: $tType,X3: 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 @ X3 @ Y ) @ R2 )
=> ( ( Xs
= ( append @ A @ Us @ ( cons @ A @ X3 @ 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_7322_lexord__append__left__rightI,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),U: list @ A,X3: 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 @ X3 ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y ) ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_left_rightI
thf(fact_7323_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: 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 @ Zs2 ) ) @ ( lexord @ A @ R2 ) )
= ( ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R2 ) )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_7324_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W2: list @ A,R2: set @ ( product_prod @ A @ A ),V2: list @ A,Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ 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 @ V2 ) @ ( append @ A @ W2 @ Z2 ) ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_sufI
thf(fact_7325_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ A,Ys: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X3 @ Xs ) @ Ys )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_7326_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list @ A,X3: 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 @ X3 @ ( 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 ) )
& ( X3 = Y ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_7327_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F2: B > A,A2: A,Xs: list @ B,Ys: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ ( append @ B @ Xs @ Ys ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A2 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ Ys ) ) ) ) ) ).
% horner_sum_append
thf(fact_7328_nths__Cons,axiom,
! [A: $tType,X3: A,L: list @ A,A3: set @ nat] :
( ( nths @ A @ ( cons @ A @ X3 @ L ) @ A3 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A3 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( suc @ J3 ) @ A3 ) ) ) ) ) ).
% nths_Cons
thf(fact_7329_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,Zs: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N3 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N3 @ Zs ) )
= ( append @ A @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_7330_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys3: list @ A] :
? [Us2: list @ A,Z6: A,Z7: A,Vs: list @ A] :
( ( Xs2
= ( append @ A @ Us2 @ ( cons @ A @ Z6 @ Vs ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ Z7 ) @ R5 )
& ( Ys3
= ( append @ A @ Us2 @ ( cons @ A @ Z7 @ Vs ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_7331_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
@ ^ [X: list @ A,Y2: list @ A] :
? [A4: A,V5: list @ A] :
( ( Y2
= ( append @ A @ X @ ( cons @ A @ A4 @ V5 ) ) )
| ? [U2: list @ A,B4: A,C4: A,W3: list @ A,Z6: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C4 ) @ R5 )
& ( X
= ( append @ A @ U2 @ ( cons @ A @ B4 @ W3 ) ) )
& ( Y2
= ( append @ A @ U2 @ ( cons @ A @ C4 @ Z6 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_7332_take__Suc__conv__app__nth,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ ( suc @ I ) @ Xs )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( nil @ A ) ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_7333_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ? [Xys2: list @ A,X: A,Y2: A,Xs6: list @ A,Ys7: list @ A] :
( ( Xs2
= ( append @ A @ Xys2 @ ( cons @ A @ X @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y2 @ Ys7 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R5 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_7334_nth__repl,axiom,
! [A: $tType,M: nat,Xs: list @ A,N2: nat,X3: A] :
( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( M != N2 )
=> ( ( nth @ A @ ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( drop @ A @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ Xs ) ) ) @ M )
= ( nth @ A @ Xs @ M ) ) ) ) ) ).
% nth_repl
thf(fact_7335_pos__n__replace,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Y: A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( append @ A @ ( cons @ A @ Y @ ( nil @ A ) ) @ ( drop @ A @ ( suc @ N2 ) @ Xs ) ) ) ) ) ) ).
% pos_n_replace
thf(fact_7336_drop0,axiom,
! [A: $tType] :
( ( drop @ A @ ( zero_zero @ nat ) )
= ( ^ [X: list @ A] : X ) ) ).
% drop0
thf(fact_7337_drop__drop,axiom,
! [A: $tType,N2: nat,M: nat,Xs: list @ A] :
( ( drop @ A @ N2 @ ( drop @ A @ M @ Xs ) )
= ( drop @ A @ ( plus_plus @ nat @ N2 @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_7338_drop__Suc__Cons,axiom,
! [A: $tType,N2: nat,X3: A,Xs: list @ A] :
( ( drop @ A @ ( suc @ N2 ) @ ( cons @ A @ X3 @ Xs ) )
= ( drop @ A @ N2 @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_7339_length__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( drop @ A @ N2 @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ).
% length_drop
thf(fact_7340_append__take__drop__id,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( drop @ A @ N2 @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_7341_drop__update__cancel,axiom,
! [A: $tType,N2: nat,M: nat,Xs: list @ A,X3: A] :
( ( ord_less @ nat @ N2 @ M )
=> ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N2 @ X3 ) )
= ( drop @ A @ M @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_7342_drop__replicate,axiom,
! [A: $tType,I: nat,K: nat,X3: A] :
( ( drop @ A @ I @ ( replicate @ A @ K @ X3 ) )
= ( replicate @ A @ ( minus_minus @ nat @ K @ I ) @ X3 ) ) ).
% drop_replicate
thf(fact_7343_drop__eq__Nil2,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( nil @ A )
= ( drop @ A @ N2 @ Xs ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ).
% drop_eq_Nil2
thf(fact_7344_drop__eq__Nil,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( drop @ A @ N2 @ Xs )
= ( nil @ A ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ).
% drop_eq_Nil
thf(fact_7345_drop__all,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 )
=> ( ( drop @ A @ N2 @ Xs )
= ( nil @ A ) ) ) ).
% drop_all
thf(fact_7346_drop__append,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Ys: list @ A] :
( ( drop @ A @ N2 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( drop @ A @ N2 @ Xs ) @ ( drop @ A @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_7347_drop__Cons__numeral,axiom,
! [A: $tType,V2: num,X3: A,Xs: list @ A] :
( ( drop @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X3 @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_7348_nth__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A,I: nat] :
( ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( drop @ A @ N2 @ Xs ) @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ N2 @ I ) ) ) ) ).
% nth_drop
thf(fact_7349_nth__via__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ( drop @ A @ N2 @ Xs )
= ( cons @ A @ Y @ Ys ) )
=> ( ( nth @ A @ Xs @ N2 )
= Y ) ) ).
% nth_via_drop
thf(fact_7350_set__drop__subset__set__drop,axiom,
! [A: $tType,N2: nat,M: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ M @ Xs ) ) @ ( set2 @ A @ ( drop @ A @ N2 @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_7351_drop__zip,axiom,
! [A: $tType,B: $tType,N2: nat,Xs: list @ A,Ys: list @ B] :
( ( drop @ ( product_prod @ A @ B ) @ N2 @ ( zip @ A @ B @ Xs @ Ys ) )
= ( zip @ A @ B @ ( drop @ A @ N2 @ Xs ) @ ( drop @ B @ N2 @ Ys ) ) ) ).
% drop_zip
thf(fact_7352_drop__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( drop @ A @ ( zero_zero @ nat ) @ Xs )
= Xs ) ).
% drop_0
thf(fact_7353_drop__Nil,axiom,
! [A: $tType,N2: nat] :
( ( drop @ A @ N2 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% drop_Nil
thf(fact_7354_in__set__dropD,axiom,
! [A: $tType,X3: A,N2: nat,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( drop @ A @ N2 @ Xs ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% in_set_dropD
thf(fact_7355_take__drop,axiom,
! [A: $tType,N2: nat,M: nat,Xs: list @ A] :
( ( take @ A @ N2 @ ( drop @ A @ M @ Xs ) )
= ( drop @ A @ M @ ( take @ A @ ( plus_plus @ nat @ N2 @ M ) @ Xs ) ) ) ).
% take_drop
thf(fact_7356_drop__take,axiom,
! [A: $tType,N2: nat,M: nat,Xs: list @ A] :
( ( drop @ A @ N2 @ ( take @ A @ M @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ M @ N2 ) @ ( drop @ A @ N2 @ Xs ) ) ) ).
% drop_take
thf(fact_7357_distinct__drop,axiom,
! [A: $tType,Xs: list @ A,I: nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( drop @ A @ I @ Xs ) ) ) ).
% distinct_drop
thf(fact_7358_set__drop__subset,axiom,
! [A: $tType,N2: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ N2 @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_drop_subset
thf(fact_7359_drop__eq__nths,axiom,
! [A: $tType] :
( ( drop @ A )
= ( ^ [N: nat,Xs2: list @ A] : ( nths @ A @ Xs2 @ ( collect @ nat @ ( ord_less_eq @ nat @ N ) ) ) ) ) ).
% drop_eq_nths
thf(fact_7360_drop__map,axiom,
! [A: $tType,B: $tType,N2: nat,F2: B > A,Xs: list @ B] :
( ( drop @ A @ N2 @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( drop @ B @ N2 @ Xs ) ) ) ).
% drop_map
thf(fact_7361_take__add,axiom,
! [A: $tType,I: nat,J: nat,Xs: list @ A] :
( ( take @ A @ ( plus_plus @ nat @ I @ J ) @ Xs )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( take @ A @ J @ ( drop @ A @ I @ Xs ) ) ) ) ).
% take_add
thf(fact_7362_append__eq__conv__conj,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Zs2 )
= ( ( Xs
= ( take @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs2 ) )
& ( Ys
= ( drop @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs2 ) ) ) ) ).
% append_eq_conv_conj
thf(fact_7363_drop__update__swap,axiom,
! [A: $tType,M: nat,N2: nat,Xs: list @ A,X3: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N2 @ X3 ) )
= ( list_update @ A @ ( drop @ A @ M @ Xs ) @ ( minus_minus @ nat @ N2 @ M ) @ X3 ) ) ) ).
% drop_update_swap
thf(fact_7364_drop__Cons,axiom,
! [A: $tType,N2: nat,X3: A,Xs: list @ A] :
( ( drop @ A @ N2 @ ( cons @ A @ X3 @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ X3 @ Xs )
@ ^ [M2: nat] : ( drop @ A @ M2 @ Xs )
@ N2 ) ) ).
% drop_Cons
thf(fact_7365_nths__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A,I6: set @ nat] :
( ( nths @ A @ ( drop @ A @ N2 @ Xs ) @ I6 )
= ( nths @ A @ Xs @ ( image @ nat @ nat @ ( plus_plus @ nat @ N2 ) @ I6 ) ) ) ).
% nths_drop
thf(fact_7366_drop__Cons_H,axiom,
! [A: $tType,N2: nat,X3: A,Xs: list @ A] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N2 @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ Xs ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N2 @ ( cons @ A @ X3 @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_7367_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_7368_zip__append2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ B] :
( ( zip @ A @ B @ Xs @ ( append @ B @ Ys @ Zs2 ) )
= ( 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 ) @ Zs2 ) ) ) ).
% zip_append2
thf(fact_7369_zip__append1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ B] :
( ( zip @ A @ B @ ( append @ A @ Xs @ Ys ) @ Zs2 )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ ( take @ B @ ( size_size @ ( list @ A ) @ Xs ) @ Zs2 ) ) @ ( zip @ A @ B @ Ys @ ( drop @ B @ ( size_size @ ( list @ A ) @ Xs ) @ Zs2 ) ) ) ) ).
% zip_append1
thf(fact_7370_Cons__nth__drop__Suc,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) )
= ( drop @ A @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_7371_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs2: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Vs2 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I @ Vs2 ) ) @ ( set2 @ A @ ( drop @ A @ J @ Vs2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_take_disj_set_drop_if_distinct
thf(fact_7372_id__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( Xs
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_7373_upd__conv__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A,A2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ Xs @ I @ A2 )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ A2 @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_7374_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),N: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= N )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N )
& ? [Xys2: list @ A,X: A,Y2: A,Xs6: list @ A,Ys7: list @ A] :
( ( Xs2
= ( append @ A @ Xys2 @ ( cons @ A @ X @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y2 @ Ys7 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R5 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_7375_list__encode_Oelims,axiom,
! [X3: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X3 )
= Y )
=> ( ( ( X3
= ( nil @ nat ) )
=> ( Y
!= ( zero_zero @ nat ) ) )
=> ~ ! [X5: nat,Xs3: list @ nat] :
( ( X3
= ( cons @ nat @ X5 @ Xs3 ) )
=> ( Y
!= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_7376_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_7377_list__encode_Osimps_I1_J,axiom,
( ( nat_list_encode @ ( nil @ nat ) )
= ( zero_zero @ nat ) ) ).
% list_encode.simps(1)
thf(fact_7378_lexn__length,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),N2: nat] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexn @ A @ R2 @ N2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= N2 )
& ( ( size_size @ ( list @ A ) @ Ys )
= N2 ) ) ) ).
% lexn_length
thf(fact_7379_lex__def,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] : ( complete_Sup_Sup @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image @ nat @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( lexn @ A @ R5 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% lex_def
thf(fact_7380_list__encode_Osimps_I2_J,axiom,
! [X3: nat,Xs: list @ nat] :
( ( nat_list_encode @ ( cons @ nat @ X3 @ Xs ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_7381_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_7382_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_7383_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,X24: list @ A] : ( H2 @ ( F22 @ X15 @ X24 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_7384_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_7385_remdups__adj__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) )
= ( case_list @ ( list @ A ) @ A @ ( cons @ A @ X3 @ ( nil @ A ) )
@ ^ [Y2: A,Xs2: list @ A] : ( if @ ( list @ A ) @ ( X3 = Y2 ) @ ( cons @ A @ Y2 @ Xs2 ) @ ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Xs2 ) ) )
@ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_7386_min__list_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A] :
( ( min_list @ A @ ( cons @ A @ X3 @ Xs ) )
= ( case_list @ A @ A @ X3
@ ^ [A4: A,List2: list @ A] : ( ord_min @ A @ X3 @ ( min_list @ A @ Xs ) )
@ Xs ) ) ) ).
% min_list.simps
thf(fact_7387_transpose_Oelims,axiom,
! [A: $tType,X3: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X3 )
= Y )
=> ( ( ( X3
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X3
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( Y
!= ( transpose @ A @ Xss2 ) ) )
=> ~ ! [X5: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
( ( X3
= ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) )
=> ( Y
!= ( cons @ ( list @ A )
@ ( cons @ A @ X5
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs3
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_7388_transpose_Osimps_I3_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X3
@ ( 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_7389_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_7390_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_7391_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_7392_transpose_Osimps_I1_J,axiom,
! [A: $tType] :
( ( transpose @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ ( list @ A ) ) ) ).
% transpose.simps(1)
thf(fact_7393_transpose__map__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ ( list @ B )] :
( ( transpose @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ Xs ) )
= ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ ( transpose @ B @ Xs ) ) ) ).
% transpose_map_map
thf(fact_7394_transpose__empty,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( ( transpose @ A @ Xs )
= ( nil @ ( list @ A ) ) )
= ( ! [X: list @ A] :
( ( member @ ( list @ A ) @ X @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( X
= ( nil @ A ) ) ) ) ) ).
% transpose_empty
thf(fact_7395_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 ) )
@ ^ [Z6: A,Zs3: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z6 @ Y ) @ ( zip @ A @ B @ Zs3 @ Ys ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_7396_zip__Cons1,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X3 @ Xs ) @ Ys )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Y2: B,Ys3: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ ( zip @ A @ B @ Xs @ Ys3 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_7397_upto_Opsimps,axiom,
! [I: int,J: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I @ J ) )
=> ( ( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) )
& ( ~ ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ) ) ).
% upto.psimps
thf(fact_7398_upto_Opelims,axiom,
! [X3: int,Xa2: int,Y: list @ int] :
( ( ( upto @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X3 @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X3 @ Xa2 )
=> ( Y
= ( cons @ int @ X3 @ ( upto @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ X3 @ Xa2 )
=> ( Y
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X3 @ Xa2 ) ) ) ) ) ).
% upto.pelims
thf(fact_7399_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less @ int @ J @ I )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ).
% upto_empty
thf(fact_7400_upto__Nil2,axiom,
! [I: int,J: int] :
( ( ( nil @ int )
= ( upto @ I @ J ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil2
thf(fact_7401_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= ( nil @ int ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil
thf(fact_7402_upto__single,axiom,
! [I: int] :
( ( upto @ I @ I )
= ( cons @ int @ I @ ( nil @ int ) ) ) ).
% upto_single
thf(fact_7403_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) @ J )
=> ( ( nth @ int @ ( upto @ I @ J ) @ K )
= ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) ) ) ).
% nth_upto
thf(fact_7404_length__upto,axiom,
! [I: int,J: int] :
( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ).
% length_upto
thf(fact_7405_upto__rec__numeral_I1_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N2 ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(1)
thf(fact_7406_upto__rec__numeral_I2_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(2)
thf(fact_7407_upto__rec__numeral_I3_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N2 ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(3)
thf(fact_7408_upto__rec__numeral_I4_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(4)
thf(fact_7409_atLeastAtMost__upto,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I3: int,J3: int] : ( set2 @ int @ ( upto @ I3 @ J3 ) ) ) ) ).
% atLeastAtMost_upto
thf(fact_7410_distinct__upto,axiom,
! [I: int,J: int] : ( distinct @ int @ ( upto @ I @ J ) ) ).
% distinct_upto
thf(fact_7411_upto__aux__def,axiom,
( upto_aux
= ( ^ [I3: int,J3: int] : ( append @ int @ ( upto @ I3 @ J3 ) ) ) ) ).
% upto_aux_def
thf(fact_7412_upto__code,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( upto_aux @ I3 @ J3 @ ( nil @ int ) ) ) ) ).
% upto_code
thf(fact_7413_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_7414_upto__split1,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_7415_atLeastLessThan__upto,axiom,
( ( set_or7035219750837199246ssThan @ int )
= ( ^ [I3: int,J3: int] : ( set2 @ int @ ( upto @ I3 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_7416_greaterThanAtMost__upto,axiom,
( ( set_or3652927894154168847AtMost @ int )
= ( ^ [I3: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_7417_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_7418_upto_Osimps,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I3 @ J3 ) @ ( cons @ int @ I3 @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_7419_upto_Oelims,axiom,
! [X3: int,Xa2: int,Y: list @ int] :
( ( ( upto @ X3 @ Xa2 )
= Y )
=> ( ( ( ord_less_eq @ int @ X3 @ Xa2 )
=> ( Y
= ( cons @ int @ X3 @ ( upto @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ X3 @ Xa2 )
=> ( Y
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_7420_upto__rec2,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( nil @ int ) ) ) ) ) ).
% upto_rec2
thf(fact_7421_greaterThanLessThan__upto,axiom,
( ( set_or5935395276787703475ssThan @ int )
= ( ^ [I3: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_7422_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_7423_transpose_Opelims,axiom,
! [A: $tType,X3: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X3 )
= Y )
=> ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X3 )
=> ( ( ( X3
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( nil @ ( list @ A ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X3
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( Y
= ( transpose @ A @ Xss2 ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ~ ! [X5: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
( ( X3
= ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) )
=> ( ( Y
= ( cons @ ( list @ A )
@ ( cons @ A @ X5
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs3
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_7424_transpose_Opsimps_I3_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X3
@ ( 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_7425_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_7426_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_7427_transpose_Opinduct,axiom,
! [A: $tType,A0: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > $o] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ A0 )
=> ( ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
=> ( P @ ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( P @ Xss2 )
=> ( P @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ( ! [X5: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) )
=> ( ( P
@ ( cons @ ( list @ A ) @ Xs3
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) )
=> ( P @ ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_7428_remdups__adj_Opelims,axiom,
! [A: $tType,X3: list @ A,Y: list @ A] :
( ( ( remdups_adj @ A @ X3 )
= Y )
=> ( ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ X3 )
=> ( ( ( X3
= ( nil @ A ) )
=> ( ( Y
= ( nil @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( nil @ A ) ) ) )
=> ( ! [X5: A] :
( ( X3
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ( Y
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) )
=> ~ ! [X5: A,Y4: A,Xs3: list @ A] :
( ( X3
= ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Xs3 ) ) )
=> ( ( ( ( X5 = Y4 )
=> ( Y
= ( remdups_adj @ A @ ( cons @ A @ X5 @ Xs3 ) ) ) )
& ( ( X5 != Y4 )
=> ( Y
= ( cons @ A @ X5 @ ( remdups_adj @ A @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_7429_take__hd__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( cons @ A @ ( hd @ A @ ( drop @ A @ N2 @ Xs ) ) @ ( nil @ A ) ) )
= ( take @ A @ ( suc @ N2 ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_7430_hd__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] :
( ( hd @ A @ ( remdups_adj @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ).
% hd_remdups_adj
thf(fact_7431_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_7432_hd__replicate,axiom,
! [A: $tType,N2: nat,X3: A] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N2 @ X3 ) )
= X3 ) ) ).
% hd_replicate
thf(fact_7433_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_7434_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_7435_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_7436_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_7437_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_7438_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_7439_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_7440_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_7441_hd__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F2 @ Xs ) )
= ( F2 @ ( hd @ A @ Xs ) ) ) ) ).
% hd_map
thf(fact_7442_list_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A2: list @ A,F2: A > B] :
( ( A2
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F2 @ A2 ) )
= ( F2 @ ( hd @ A @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_7443_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_7444_hd__drop__conv__nth,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hd @ A @ ( drop @ A @ N2 @ Xs ) )
= ( nth @ A @ Xs @ N2 ) ) ) ).
% hd_drop_conv_nth
thf(fact_7445_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_7446_list__encode_Opelims,axiom,
! [X3: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X3 )
= Y )
=> ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X3 )
=> ( ( ( X3
= ( nil @ nat ) )
=> ( ( Y
= ( zero_zero @ nat ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) )
=> ~ ! [X5: nat,Xs3: list @ nat] :
( ( X3
= ( cons @ nat @ X5 @ Xs3 ) )
=> ( ( Y
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X5 @ Xs3 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_7447_horner__sum__bit__eq__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N2: 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 ) @ N2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ).
% horner_sum_bit_eq_take_bit
thf(fact_7448_remdups__upt,axiom,
! [M: nat,N2: nat] :
( ( remdups @ nat @ ( upt @ M @ N2 ) )
= ( upt @ M @ N2 ) ) ).
% remdups_upt
thf(fact_7449_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( hd @ nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_7450_drop__upt,axiom,
! [M: nat,I: nat,J: nat] :
( ( drop @ nat @ M @ ( upt @ I @ J ) )
= ( upt @ ( plus_plus @ nat @ I @ M ) @ J ) ) ).
% drop_upt
thf(fact_7451_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size @ ( list @ nat ) @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ I ) ) ).
% length_upt
thf(fact_7452_take__upt,axiom,
! [I: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ M ) @ N2 )
=> ( ( take @ nat @ M @ ( upt @ I @ N2 ) )
= ( upt @ I @ ( plus_plus @ nat @ I @ M ) ) ) ) ).
% take_upt
thf(fact_7453_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( upt @ I @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_7454_sorted__list__of__set__range,axiom,
! [M: nat,N2: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
= ( upt @ M @ N2 ) ) ).
% sorted_list_of_set_range
thf(fact_7455_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= ( nil @ nat ) )
= ( ( J
= ( zero_zero @ nat ) )
| ( ord_less_eq @ nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_7456_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J )
=> ( ( nth @ nat @ ( upt @ I @ J ) @ K )
= ( plus_plus @ nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_7457_map__fst__enumerate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) )
= ( upt @ N2 @ ( plus_plus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% map_fst_enumerate
thf(fact_7458_upt__rec__numeral,axiom,
! [M: num,N2: num] :
( ( ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) )
= ( cons @ nat @ ( numeral_numeral @ nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N2 ) ) ) ) )
& ( ~ ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) )
= ( nil @ nat ) ) ) ) ).
% upt_rec_numeral
thf(fact_7459_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_7460_upt__conv__Cons__Cons,axiom,
! [M: nat,N2: nat,Ns: list @ nat,Q4: nat] :
( ( ( cons @ nat @ M @ ( cons @ nat @ N2 @ Ns ) )
= ( upt @ M @ Q4 ) )
= ( ( cons @ nat @ N2 @ Ns )
= ( upt @ ( suc @ M ) @ Q4 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_7461_enumerate__map__upt,axiom,
! [A: $tType,N2: nat,F2: nat > A,M: nat] :
( ( enumerate @ A @ N2 @ ( map @ nat @ A @ F2 @ ( upt @ N2 @ M ) ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [K3: nat] : ( product_Pair @ nat @ A @ K3 @ ( F2 @ K3 ) )
@ ( upt @ N2 @ M ) ) ) ).
% enumerate_map_upt
thf(fact_7462_map__replicate__trivial,axiom,
! [A: $tType,X3: A,I: nat] :
( ( map @ nat @ A
@ ^ [I3: nat] : X3
@ ( upt @ ( zero_zero @ nat ) @ I ) )
= ( replicate @ A @ I @ X3 ) ) ).
% map_replicate_trivial
thf(fact_7463_map__add__upt,axiom,
! [N2: nat,M: nat] :
( ( map @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ N2 )
@ ( upt @ ( zero_zero @ nat ) @ M ) )
= ( upt @ N2 @ ( plus_plus @ nat @ M @ N2 ) ) ) ).
% map_add_upt
thf(fact_7464_map__Suc__upt,axiom,
! [M: nat,N2: nat] :
( ( map @ nat @ nat @ suc @ ( upt @ M @ N2 ) )
= ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% map_Suc_upt
thf(fact_7465_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) ) ) ) ).
% atMost_upto
thf(fact_7466_greaterThanAtMost__upt,axiom,
( ( set_or3652927894154168847AtMost @ nat )
= ( ^ [N: nat,M2: nat] : ( set2 @ nat @ ( upt @ ( suc @ N ) @ ( suc @ M2 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_7467_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast_upt
thf(fact_7468_atLeastAtMost__upt,axiom,
( ( set_or1337092689740270186AtMost @ nat )
= ( ^ [N: nat,M2: nat] : ( set2 @ nat @ ( upt @ N @ ( suc @ M2 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_7469_atLeastLessThan__upt,axiom,
( ( set_or7035219750837199246ssThan @ nat )
= ( ^ [I3: nat,J3: nat] : ( set2 @ nat @ ( upt @ I3 @ J3 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_7470_greaterThanLessThan__upt,axiom,
( ( set_or5935395276787703475ssThan @ nat )
= ( ^ [N: nat,M2: nat] : ( set2 @ nat @ ( upt @ ( suc @ N ) @ M2 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_7471_distinct__upt,axiom,
! [I: nat,J: nat] : ( distinct @ nat @ ( upt @ I @ J ) ) ).
% distinct_upt
thf(fact_7472_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons @ nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_7473_enumerate__eq__zip,axiom,
! [A: $tType] :
( ( enumerate @ A )
= ( ^ [N: nat,Xs2: list @ A] : ( zip @ nat @ A @ ( upt @ N @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) @ Xs2 ) ) ) ).
% enumerate_eq_zip
thf(fact_7474_map__upt__Suc,axiom,
! [A: $tType,F2: nat > A,N2: nat] :
( ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( cons @ A @ ( F2 @ ( zero_zero @ nat ) )
@ ( map @ nat @ A
@ ^ [I3: nat] : ( F2 @ ( suc @ I3 ) )
@ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% map_upt_Suc
thf(fact_7475_map__decr__upt,axiom,
! [M: nat,N2: nat] :
( ( map @ nat @ nat
@ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( upt @ M @ N2 ) ) ).
% map_decr_upt
thf(fact_7476_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_7477_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus @ nat @ J @ K ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_7478_nth__map__upt,axiom,
! [A: $tType,I: nat,N2: nat,M: nat,F2: nat > A] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N2 @ M ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F2 @ ( upt @ M @ N2 ) ) @ I )
= ( F2 @ ( plus_plus @ nat @ M @ I ) ) ) ) ).
% nth_map_upt
thf(fact_7479_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X3: nat,Xs: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X3 @ Xs ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X3 )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_7480_upt__rec,axiom,
( upt
= ( ^ [I3: nat,J3: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I3 @ J3 ) @ ( cons @ nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_7481_enumerate__replicate__eq,axiom,
! [A: $tType,N2: nat,M: nat,A2: A] :
( ( enumerate @ A @ N2 @ ( replicate @ A @ M @ A2 ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [Q5: nat] : ( product_Pair @ nat @ A @ Q5 @ A2 )
@ ( upt @ N2 @ ( plus_plus @ nat @ N2 @ M ) ) ) ) ).
% enumerate_replicate_eq
thf(fact_7482_map__upt__eqI,axiom,
! [A: $tType,Xs: list @ A,N2: nat,M: nat,F2: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( minus_minus @ nat @ N2 @ M ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I4 )
= ( F2 @ ( plus_plus @ nat @ M @ I4 ) ) ) )
=> ( ( map @ nat @ A @ F2 @ ( upt @ M @ N2 ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_7483_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( nil @ nat ) ) ) ) ).
% upt_Suc
thf(fact_7484_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) ) ).
% upt_Suc_append
thf(fact_7485_transpose__rectangle,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N2: nat] :
( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( N2
= ( zero_zero @ nat ) ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I4 ) )
= N2 ) )
=> ( ( transpose @ A @ Xs )
= ( map @ nat @ ( list @ A )
@ ^ [I3: nat] :
( map @ nat @ A
@ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J3 ) @ I3 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% transpose_rectangle
thf(fact_7486_extract__SomeE,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A,Y: A,Zs2: list @ A] :
( ( ( extract @ A @ P @ 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 @ Zs2 ) ) ) )
=> ( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs2 ) ) )
& ( P @ Y )
& ~ ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ys ) )
& ( P @ X2 ) ) ) ) ).
% extract_SomeE
thf(fact_7487_extract__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A,Y: A,Zs2: list @ A] :
( ( ( extract @ A @ P @ 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 @ Zs2 ) ) ) )
= ( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs2 ) ) )
& ( P @ Y )
& ~ ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Ys ) )
& ( P @ X ) ) ) ) ).
% extract_Some_iff
thf(fact_7488_extract__None__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( extract @ A @ P @ Xs )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) )
= ( ~ ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( P @ X ) ) ) ) ).
% extract_None_iff
thf(fact_7489_extract__Nil__code,axiom,
! [A: $tType,P: A > $o] :
( ( extract @ A @ P @ ( nil @ A ) )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
% extract_Nil_code
thf(fact_7490_extract__Cons__code,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ( ( P @ X3 )
=> ( ( extract @ A @ P @ ( cons @ A @ X3 @ 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 ) @ X3 @ Xs ) ) ) ) )
& ( ~ ( P @ X3 )
=> ( ( extract @ A @ P @ ( cons @ A @ X3 @ 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 ) ) ) )
@ ^ [Y2: 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 @ X3 @ Ys3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y2 @ Zs3 ) ) ) ) )
@ ( extract @ A @ P @ Xs ) ) ) ) ) ).
% extract_Cons_code
thf(fact_7491_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list @ A,X8: set @ A,F2: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X8 )
=> ( ( finite_finite2 @ A @ X8 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X ) @ ( F2 @ X ) )
@ X8 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_7492_nth__transpose,axiom,
! [A: $tType,I: nat,Xs: list @ ( list @ A )] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I )
= ( map @ ( list @ A ) @ A
@ ^ [Xs2: list @ A] : ( nth @ A @ Xs2 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_7493_filter__filter,axiom,
! [A: $tType,P: A > $o,Q: A > $o,Xs: list @ A] :
( ( filter2 @ A @ P @ ( filter2 @ A @ Q @ Xs ) )
= ( filter2 @ A
@ ^ [X: A] :
( ( Q @ X )
& ( P @ X ) )
@ Xs ) ) ).
% filter_filter
thf(fact_7494_filter__True,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ( filter2 @ A @ P @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_7495_filter__append,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A] :
( ( filter2 @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( filter2 @ A @ P @ Xs ) @ ( filter2 @ A @ P @ Ys ) ) ) ).
% filter_append
thf(fact_7496_remove1__filter__not,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ~ ( P @ X3 )
=> ( ( remove1 @ A @ X3 @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ) ).
% remove1_filter_not
thf(fact_7497_removeAll__filter__not,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ~ ( P @ X3 )
=> ( ( removeAll @ A @ X3 @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ) ).
% removeAll_filter_not
thf(fact_7498_set__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( set2 @ A @ ( filter2 @ A @ P @ Xs ) )
= ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_7499_sum__list_ONil,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
= ( zero_zero @ A ) ) ) ).
% sum_list.Nil
thf(fact_7500_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [Ns: list @ A] :
( ( ( groups8242544230860333062m_list @ A @ Ns )
= ( zero_zero @ A ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Ns ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_7501_sum__list_OCons,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [X3: A,Xs: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( cons @ A @ X3 @ Xs ) )
= ( plus_plus @ A @ X3 @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% sum_list.Cons
thf(fact_7502_filter__False,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ~ ( P @ X5 ) )
=> ( ( filter2 @ A @ P @ Xs )
= ( nil @ A ) ) ) ).
% filter_False
thf(fact_7503_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_7504_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X: B] : ( zero_zero @ A )
@ Xs ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_7505_length__filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ ( map @ B @ A @ F2 @ Xs ) ) )
= ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ).
% length_filter_map
thf(fact_7506_sum__list__upt,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups8242544230860333062m_list @ nat @ ( upt @ M @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% sum_list_upt
thf(fact_7507_filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] :
( ( filter2 @ A @ P @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ).
% filter_map
thf(fact_7508_sum__list__filter__le__nat,axiom,
! [A: $tType,F2: A > nat,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ ( filter2 @ A @ P @ Xs ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) ) ) ).
% sum_list_filter_le_nat
thf(fact_7509_sum__list__addf,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: B > A,G: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ Xs ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum_list_addf
thf(fact_7510_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F2: B > A,P: B > $o,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( F2 @ X ) @ ( zero_zero @ A ) )
@ Xs ) ) ) ) ).
% sum_list_map_filter'
thf(fact_7511_filter__concat,axiom,
! [A: $tType,P4: A > $o,Xs: list @ ( list @ A )] :
( ( filter2 @ A @ P4 @ ( concat @ A @ Xs ) )
= ( concat @ A @ ( map @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ P4 ) @ Xs ) ) ) ).
% filter_concat
thf(fact_7512_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B,P: B > $o,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ Xs ) )
=> ( ~ ( P @ X5 )
=> ( ( F2 @ X5 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_7513_distinct__map__filter,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,P: B > $o] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) )
=> ( distinct @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) ) ) ).
% distinct_map_filter
thf(fact_7514_length__concat,axiom,
! [B: $tType,Xss: list @ ( list @ B )] :
( ( size_size @ ( list @ B ) @ ( concat @ B @ Xss ) )
= ( groups8242544230860333062m_list @ nat @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) ) ) ).
% length_concat
thf(fact_7515_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X5 @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_7516_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 ) )
=> ( ( ( groups8242544230860333062m_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_7517_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_7518_empty__filter__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( nil @ A )
= ( filter2 @ A @ P @ Xs ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( P @ X ) ) ) ) ).
% empty_filter_conv
thf(fact_7519_filter__empty__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Xs )
= ( nil @ A ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( P @ X ) ) ) ) ).
% filter_empty_conv
thf(fact_7520_Cons__eq__filterD,axiom,
! [A: $tType,X3: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X3 @ Xs )
= ( filter2 @ A @ P @ Ys ) )
=> ? [Us3: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X3 @ Vs3 ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Us3 ) )
=> ~ ( P @ X2 ) )
& ( P @ X3 )
& ( Xs
= ( filter2 @ A @ P @ Vs3 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_7521_filter__eq__ConsD,axiom,
! [A: $tType,P: A > $o,Ys: list @ A,X3: A,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Ys )
= ( cons @ A @ X3 @ Xs ) )
=> ? [Us3: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X3 @ Vs3 ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Us3 ) )
=> ~ ( P @ X2 ) )
& ( P @ X3 )
& ( Xs
= ( filter2 @ A @ P @ Vs3 ) ) ) ) ).
% filter_eq_ConsD
thf(fact_7522_Cons__eq__filter__iff,axiom,
! [A: $tType,X3: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X3 @ Xs )
= ( filter2 @ A @ P @ Ys ) )
= ( ? [Us2: list @ A,Vs: list @ A] :
( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ X3 @ Vs ) ) )
& ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Us2 ) )
=> ~ ( P @ X ) )
& ( P @ X3 )
& ( Xs
= ( filter2 @ A @ P @ Vs ) ) ) ) ) ).
% Cons_eq_filter_iff
thf(fact_7523_filter__eq__Cons__iff,axiom,
! [A: $tType,P: A > $o,Ys: list @ A,X3: A,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Ys )
= ( cons @ A @ X3 @ Xs ) )
= ( ? [Us2: list @ A,Vs: list @ A] :
( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ X3 @ Vs ) ) )
& ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Us2 ) )
=> ~ ( P @ X ) )
& ( P @ X3 )
& ( Xs
= ( filter2 @ A @ P @ Vs ) ) ) ) ) ).
% filter_eq_Cons_iff
thf(fact_7524_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,Y: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert @ A @ Y @ ( set2 @ A @ Xs ) ) )
=> ( ( filter2 @ A
@ ^ [X: A] :
( ( F2 @ Y )
= ( F2 @ X ) )
@ Xs )
= ( filter2 @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ Y )
@ Xs ) ) ) ).
% inj_on_filter_key_eq
thf(fact_7525_member__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% member_le_sum_list
thf(fact_7526_length__filter__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_filter_le
thf(fact_7527_filter__is__subset,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( filter2 @ A @ P @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% filter_is_subset
thf(fact_7528_distinct__filter,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( filter2 @ A @ P @ Xs ) ) ) ).
% distinct_filter
thf(fact_7529_length__filter__less,axiom,
! [A: $tType,X3: A,Xs: list @ A,P: A > $o] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X3 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_7530_filter__set,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( filter3 @ A @ P @ ( set2 @ A @ Xs ) )
= ( set2 @ A @ ( filter2 @ A @ P @ Xs ) ) ) ).
% filter_set
thf(fact_7531_replicate__length__filter,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X3 )
@ Xs ) )
@ X3 )
= ( filter2 @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X3 )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_7532_filter__id__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Xs )
= Xs )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) ) ) ) ).
% filter_id_conv
thf(fact_7533_filter__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( filter2 @ A @ P @ Xs )
= ( filter2 @ A @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_7534_inter__set__filter,axiom,
! [A: $tType,A3: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( set2 @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X: A] : ( member @ A @ X @ A3 )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_7535_sum__length__filter__compl,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) )
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ^ [X: A] :
~ ( P @ X )
@ Xs ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_7536_remdups__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( remdups @ A @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ ( remdups @ A @ Xs ) ) ) ).
% remdups_filter
thf(fact_7537_filter__replicate,axiom,
! [A: $tType,P: A > $o,X3: A,N2: nat] :
( ( ( P @ X3 )
=> ( ( filter2 @ A @ P @ ( replicate @ A @ N2 @ X3 ) )
= ( replicate @ A @ N2 @ X3 ) ) )
& ( ~ ( P @ X3 )
=> ( ( filter2 @ A @ P @ ( replicate @ A @ N2 @ X3 ) )
= ( nil @ A ) ) ) ) ).
% filter_replicate
thf(fact_7538_filter_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( filter2 @ A @ P @ ( nil @ A ) )
= ( nil @ A ) ) ).
% filter.simps(1)
thf(fact_7539_filter_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ( ( P @ X3 )
=> ( ( filter2 @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( filter2 @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X3 )
=> ( ( filter2 @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_7540_filter__shuffles,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A] :
( ( image @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ P ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( shuffles @ A @ ( filter2 @ A @ P @ Xs ) @ ( filter2 @ A @ P @ Ys ) ) ) ).
% filter_shuffles
thf(fact_7541_removeAll__filter__not__eq,axiom,
! [A: $tType] :
( ( removeAll @ A )
= ( ^ [X: A] :
( filter2 @ A
@ ^ [Y2: A] : X != Y2 ) ) ) ).
% removeAll_filter_not_eq
thf(fact_7542_filter__remove1,axiom,
! [A: $tType,Q: A > $o,X3: A,Xs: list @ A] :
( ( filter2 @ A @ Q @ ( remove1 @ A @ X3 @ Xs ) )
= ( remove1 @ A @ X3 @ ( filter2 @ A @ Q @ Xs ) ) ) ).
% filter_remove1
thf(fact_7543_partition__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( member @ ( list @ A ) @ Xs
@ ( shuffles @ A @ ( filter2 @ A @ P @ Xs )
@ ( filter2 @ A
@ ^ [X: A] :
~ ( P @ X )
@ Xs ) ) ) ).
% partition_in_shuffles
thf(fact_7544_filter__insort__triv,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,X3: B,F2: B > A,Xs: list @ B] :
( ~ ( P @ X3 )
=> ( ( filter2 @ B @ P @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) )
= ( filter2 @ B @ P @ Xs ) ) ) ) ).
% filter_insort_triv
thf(fact_7545_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_7546_filter__in__nths,axiom,
! [A: $tType,Xs: list @ A,S: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ( filter2 @ A
@ ^ [X: A] : ( member @ A @ X @ ( set2 @ A @ ( nths @ A @ Xs @ S ) ) )
@ Xs )
= ( nths @ A @ Xs @ S ) ) ) ).
% filter_in_nths
thf(fact_7547_sum__list__replicate,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat,C2: A] :
( ( groups8242544230860333062m_list @ A @ ( replicate @ A @ N2 @ C2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ C2 ) ) ) ).
% sum_list_replicate
thf(fact_7548_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs: list @ A,F2: A > B,G: A > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ).
% sum_list_mono
thf(fact_7549_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
@ ^ [X: A] : X
@ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_sum_list_conv_Sum
thf(fact_7550_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
@ ^ [X: A] : X != Y
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_7551_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X: A] :
~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
@ Zs2 )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_7552_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X: A] : ( member @ A @ X @ ( set2 @ A @ Xs ) )
@ Zs2 )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_7553_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X: A] :
~ ( member @ A @ X @ ( set2 @ A @ Ys ) )
@ Zs2 )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_7554_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X: A] : ( member @ A @ X @ ( set2 @ A @ Ys ) )
@ Zs2 )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_7555_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P3: A > $o,Xs2: list @ A] :
( nths @ A @ Xs2
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P3 @ ( nth @ A @ Xs2 @ I3 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_7556_length__filter__conv__card,axiom,
! [A: $tType,P4: A > $o,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P4 @ Xs ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P4 @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_7557_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( strict9044650504122735259up_add @ B ) )
=> ! [Xs: list @ A,F2: A > B,G: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ B @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_7558_elem__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [K: nat,Ns: list @ A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Ns ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Ns @ K ) @ ( groups8242544230860333062m_list @ A @ Ns ) ) ) ) ).
% elem_le_sum_list
thf(fact_7559_sum__list__distinct__conv__sum__set,axiom,
! [C: $tType,B: $tType] :
( ( comm_monoid_add @ C )
=> ! [Xs: list @ B,F2: B > C] :
( ( distinct @ B @ Xs )
=> ( ( groups8242544230860333062m_list @ C @ ( map @ B @ C @ F2 @ Xs ) )
= ( groups7311177749621191930dd_sum @ B @ C @ F2 @ ( set2 @ B @ Xs ) ) ) ) ) ).
% sum_list_distinct_conv_sum_set
thf(fact_7560_sum_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ B,G: B > A] :
( ( distinct @ B @ Xs )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set2 @ B @ Xs ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum.distinct_set_conv_list
thf(fact_7561_sum__list__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [X3: B,Xs: list @ B,F2: B > A] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) )
= ( plus_plus @ A @ ( F2 @ X3 ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( remove1 @ B @ X3 @ Xs ) ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_7562_sum__code,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,Xs: list @ B] :
( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set2 @ B @ Xs ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ ( remdups @ B @ Xs ) ) ) ) ) ).
% sum_code
thf(fact_7563_size__list__conv__sum__list,axiom,
! [B: $tType] :
( ( size_list @ B )
= ( ^ [F3: B > nat,Xs2: list @ B] : ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ B @ nat @ F3 @ Xs2 ) ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_7564_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R2: B,Xs: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X: C] : R2
@ Xs ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R2 ) ) ) ).
% sum_list_triv
thf(fact_7565_sum__list__Suc,axiom,
! [A: $tType,F2: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat
@ ( map @ A @ nat
@ ^ [X: A] : ( suc @ ( F2 @ X ) )
@ Xs ) )
= ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_7566_distinct__length__filter,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) )
= ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_length_filter
thf(fact_7567_sum__list__sum__nth,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ( ( groups8242544230860333062m_list @ B )
= ( ^ [Xs2: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ).
% sum_list_sum_nth
thf(fact_7568_card__length__sum__list__rec,axiom,
! [M: nat,N7: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N7 ) ) ) )
= ( plus_plus @ nat
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= ( minus_minus @ nat @ M @ ( one_one @ nat ) ) )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N7 ) ) ) )
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M )
& ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L2 ) @ ( one_one @ nat ) )
= N7 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_7569_card__length__sum__list,axiom,
! [M: nat,N7: nat] :
( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N7 ) ) ) )
= ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N7 @ M ) @ ( one_one @ nat ) ) @ N7 ) ) ).
% card_length_sum_list
thf(fact_7570_sum__list__map__eq__sum__count,axiom,
! [A: $tType,F2: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X ) @ ( F2 @ X ) )
@ ( set2 @ A @ Xs ) ) ) ).
% sum_list_map_eq_sum_count
thf(fact_7571_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K: nat,Xs: list @ A,X3: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X3 ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ).
% sum_list_update
thf(fact_7572_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_7573_map__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter @ A @ B )
= ( ^ [F3: A > ( option @ B ),Xs2: list @ A] :
( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F3 )
@ ( filter2 @ A
@ ^ [X: A] :
( ( F3 @ X )
!= ( none @ B ) )
@ Xs2 ) ) ) ) ).
% map_filter_def
thf(fact_7574_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_7575_tl__upt,axiom,
! [M: nat,N2: nat] :
( ( tl @ nat @ ( upt @ M @ N2 ) )
= ( upt @ ( suc @ M ) @ N2 ) ) ).
% tl_upt
thf(fact_7576_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_7577_remdups__adj__Cons__alt,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( cons @ A @ X3 @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_7578_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_7579_list_Ocollapse,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
= List ) ) ).
% list.collapse
thf(fact_7580_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_7581_tl__replicate,axiom,
! [A: $tType,N2: nat,X3: A] :
( ( tl @ A @ ( replicate @ A @ N2 @ X3 ) )
= ( replicate @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ X3 ) ) ).
% tl_replicate
thf(fact_7582_list_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A2: list @ A,F2: A > B] :
( ( A2
!= ( nil @ A ) )
=> ( ( tl @ B @ ( map @ A @ B @ F2 @ A2 ) )
= ( map @ A @ B @ F2 @ ( tl @ A @ A2 ) ) ) ) ).
% list.map_sel(2)
thf(fact_7583_map__tl,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( map @ B @ A @ F2 @ ( tl @ B @ Xs ) )
= ( tl @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ).
% map_tl
thf(fact_7584_map__of__filter__in,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Z2: A,P: B > A > $o] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ Z2 ) )
=> ( ( P @ K @ Z2 )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) @ Xs ) @ K )
= ( some @ A @ Z2 ) ) ) ) ).
% map_of_filter_in
thf(fact_7585_tl__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( case_list @ ( list @ A ) @ A @ ( tl @ A @ Ys )
@ ^ [Z6: A,Zs3: list @ A] : ( append @ A @ Zs3 @ Ys )
@ Xs ) ) ).
% tl_append
thf(fact_7586_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( tl @ A @ ( cons @ A @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_7587_Nil__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( tl @ A @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X: A] :
( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% Nil_tl
thf(fact_7588_tl__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X: A] :
( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% tl_Nil
thf(fact_7589_list_Osel_I2_J,axiom,
! [A: $tType] :
( ( tl @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% list.sel(2)
thf(fact_7590_map__filter__simps_I2_J,axiom,
! [B: $tType,A: $tType,F2: B > ( option @ A )] :
( ( map_filter @ B @ A @ F2 @ ( nil @ B ) )
= ( nil @ A ) ) ).
% map_filter_simps(2)
thf(fact_7591_tl__def,axiom,
! [A: $tType] :
( ( tl @ A )
= ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [X213: A,X224: list @ A] : X224 ) ) ).
% tl_def
thf(fact_7592_take__tl,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( take @ A @ N2 @ ( tl @ A @ Xs ) )
= ( tl @ A @ ( take @ A @ ( suc @ N2 ) @ Xs ) ) ) ).
% take_tl
thf(fact_7593_drop__Suc,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( drop @ A @ ( suc @ N2 ) @ Xs )
= ( drop @ A @ N2 @ ( tl @ A @ Xs ) ) ) ).
% drop_Suc
thf(fact_7594_tl__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( tl @ A @ ( drop @ A @ N2 @ Xs ) )
= ( drop @ A @ N2 @ ( tl @ A @ Xs ) ) ) ).
% tl_drop
thf(fact_7595_distinct__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( tl @ A @ Xs ) ) ) ).
% distinct_tl
thf(fact_7596_list_Oset__sel_I2_J,axiom,
! [A: $tType,A2: list @ A,X3: A] :
( ( A2
!= ( nil @ A ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ ( tl @ A @ A2 ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_7597_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_7598_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_7599_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),X3: B,Xs: list @ B] :
( ( map_filter @ B @ A @ F2 @ ( cons @ B @ X3 @ Xs ) )
= ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F2 @ Xs )
@ ^ [Y2: A] : ( cons @ A @ Y2 @ ( map_filter @ B @ A @ F2 @ Xs ) )
@ ( F2 @ X3 ) ) ) ).
% map_filter_simps(1)
thf(fact_7600_tl__take,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( tl @ A @ ( take @ A @ N2 @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( tl @ A @ Xs ) ) ) ).
% tl_take
thf(fact_7601_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_7602_Nitpick_Osize__list__simp_I2_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( ^ [Xs2: list @ A] :
( if @ nat
@ ( Xs2
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs2 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_7603_nth__tl,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
=> ( ( nth @ A @ ( tl @ A @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( suc @ N2 ) ) ) ) ).
% nth_tl
thf(fact_7604_remdups__adj__append,axiom,
! [A: $tType,Xs_1: list @ A,X3: A,Xs_2: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X3 @ Xs_2 ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_7605_Cons__in__shuffles__iff,axiom,
! [A: $tType,Z2: A,Zs2: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ Z2 @ Zs2 ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( ( ( Xs
!= ( nil @ A ) )
& ( ( hd @ A @ Xs )
= Z2 )
& ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ ( tl @ A @ Xs ) @ Ys ) ) )
| ( ( Ys
!= ( nil @ A ) )
& ( ( hd @ A @ Ys )
= Z2 )
& ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ ( tl @ A @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_7606_list_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
= ( ~ ( ( ( List
= ( nil @ A ) )
& ~ ( P @ F1 ) )
| ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
& ~ ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ) ).
% list.split_sel_asm
thf(fact_7607_list_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
= ( ( ( List
= ( nil @ A ) )
=> ( P @ F1 ) )
& ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
=> ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ).
% list.split_sel
thf(fact_7608_nths__shift__lemma__Suc,axiom,
! [A: $tType,P: nat > $o,Xs: list @ A,Is: list @ nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( P @ ( suc @ ( product_snd @ A @ nat @ P6 ) ) )
@ ( zip @ A @ nat @ Xs @ Is ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( P @ ( product_snd @ A @ nat @ P6 ) )
@ ( zip @ A @ nat @ Xs @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% nths_shift_lemma_Suc
thf(fact_7609_take__Suc,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( Xs
!= ( nil @ A ) )
=> ( ( take @ A @ ( suc @ N2 ) @ Xs )
= ( cons @ A @ ( hd @ A @ Xs ) @ ( take @ A @ N2 @ ( tl @ A @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_7610_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_7611_nths__shift__lemma,axiom,
! [A: $tType,A3: set @ nat,Xs: list @ A,I: nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A3 )
@ ( zip @ A @ nat @ Xs @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P6 ) @ I ) @ A3 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_7612_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType] :
( ( size_list @ A )
= ( ^ [F3: A > nat,Xs2: list @ A] :
( if @ nat
@ ( Xs2
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( plus_plus @ nat @ ( F3 @ ( hd @ A @ Xs2 ) ) @ ( size_list @ A @ F3 @ ( tl @ A @ Xs2 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_7613_nths__def,axiom,
! [A: $tType] :
( ( nths @ A )
= ( ^ [Xs2: list @ A,A6: 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 ) @ A6 )
@ ( zip @ A @ nat @ Xs2 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ) ).
% nths_def
thf(fact_7614_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F2: B > A,P: B > $o,Xs: list @ B] :
( ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) )
= ( map_filter @ B @ A
@ ^ [X: B] : ( if @ ( option @ A ) @ ( P @ X ) @ ( some @ A @ ( F2 @ X ) ) @ ( none @ A ) )
@ Xs ) ) ).
% map_filter_map_filter
thf(fact_7615_transpose__max__length,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( foldr @ ( list @ A ) @ nat
@ ^ [Xs2: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) )
@ ( transpose @ A @ Xs )
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [X: list @ A] :
( X
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_7616_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Xss: list @ ( list @ B )] :
( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs ) )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [Xs2: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs2 ) )
@ Xss
@ ( zero_zero @ nat ) ) )
= ( suc
@ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [X: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys3: list @ B] :
( Ys3
!= ( nil @ B ) )
@ Xss )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_7617_foldr__append,axiom,
! [B: $tType,A: $tType,F2: B > A > A,Xs: list @ B,Ys: list @ B,A2: A] :
( ( foldr @ B @ A @ F2 @ ( append @ B @ Xs @ Ys ) @ A2 )
= ( foldr @ B @ A @ F2 @ Xs @ ( foldr @ B @ A @ F2 @ Ys @ A2 ) ) ) ).
% foldr_append
thf(fact_7618_foldr__replicate,axiom,
! [A: $tType,B: $tType,F2: B > A > A,N2: nat,X3: B] :
( ( foldr @ B @ A @ F2 @ ( replicate @ B @ N2 @ X3 ) )
= ( compow @ ( A > A ) @ N2 @ ( F2 @ X3 ) ) ) ).
% foldr_replicate
thf(fact_7619_foldr__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: B > A > A,F2: C > B,Xs: list @ C,A2: A] :
( ( foldr @ B @ A @ G @ ( map @ C @ B @ F2 @ Xs ) @ A2 )
= ( foldr @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G @ F2 ) @ Xs @ A2 ) ) ).
% foldr_map
thf(fact_7620_foldr__Cons,axiom,
! [B: $tType,A: $tType,F2: A > B > B,X3: A,Xs: list @ A] :
( ( foldr @ A @ B @ F2 @ ( cons @ A @ X3 @ Xs ) )
= ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( foldr @ A @ B @ F2 @ Xs ) ) ) ).
% foldr_Cons
thf(fact_7621_foldr__cong,axiom,
! [B: $tType,A: $tType,A2: A,B2: A,L: list @ B,K: list @ B,F2: B > A > A,G: B > A > A] :
( ( A2 = B2 )
=> ( ( L = K )
=> ( ! [A5: A,X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ L ) )
=> ( ( F2 @ X5 @ A5 )
= ( G @ X5 @ A5 ) ) )
=> ( ( foldr @ B @ A @ F2 @ L @ A2 )
= ( foldr @ B @ A @ G @ K @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_7622_foldr__Nil,axiom,
! [A: $tType,B: $tType,F2: A > B > B] :
( ( foldr @ A @ B @ F2 @ ( nil @ A ) )
= ( id @ B ) ) ).
% foldr_Nil
thf(fact_7623_foldr__filter,axiom,
! [A: $tType,B: $tType,F2: B > A > A,P: B > $o,Xs: list @ B] :
( ( foldr @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) )
= ( foldr @ B @ A
@ ^ [X: B] : ( if @ ( A > A ) @ ( P @ X ) @ ( F2 @ X ) @ ( id @ A ) )
@ Xs ) ) ).
% foldr_filter
thf(fact_7624_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( ^ [Xs2: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs2 @ ( zero_zero @ A ) ) ) ) ) ).
% sum_list.eq_foldr
thf(fact_7625_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F3: B > A,A4: A,Xs2: list @ B] :
( foldr @ B @ A
@ ^ [X: B,B4: A] : ( plus_plus @ A @ ( F3 @ X ) @ ( times_times @ A @ A4 @ B4 ) )
@ Xs2
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_7626_length__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( foldr @ ( list @ A ) @ nat
@ ^ [Xs2: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) )
@ Xs
@ ( zero_zero @ nat ) ) ) ).
% length_transpose
thf(fact_7627_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F2: nat > B,Ns: list @ nat] :
( ! [X5: nat,Y4: nat] :
( ( ord_less_eq @ nat @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F2 @ Ns ) ) ) ) ) ) ).
% sorted_wrt_less_sum_mono_lowerbound
thf(fact_7628_length__product__lists,axiom,
! [B: $tType,Xss: list @ ( list @ B )] :
( ( size_size @ ( list @ ( list @ B ) ) @ ( product_lists @ B @ Xss ) )
= ( foldr @ nat @ nat @ ( times_times @ nat ) @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) @ ( one_one @ nat ) ) ) ).
% length_product_lists
thf(fact_7629_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,G: ( list @ B ) > A,Xs: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F2
@ ( filter2 @ B
@ ^ [X: B] :
( ( F2 @ X )
= ( G @ Xs ) )
@ Xs ) ) ) ) ).
% sorted_map_same
thf(fact_7630_sorted__map,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X: B,Y2: B] : ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y2 ) )
@ Xs ) ) ) ).
% sorted_map
thf(fact_7631_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,P: B > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ).
% sorted_filter
thf(fact_7632_sorted__wrt__map,axiom,
! [A: $tType,B: $tType,R: A > A > $o,F2: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ R @ ( map @ B @ A @ F2 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X: B,Y2: B] : ( R @ ( F2 @ X ) @ ( F2 @ Y2 ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_7633_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X3: B,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ).
% sorted_insort_key
thf(fact_7634_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,X3: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( remove1 @ B @ X3 @ Xs ) ) ) ) ) ).
% sorted_map_remove1
thf(fact_7635_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G: ( list @ A ) > A,Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter2 @ A
@ ^ [X: A] :
( X
= ( G @ Xs ) )
@ Xs ) ) ) ).
% sorted_same
thf(fact_7636_sorted__wrt__filter,axiom,
! [A: $tType,F2: A > A > $o,Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ F2 @ Xs )
=> ( sorted_wrt @ A @ F2 @ ( filter2 @ A @ P @ Xs ) ) ) ).
% sorted_wrt_filter
thf(fact_7637_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_7638_sorted__upt,axiom,
! [M: nat,N2: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M @ N2 ) ) ).
% sorted_upt
thf(fact_7639_sorted__wrt__upt,axiom,
! [M: nat,N2: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M @ N2 ) ) ).
% sorted_wrt_upt
thf(fact_7640_sorted__wrt__true,axiom,
! [A: $tType,Xs: list @ A] :
( sorted_wrt @ A
@ ^ [Uu3: A,Uv3: A] : $true
@ Xs ) ).
% sorted_wrt_true
thf(fact_7641_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_7642_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_7643_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X3
@ Xs ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted_insort
thf(fact_7644_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_7645_sorted__replicate,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N2: nat,X3: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N2 @ X3 ) ) ) ).
% sorted_replicate
thf(fact_7646_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_7647_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_7648_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_7649_sorted2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A,Zs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Zs2 ) ) )
= ( ( ord_less_eq @ A @ X3 @ Y )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y @ Zs2 ) ) ) ) ) ).
% sorted2
thf(fact_7650_sorted1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% sorted1
thf(fact_7651_sorted__wrt1,axiom,
! [A: $tType,P: A > A > $o,X3: A] : ( sorted_wrt @ A @ P @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% sorted_wrt1
thf(fact_7652_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ).
% strict_sorted_simps(1)
thf(fact_7653_sorted__wrt_Osimps_I1_J,axiom,
! [A: $tType,P: A > A > $o] : ( sorted_wrt @ A @ P @ ( nil @ A ) ) ).
% sorted_wrt.simps(1)
thf(fact_7654_sorted0,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% sorted0
thf(fact_7655_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_7656_sorted__take,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N2: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( take @ A @ N2 @ Xs ) ) ) ) ).
% sorted_take
thf(fact_7657_sorted__wrt__take,axiom,
! [A: $tType,F2: A > A > $o,Xs: list @ A,N2: nat] :
( ( sorted_wrt @ A @ F2 @ Xs )
=> ( sorted_wrt @ A @ F2 @ ( take @ A @ N2 @ Xs ) ) ) ).
% sorted_wrt_take
thf(fact_7658_sorted__drop,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N2: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( drop @ A @ N2 @ Xs ) ) ) ) ).
% sorted_drop
thf(fact_7659_sorted__wrt__drop,axiom,
! [A: $tType,F2: A > A > $o,Xs: list @ A,N2: nat] :
( ( sorted_wrt @ A @ F2 @ Xs )
=> ( sorted_wrt @ A @ F2 @ ( drop @ A @ N2 @ Xs ) ) ) ).
% sorted_wrt_drop
thf(fact_7660_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_7661_sorted__wrt__mono__rel,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o,Q: A > A > $o] :
( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) )
=> ( ( P @ X5 @ Y4 )
=> ( Q @ X5 @ Y4 ) ) ) )
=> ( ( sorted_wrt @ A @ P @ Xs )
=> ( sorted_wrt @ A @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_7662_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P3: A > A > $o,Xs2: list @ A] :
! [I3: nat,J3: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P3 @ ( nth @ A @ Xs2 @ I3 ) @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_7663_sorted__wrt__nth__less,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ P @ Xs )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_7664_sorted__wrt01,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_7665_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_7666_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_7667_sorted__wrt__append,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( ( sorted_wrt @ A @ P @ Xs )
& ( sorted_wrt @ A @ P @ Ys )
& ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Ys ) )
=> ( P @ X @ Y2 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_7668_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 )
& ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ).
% sorted_append
thf(fact_7669_sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X3 @ Ys ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X3 @ X ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys ) ) ) ) ).
% sorted_simps(2)
thf(fact_7670_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X3 @ Ys ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ord_less @ A @ X3 @ X ) )
& ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys ) ) ) ) ).
% strict_sorted_simps(2)
thf(fact_7671_sorted__wrt_Oelims_I3_J,axiom,
! [A: $tType,X3: A > A > $o,Xa2: list @ A] :
( ~ ( sorted_wrt @ A @ X3 @ Xa2 )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ Ys4 ) )
=> ( ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys4 ) )
=> ( X3 @ X5 @ Xa3 ) )
& ( sorted_wrt @ A @ X3 @ Ys4 ) ) ) ) ).
% sorted_wrt.elims(3)
thf(fact_7672_sorted__wrt_Osimps_I2_J,axiom,
! [A: $tType,P: A > A > $o,X3: A,Ys: list @ A] :
( ( sorted_wrt @ A @ P @ ( cons @ A @ X3 @ Ys ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( P @ X3 @ X ) )
& ( sorted_wrt @ A @ P @ Ys ) ) ) ).
% sorted_wrt.simps(2)
thf(fact_7673_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_7674_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I3: nat,J3: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_7675_sorted__wrt_Oelims_I1_J,axiom,
! [A: $tType,X3: A > A > $o,Xa2: list @ A,Y: $o] :
( ( ( sorted_wrt @ A @ X3 @ Xa2 )
= Y )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ~ Y )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ Ys4 ) )
=> ( Y
= ( ~ ( ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Ys4 ) )
=> ( X3 @ X5 @ Y2 ) )
& ( sorted_wrt @ A @ X3 @ Ys4 ) ) ) ) ) ) ) ).
% sorted_wrt.elims(1)
thf(fact_7676_sorted__wrt_Oelims_I2_J,axiom,
! [A: $tType,X3: A > A > $o,Xa2: list @ A] :
( ( sorted_wrt @ A @ X3 @ Xa2 )
=> ( ( Xa2
!= ( nil @ A ) )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ Ys4 ) )
=> ~ ( ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ( X3 @ X5 @ Xa ) )
& ( sorted_wrt @ A @ X3 @ Ys4 ) ) ) ) ) ).
% sorted_wrt.elims(2)
thf(fact_7677_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [X5: list @ A] :
( ( ( set2 @ A @ X5 )
= A3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X5 )
& ( distinct @ A @ X5 )
& ! [Y5: list @ A] :
( ( ( ( set2 @ A @ Y5 )
= A3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y5 )
& ( distinct @ A @ Y5 ) )
=> ( Y5 = X5 ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_7678_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_7679_filter__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,P: B > $o,X3: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
=> ( ( P @ X3 )
=> ( ( filter2 @ B @ P @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) )
= ( linorder_insort_key @ B @ A @ F2 @ X3 @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ) ).
% filter_insort
thf(fact_7680_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_7681_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
@ ^ [X: A] : X
@ A2
@ ( remove1 @ A @ A2 @ Xs ) )
= Xs ) ) ) ) ).
% insort_remove1
thf(fact_7682_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I3: nat] :
( ( ord_less @ nat @ ( suc @ I3 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ ( suc @ I3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_7683_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ~ ! [L4: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L4 )
=> ( ( ( set2 @ A @ L4 )
= A3 )
=> ( ( size_size @ ( list @ A ) @ L4 )
!= ( finite_card @ A @ A3 ) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_7684_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_7685_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_7686_sorted__wrt__less__idx,axiom,
! [Ns: list @ nat,I: nat] :
( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ nat ) @ Ns ) )
=> ( ord_less_eq @ nat @ I @ ( nth @ nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_7687_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ( ( find @ A @ P @ Xs )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( P @ X ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_7688_sorted__enumerate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) ) ) ).
% sorted_enumerate
thf(fact_7689_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,Ys: list @ B] :
( ( inj_on @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Ys ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F2 @ Ys ) )
=> ( ( ( set2 @ B @ Xs )
= ( set2 @ B @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_7690_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,L: list @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
& ( ( set2 @ A @ L )
= A3 )
& ( ( size_size @ ( list @ A ) @ L )
= ( finite_card @ A @ A3 ) ) )
= ( ( linord4507533701916653071of_set @ A @ A3 )
= L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_7691_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X5 @ A2 ) )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ A2
@ Xs )
= ( append @ A @ Xs @ ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_insort_is_snoc
thf(fact_7692_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [A2: B,Xs: list @ B,F2: B > A] :
( ( member @ B @ A2 @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
=> ( ( ( hd @ B
@ ( filter2 @ B
@ ^ [X: B] :
( ( F2 @ A2 )
= ( F2 @ X ) )
@ Xs ) )
= A2 )
=> ( ( linorder_insort_key @ B @ A @ F2 @ A2 @ ( remove1 @ B @ A2 @ Xs ) )
= Xs ) ) ) ) ) ).
% insort_key_remove1
thf(fact_7693_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat,J: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( ord_less @ nat @ J
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I ) @ J )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J ) @ I ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_7694_transpose__column,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( map @ ( list @ A ) @ A
@ ^ [Ys3: list @ A] : ( nth @ A @ Ys3 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ).
% transpose_column
thf(fact_7695_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_7696_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_7697_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rev @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rev_is_Nil_conv
thf(fact_7698_Nil__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( rev @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% Nil_is_rev_conv
thf(fact_7699_set__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rev @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rev
thf(fact_7700_length__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rev
thf(fact_7701_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_7702_distinct__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rev @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_rev
thf(fact_7703_rev__replicate,axiom,
! [A: $tType,N2: nat,X3: A] :
( ( rev @ A @ ( replicate @ A @ N2 @ X3 ) )
= ( replicate @ A @ N2 @ X3 ) ) ).
% rev_replicate
thf(fact_7704_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_7705_inj__on__rev,axiom,
! [A: $tType,A3: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( rev @ A ) @ A3 ) ).
% inj_on_rev
thf(fact_7706_singleton__rev__conv,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( ( cons @ A @ X3 @ ( nil @ A ) )
= ( rev @ A @ Xs ) )
= ( ( cons @ A @ X3 @ ( nil @ A ) )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_7707_rev__singleton__conv,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% rev_singleton_conv
thf(fact_7708_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_7709_foldl__conv__foldr,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F3: A > B > A,A4: A,Xs2: list @ B] :
( foldr @ B @ A
@ ^ [X: B,Y2: A] : ( F3 @ Y2 @ X )
@ ( rev @ B @ Xs2 )
@ A4 ) ) ) ).
% foldl_conv_foldr
thf(fact_7710_foldr__conv__foldl,axiom,
! [A: $tType,B: $tType] :
( ( foldr @ B @ A )
= ( ^ [F3: B > A > A,Xs2: list @ B,A4: A] :
( foldl @ A @ B
@ ^ [X: A,Y2: B] : ( F3 @ Y2 @ X )
@ A4
@ ( rev @ B @ Xs2 ) ) ) ) ).
% foldr_conv_foldl
thf(fact_7711_sorted__wrt__rev,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( sorted_wrt @ A @ P @ ( rev @ A @ Xs ) )
= ( sorted_wrt @ A
@ ^ [X: A,Y2: A] : ( P @ Y2 @ X )
@ Xs ) ) ).
% sorted_wrt_rev
thf(fact_7712_sorted__upto,axiom,
! [M: int,N2: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M @ N2 ) ) ).
% sorted_upto
thf(fact_7713_sorted__wrt__upto,axiom,
! [I: int,J: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I @ J ) ) ).
% sorted_wrt_upto
thf(fact_7714_rev_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rev @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rev.simps(1)
thf(fact_7715_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= Ys )
= ( Xs
= ( rev @ A @ Ys ) ) ) ).
% rev_swap
thf(fact_7716_rev_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( rev @ A @ ( cons @ A @ X3 @ Xs ) )
= ( append @ A @ ( rev @ A @ Xs ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% rev.simps(2)
thf(fact_7717_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_7718_rev__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( rev @ A @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ ( rev @ A @ Xs ) ) ) ).
% rev_filter
thf(fact_7719_rev__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( rev @ A @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( rev @ B @ Xs ) ) ) ).
% rev_map
thf(fact_7720_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_7721_take__rev,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( take @ A @ N2 @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) @ Xs ) ) ) ).
% take_rev
thf(fact_7722_rev__take,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( rev @ A @ ( take @ A @ I @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I ) @ ( rev @ A @ Xs ) ) ) ).
% rev_take
thf(fact_7723_rev__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( rev @ A @ ( drop @ A @ I @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I ) @ ( rev @ A @ Xs ) ) ) ).
% rev_drop
thf(fact_7724_drop__rev,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( drop @ A @ N2 @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) @ Xs ) ) ) ).
% drop_rev
thf(fact_7725_rev__nth,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rev @ A @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N2 ) ) ) ) ) ).
% rev_nth
thf(fact_7726_rev__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,Y: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs @ K @ Y ) )
= ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) @ ( one_one @ nat ) ) @ Y ) ) ) ).
% rev_update
thf(fact_7727_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_7728_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I3: nat] :
( ( ord_less @ nat @ ( suc @ I3 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I3 ) ) @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_7729_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_7730_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J3 ) @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_7731_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_7732_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_7733_transpose__column__length,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ) ).
% transpose_column_length
thf(fact_7734_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ~ ! [L4: list @ B] :
( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F2 @ L4 ) )
=> ( ( ( set2 @ B @ L4 )
= A3 )
=> ( ( size_size @ ( list @ B ) @ L4 )
!= ( finite_card @ B @ A3 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_7735_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 )
@ ^ [X: list @ A] :
( X
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_transpose
thf(fact_7736_takeWhile__idem,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( takeWhile @ A @ P @ ( takeWhile @ A @ P @ Xs ) )
= ( takeWhile @ A @ P @ Xs ) ) ).
% takeWhile_idem
thf(fact_7737_takeWhile__eq__all__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( takeWhile @ A @ P @ Xs )
= Xs )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) ) ) ) ).
% takeWhile_eq_all_conv
thf(fact_7738_takeWhile__append1,axiom,
! [A: $tType,X3: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X3 )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% takeWhile_append1
thf(fact_7739_takeWhile__append2,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( takeWhile @ A @ P @ Ys ) ) ) ) ).
% takeWhile_append2
thf(fact_7740_takeWhile__replicate,axiom,
! [A: $tType,P: A > $o,X3: A,N2: nat] :
( ( ( P @ X3 )
=> ( ( takeWhile @ A @ P @ ( replicate @ A @ N2 @ X3 ) )
= ( replicate @ A @ N2 @ X3 ) ) )
& ( ~ ( P @ X3 )
=> ( ( takeWhile @ A @ P @ ( replicate @ A @ N2 @ X3 ) )
= ( nil @ A ) ) ) ) ).
% takeWhile_replicate
thf(fact_7741_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_7742_sorted__takeWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% sorted_takeWhile
thf(fact_7743_nth__length__takeWhile,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P @ ( nth @ A @ Xs @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_7744_takeWhile__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) )
=> ( ( nth @ A @ ( takeWhile @ A @ P @ Xs ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% takeWhile_nth
thf(fact_7745_takeWhile__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] :
( ( takeWhile @ A @ P @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( takeWhile @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ).
% takeWhile_map
thf(fact_7746_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) )
=> ( distinct @ B @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_7747_zip__takeWhile__snd,axiom,
! [A: $tType,B: $tType,Xs: list @ A,P: B > $o,Ys: list @ B] :
( ( zip @ A @ B @ Xs @ ( takeWhile @ B @ P @ Ys ) )
= ( takeWhile @ ( product_prod @ A @ B ) @ ( comp @ B @ $o @ ( product_prod @ A @ B ) @ P @ ( product_snd @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_takeWhile_snd
thf(fact_7748_zip__takeWhile__fst,axiom,
! [A: $tType,B: $tType,P: A > $o,Xs: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( takeWhile @ A @ P @ Xs ) @ Ys )
= ( takeWhile @ ( product_prod @ A @ B ) @ ( comp @ A @ $o @ ( product_prod @ A @ B ) @ P @ ( product_fst @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_takeWhile_fst
thf(fact_7749_takeWhile__tail,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A,L: list @ A] :
( ~ ( P @ X3 )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ ( cons @ A @ X3 @ L ) ) )
= ( takeWhile @ A @ P @ Xs ) ) ) ).
% takeWhile_tail
thf(fact_7750_folding__insort__key_Oinj__on,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( inj_on @ B @ A @ F2 @ S3 ) ) ).
% folding_insort_key.inj_on
thf(fact_7751_takeWhile_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ( ( P @ X3 )
=> ( ( takeWhile @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( takeWhile @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X3 )
=> ( ( takeWhile @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( nil @ A ) ) ) ) ).
% takeWhile.simps(2)
thf(fact_7752_takeWhile_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( takeWhile @ A @ P @ ( nil @ A ) )
= ( nil @ A ) ) ).
% takeWhile.simps(1)
thf(fact_7753_set__takeWhileD,axiom,
! [A: $tType,X3: A,P: A > $o,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( takeWhile @ A @ P @ Xs ) ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) ) ).
% set_takeWhileD
thf(fact_7754_takeWhile__cong,axiom,
! [A: $tType,L: list @ A,K: list @ A,P: A > $o,Q: A > $o] :
( ( L = K )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ L ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( takeWhile @ A @ P @ L )
= ( takeWhile @ A @ Q @ K ) ) ) ) ).
% takeWhile_cong
thf(fact_7755_takeWhile__eq__take,axiom,
! [A: $tType] :
( ( takeWhile @ A )
= ( ^ [P3: A > $o,Xs2: list @ A] : ( take @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs2 ) ) @ Xs2 ) ) ) ).
% takeWhile_eq_take
thf(fact_7756_distinct__takeWhile,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( takeWhile @ A @ P @ Xs ) ) ) ).
% distinct_takeWhile
thf(fact_7757_length__takeWhile__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_takeWhile_le
thf(fact_7758_takeWhile__eq__Nil__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( takeWhile @ A @ P @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ~ ( P @ ( hd @ A @ Xs ) ) ) ) ).
% takeWhile_eq_Nil_iff
thf(fact_7759_takeWhile__append,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( takeWhile @ A @ P @ Ys ) ) ) )
& ( ~ ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% takeWhile_append
thf(fact_7760_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ J )
=> ( P @ ( nth @ A @ Xs @ I4 ) ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_7761_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N2: nat,Xs: list @ A,P: A > $o] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
=> ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I4 ) ) ) )
=> ( ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P @ ( nth @ A @ Xs @ N2 ) ) )
=> ( ( takeWhile @ A @ P @ Xs )
= ( take @ A @ N2 @ Xs ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_7762_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 ) )
@ ^ [X: A] : X ) ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_7763_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,T2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F2 @ Xs ) ) )
=> ( ( filter2 @ B
@ ^ [X: B] : ( ord_less @ A @ T2 @ ( F2 @ X ) )
@ Xs )
= ( takeWhile @ B
@ ^ [X: B] : ( ord_less @ A @ T2 @ ( F2 @ X ) )
@ Xs ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_7764_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B,L: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F2 @ L ) )
& ( ( set2 @ B @ L )
= A3 )
& ( ( size_size @ ( list @ B ) @ L )
= ( finite_card @ B @ A3 ) ) )
= ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_7765_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,X3: B,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X3 @ A3 ) @ S3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( remove1 @ B @ X3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
thf(fact_7766_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_7767_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B,B3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ S3 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 )
= ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ B3 ) )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( A3 = B3 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_inject
thf(fact_7768_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( bot_bot @ ( set @ B ) ) )
= ( nil @ B ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_empty
thf(fact_7769_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( set2 @ B @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) )
= A3 ) ) ) ) ).
% folding_insort_key.set_sorted_key_list_of_set
thf(fact_7770_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( ( size_size @ ( list @ B ) @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) )
= ( finite_card @ B @ A3 ) ) ) ) ).
% folding_insort_key.length_sorted_key_list_of_set
thf(fact_7771_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( distinct @ A @ ( map @ B @ A @ F2 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_7772_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F2 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_7773_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( sorted_wrt @ A @ Less_eq2 @ ( map @ B @ A @ F2 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_7774_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 )
= ( nil @ B ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
thf(fact_7775_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( set2 @ B @ Xs ) @ S3 )
=> ( ( sorted_wrt @ A @ Less_eq2 @ ( map @ B @ A @ F2 @ Xs ) )
=> ( ( distinct @ B @ Xs )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( set2 @ B @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_7776_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,X3: B,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X3 @ A3 ) @ S3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( insert @ B @ X3 @ A3 ) )
= ( insort_key @ A @ B @ Less_eq2 @ F2 @ X3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
thf(fact_7777_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,X3: B,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X3 @ A3 ) @ S3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X3 @ A3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( insert @ B @ X3 @ A3 ) )
= ( insort_key @ A @ B @ Less_eq2 @ F2 @ X3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert
thf(fact_7778_linorder_Oinsort__key_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( insort_key @ A @ B )
= ( insort_key @ A @ B ) ) ).
% linorder.insort_key.cong
thf(fact_7779_folding__insort__key_Oinsort__key__commute,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,X3: B,Y: B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 )
=> ( ( member @ B @ X3 @ S3 )
=> ( ( member @ B @ Y @ S3 )
=> ( ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq2 @ F2 @ Y ) @ ( insort_key @ A @ B @ Less_eq2 @ F2 @ X3 ) )
= ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq2 @ F2 @ X3 ) @ ( insort_key @ A @ B @ Less_eq2 @ F2 @ Y ) ) ) ) ) ) ).
% folding_insort_key.insort_key_commute
thf(fact_7780_extract__def,axiom,
! [A: $tType] :
( ( extract @ A )
= ( ^ [P3: A > $o,Xs2: list @ A] :
( case_list @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ A @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y2: A,Ys3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( takeWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs2 ) @ ( product_Pair @ A @ ( list @ A ) @ Y2 @ Ys3 ) ) )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs2 ) ) ) ) ).
% extract_def
thf(fact_7781_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_7782_dropWhile__idem,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( dropWhile @ A @ P @ ( dropWhile @ A @ P @ Xs ) )
= ( dropWhile @ A @ P @ Xs ) ) ).
% dropWhile_idem
thf(fact_7783_dropWhile__eq__Nil__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P @ Xs )
= ( nil @ A ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) ) ) ) ).
% dropWhile_eq_Nil_conv
thf(fact_7784_dropWhile__append2,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( dropWhile @ A @ P @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_7785_dropWhile__append1,axiom,
! [A: $tType,X3: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X3 )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_7786_dropWhile__replicate,axiom,
! [A: $tType,P: A > $o,X3: A,N2: nat] :
( ( ( P @ X3 )
=> ( ( dropWhile @ A @ P @ ( replicate @ A @ N2 @ X3 ) )
= ( nil @ A ) ) )
& ( ~ ( P @ X3 )
=> ( ( dropWhile @ A @ P @ ( replicate @ A @ N2 @ X3 ) )
= ( replicate @ A @ N2 @ X3 ) ) ) ) ).
% dropWhile_replicate
thf(fact_7787_takeWhile__dropWhile__id,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( append @ A @ ( takeWhile @ A @ P @ Xs ) @ ( dropWhile @ A @ P @ Xs ) )
= Xs ) ).
% takeWhile_dropWhile_id
thf(fact_7788_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: 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 @ Zs2 ) ) @ ( lexord @ A @ R2 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_7789_sorted__dropWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( dropWhile @ A @ P @ Xs ) ) ) ) ).
% sorted_dropWhile
thf(fact_7790_hd__dropWhile,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P @ Xs )
!= ( nil @ A ) )
=> ~ ( P @ ( hd @ A @ ( dropWhile @ A @ P @ Xs ) ) ) ) ).
% hd_dropWhile
thf(fact_7791_dropWhile__eq__self__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P @ Xs )
= Xs )
= ( ( Xs
= ( nil @ A ) )
| ~ ( P @ ( hd @ A @ Xs ) ) ) ) ).
% dropWhile_eq_self_iff
thf(fact_7792_length__dropWhile__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_dropWhile_le
thf(fact_7793_distinct__dropWhile,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( dropWhile @ A @ P @ Xs ) ) ) ).
% distinct_dropWhile
thf(fact_7794_set__dropWhileD,axiom,
! [A: $tType,X3: A,P: A > $o,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( dropWhile @ A @ P @ Xs ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% set_dropWhileD
thf(fact_7795_dropWhile__cong,axiom,
! [A: $tType,L: list @ A,K: list @ A,P: A > $o,Q: A > $o] :
( ( L = K )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ L ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( dropWhile @ A @ P @ L )
= ( dropWhile @ A @ Q @ K ) ) ) ) ).
% dropWhile_cong
thf(fact_7796_irreflI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R )
=> ( irrefl @ A @ R ) ) ).
% irreflI
thf(fact_7797_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [A4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R5 ) ) ) ).
% irrefl_def
thf(fact_7798_dropWhile_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( dropWhile @ A @ P @ ( nil @ A ) )
= ( nil @ A ) ) ).
% dropWhile.simps(1)
thf(fact_7799_dropWhile_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ( ( P @ X3 )
=> ( ( dropWhile @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( dropWhile @ A @ P @ Xs ) ) )
& ( ~ ( P @ X3 )
=> ( ( dropWhile @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ Xs ) ) ) ) ).
% dropWhile.simps(2)
thf(fact_7800_remdups__adj__Cons_H,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y2: A] : Y2 = X3
@ Xs ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_7801_irrefl__lex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R2 )
=> ( irrefl @ ( list @ A ) @ ( lex @ A @ R2 ) ) ) ).
% irrefl_lex
thf(fact_7802_lexord__irrefl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R )
=> ( irrefl @ ( list @ A ) @ ( lexord @ A @ R ) ) ) ).
% lexord_irrefl
thf(fact_7803_dropWhile__append3,axiom,
! [A: $tType,P: A > $o,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( P @ Y )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% dropWhile_append3
thf(fact_7804_dropWhile__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] :
( ( dropWhile @ A @ P @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( dropWhile @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ).
% dropWhile_map
thf(fact_7805_dropWhile__eq__Cons__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ( dropWhile @ A @ P @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( ( Xs
= ( append @ A @ ( takeWhile @ A @ P @ Xs ) @ ( cons @ A @ Y @ Ys ) ) )
& ~ ( P @ Y ) ) ) ).
% dropWhile_eq_Cons_conv
thf(fact_7806_takeWhile__eq__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ ( dropWhile @ A @ P @ Xs ) ) )
=> ~ ( P @ X5 ) )
=> ( ( takeWhile @ A @ P @ Xs )
= ( filter2 @ A @ P @ Xs ) ) ) ).
% takeWhile_eq_filter
thf(fact_7807_dropWhile__eq__drop,axiom,
! [A: $tType] :
( ( dropWhile @ A )
= ( ^ [P3: A > $o,Xs2: list @ A] : ( drop @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs2 ) ) @ Xs2 ) ) ) ).
% dropWhile_eq_drop
thf(fact_7808_dropWhile__append,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( dropWhile @ A @ P @ Ys ) ) )
& ( ~ ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append
thf(fact_7809_lexl__not__refl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: list @ A] :
( ( irrefl @ A @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ X3 ) @ ( lex @ A @ R2 ) ) ) ).
% lexl_not_refl
thf(fact_7810_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P3: A > $o,Xs2: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X: A,Xa4: list @ A] : ( some @ A @ X )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs2 ) ) ) ) ).
% find_dropWhile
thf(fact_7811_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
@ ^ [X: A] : X = Y
@ Ys ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_7812_tl__remdups__adj,axiom,
! [A: $tType,Ys: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( tl @ A @ ( remdups_adj @ A @ Ys ) )
= ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X: A] :
( X
= ( hd @ A @ Ys ) )
@ ( tl @ A @ Ys ) ) ) ) ) ).
% tl_remdups_adj
thf(fact_7813_dropWhile__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
=> ( ( nth @ A @ ( dropWhile @ A @ P @ Xs ) @ J )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% dropWhile_nth
thf(fact_7814_dropWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( dropWhile @ A
@ ^ [Y2: A] : Y2 != X3
@ ( rev @ A @ Xs ) )
= ( cons @ A @ X3
@ ( rev @ A
@ ( takeWhile @ A
@ ^ [Y2: A] : Y2 != X3
@ Xs ) ) ) ) ) ) ).
% dropWhile_neq_rev
thf(fact_7815_takeWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( takeWhile @ A
@ ^ [Y2: A] : Y2 != X3
@ ( rev @ A @ Xs ) )
= ( rev @ A
@ ( tl @ A
@ ( dropWhile @ A
@ ^ [Y2: A] : Y2 != X3
@ Xs ) ) ) ) ) ) ).
% takeWhile_neq_rev
thf(fact_7816_partition__filter__conv,axiom,
! [A: $tType] :
( ( partition @ A )
= ( ^ [F3: A > $o,Xs2: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ F3 @ Xs2 ) @ ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ F3 ) @ Xs2 ) ) ) ) ).
% partition_filter_conv
thf(fact_7817_min__list_Oelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: list @ A,Y: A] :
( ( ( min_list @ A @ X3 )
= Y )
=> ( ! [X5: A,Xs3: list @ A] :
( ( X3
= ( cons @ A @ X5 @ Xs3 ) )
=> ( Y
!= ( case_list @ A @ A @ X5
@ ^ [A4: A,List2: list @ A] : ( ord_min @ A @ X5 @ ( min_list @ A @ Xs3 ) )
@ Xs3 ) ) )
=> ~ ( ( X3
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ).
% min_list.elims
thf(fact_7818_hd__def,axiom,
! [A: $tType] :
( ( hd @ A )
= ( case_list @ A @ A @ ( undefined @ A )
@ ^ [X213: A,X224: list @ A] : X213 ) ) ).
% hd_def
thf(fact_7819_option_Othe__def,axiom,
! [A: $tType] :
( ( the2 @ A )
= ( case_option @ A @ A @ ( undefined @ A )
@ ^ [X24: A] : X24 ) ) ).
% option.the_def
thf(fact_7820_partition__filter1,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( partition @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ).
% partition_filter1
thf(fact_7821_partition_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( partition @ A @ P @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) ) ).
% partition.simps(1)
thf(fact_7822_partition__P,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Yes: list @ A,No4: list @ A] :
( ( ( partition @ A @ P @ Xs )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No4 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Yes ) )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ No4 ) )
=> ~ ( P @ X2 ) ) ) ) ).
% partition_P
thf(fact_7823_partition_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ( partition @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ^ [Yes2: list @ A,No3: list @ A] : ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Yes2 ) @ No3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ ( cons @ A @ X3 @ No3 ) ) )
@ ( partition @ A @ P @ Xs ) ) ) ).
% partition.simps(2)
thf(fact_7824_partition__filter2,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( partition @ A @ P @ Xs ) )
= ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ Xs ) ) ).
% partition_filter2
thf(fact_7825_Func__empty,axiom,
! [B: $tType,A: $tType,B3: set @ B] :
( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B3 )
= ( insert @ ( A > B )
@ ^ [X: A] : ( undefined @ B )
@ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ).
% Func_empty
thf(fact_7826_partition__set,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Yes: list @ A,No4: list @ A] :
( ( ( partition @ A @ P @ 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_7827_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X3: A > B,Xa2: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X3 @ Xa2 )
= Y )
=> ( ! [X5: A] :
( ( Xa2
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( Y != X5 ) )
=> ( ! [X5: A,Y4: A,Zs: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Zs ) ) )
=> ( Y
!= ( if @ A @ ( ord_less_eq @ B @ ( X3 @ X5 ) @ ( X3 @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y4 @ Zs ) ) ) ) @ X5 @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y4 @ Zs ) ) ) ) )
=> ~ ( ( Xa2
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_7828_min__list_Opelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: list @ A,Y: A] :
( ( ( min_list @ A @ X3 )
= Y )
=> ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X3 )
=> ( ! [X5: A,Xs3: list @ A] :
( ( X3
= ( cons @ A @ X5 @ Xs3 ) )
=> ( ( Y
= ( case_list @ A @ A @ X5
@ ^ [A4: A,List2: list @ A] : ( ord_min @ A @ X5 @ ( min_list @ A @ Xs3 ) )
@ Xs3 ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( cons @ A @ X5 @ Xs3 ) ) ) )
=> ~ ( ( X3
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ).
% min_list.pelims
thf(fact_7829_lists__length__Suc__eq,axiom,
! [A: $tType,A3: set @ A,N2: nat] :
( ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N2 ) ) ) )
= ( image @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs2: list @ A,N: A] : ( cons @ A @ N @ Xs2 ) )
@ ( product_Sigma @ ( list @ A ) @ A
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= N2 ) ) )
@ ^ [Uu3: list @ A] : A3 ) ) ) ).
% lists_length_Suc_eq
thf(fact_7830_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
= ( ( member @ A @ A2 @ A3 )
& ( member @ B @ B2 @ ( B3 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_7831_SigmaI,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,B2: B,B3: A > ( set @ B )] :
( ( member @ A @ A2 @ A3 )
=> ( ( member @ B @ B2 @ ( B3 @ A2 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) ) ) ).
% SigmaI
thf(fact_7832_Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A4: A,B4: B] :
( ( P @ A4 )
& ( Q @ B4 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [Uu3: A] : ( collect @ B @ Q ) ) ) ).
% Collect_case_prod
thf(fact_7833_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B3: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B3 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_7834_Compl__Times__UNIV2,axiom,
! [B: $tType,A: $tType,A3: set @ A] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) )
= ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A3 )
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) ) ).
% Compl_Times_UNIV2
thf(fact_7835_Compl__Times__UNIV1,axiom,
! [B: $tType,A: $tType,A3: set @ B] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : A3 ) )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : ( uminus_uminus @ ( set @ B ) @ A3 ) ) ) ).
% Compl_Times_UNIV1
thf(fact_7836_Times__empty,axiom,
! [A: $tType,B: $tType,A3: set @ A,B3: set @ B] :
( ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B3
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_7837_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A3: set @ A] :
( ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_7838_finite__SigmaI,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A3 )
=> ( finite_finite2 @ B @ ( B3 @ A5 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) ) ) ).
% finite_SigmaI
thf(fact_7839_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_7840_fst__image__times,axiom,
! [B: $tType,A: $tType,B3: set @ B,A3: set @ A] :
( ( ( B3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 ) )
= A3 ) ) ) ).
% fst_image_times
thf(fact_7841_snd__image__times,axiom,
! [B: $tType,A: $tType,A3: set @ B,B3: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu3: B] : B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu3: B] : B3 ) )
= B3 ) ) ) ).
% snd_image_times
thf(fact_7842_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_7843_insert__Times__insert,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,B2: B,B3: set @ B] :
( ( product_Sigma @ A @ B @ ( insert @ A @ A2 @ A3 )
@ ^ [Uu3: A] : ( insert @ B @ B2 @ B3 ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 )
@ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : ( insert @ B @ B2 @ B3 ) )
@ ( product_Sigma @ A @ B @ ( insert @ A @ A2 @ A3 )
@ ^ [Uu3: A] : B3 ) ) ) ) ).
% insert_Times_insert
thf(fact_7844_card__SigmaI,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( finite_finite2 @ B @ ( B3 @ X5 ) ) )
=> ( ( finite_card @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [A4: A] : ( finite_card @ B @ ( B3 @ A4 ) )
@ A3 ) ) ) ) ).
% card_SigmaI
thf(fact_7845_inj__on__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > C,A3: set @ A] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) )
= ( inj_on @ A @ C @ F2 @ A3 ) ) ).
% inj_on_apfst
thf(fact_7846_inj__on__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F2: B > C,A3: set @ B] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : A3 ) )
= ( inj_on @ B @ C @ F2 @ A3 ) ) ).
% inj_on_apsnd
thf(fact_7847_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] :
( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
= ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A3 )
& ( ( B3 @ X )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_7848_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),A3: 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 @ A3
@ ^ [Uu3: A] : A3 ) )
=> ( ( A2 = B2 )
| ( member @ A @ A2 @ A3 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_7849_SigmaE2,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
=> ~ ( ( member @ A @ A2 @ A3 )
=> ~ ( member @ B @ B2 @ ( B3 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_7850_SigmaD2,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
=> ( member @ B @ B2 @ ( B3 @ A2 ) ) ) ).
% SigmaD2
thf(fact_7851_SigmaD1,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
=> ( member @ A @ A2 @ A3 ) ) ).
% SigmaD1
thf(fact_7852_SigmaE,axiom,
! [A: $tType,B: $tType,C2: product_prod @ A @ B,A3: set @ A,B3: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ! [Y4: B] :
( ( member @ B @ Y4 @ ( B3 @ X5 ) )
=> ( C2
!= ( product_Pair @ A @ B @ X5 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_7853_times__eq__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,B3: set @ B,C5: set @ A,D6: set @ B] :
( ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 )
= ( product_Sigma @ A @ B @ C5
@ ^ [Uu3: A] : D6 ) )
= ( ( ( A3 = C5 )
& ( B3 = D6 ) )
| ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B3
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C5
= ( bot_bot @ ( set @ A ) ) )
| ( D6
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_7854_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I6: set @ A,X8: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I6 @ X8 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ I6 )
=> ( ( X8 @ X )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_7855_Sigma__Union,axiom,
! [B: $tType,A: $tType,X8: set @ ( set @ A ),B3: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ X8 ) @ B3 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image @ ( set @ A ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A6: set @ A] : ( product_Sigma @ A @ B @ A6 @ B3 )
@ X8 ) ) ) ).
% Sigma_Union
thf(fact_7856_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X3: A,C5: set @ A,A3: set @ B,B3: set @ B] :
( ( member @ A @ X3 @ C5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu3: B] : C5 )
@ ( product_Sigma @ B @ A @ B3
@ ^ [Uu3: B] : C5 ) )
= ( ord_less_eq @ ( set @ B ) @ A3 @ B3 ) ) ) ).
% Times_subset_cancel2
thf(fact_7857_Sigma__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,C5: set @ A,B3: A > ( set @ B ),D6: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( B3 @ X5 ) @ ( D6 @ X5 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ C5 @ D6 ) ) ) ) ).
% Sigma_mono
thf(fact_7858_Restr__subset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B3
@ ^ [Uu3: A] : B3 ) )
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) ) ) ) ).
% Restr_subset
thf(fact_7859_Times__Un__distrib1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A3 @ B3 )
@ ^ [Uu3: A] : C5 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B3
@ ^ [Uu3: A] : C5 ) ) ) ).
% Times_Un_distrib1
thf(fact_7860_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B ),B3: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I6
@ ^ [I3: A] : ( sup_sup @ ( set @ B ) @ ( A3 @ I3 ) @ ( B3 @ I3 ) ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I6 @ A3 ) @ ( product_Sigma @ A @ B @ I6 @ B3 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_7861_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_7862_member__product,axiom,
! [B: $tType,A: $tType,X3: product_prod @ A @ B,A3: set @ A,B3: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X3 @ ( product_product @ A @ B @ A3 @ B3 ) )
= ( member @ ( product_prod @ A @ B ) @ X3
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 ) ) ) ).
% member_product
thf(fact_7863_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A6: set @ A,B6: set @ B] :
( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 ) ) ) ).
% Product_Type.product_def
thf(fact_7864_finite__cartesian__product,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 ) ) ) ) ).
% finite_cartesian_product
thf(fact_7865_Times__Int__distrib1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A3 @ B3 )
@ ^ [Uu3: A] : C5 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B3
@ ^ [Uu3: A] : C5 ) ) ) ).
% Times_Int_distrib1
thf(fact_7866_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B ),B3: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I6
@ ^ [I3: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ I3 ) @ ( B3 @ I3 ) ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I6 @ A3 ) @ ( product_Sigma @ A @ B @ I6 @ B3 ) ) ) ).
% Sigma_Int_distrib2
thf(fact_7867_Times__Int__Times,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,C5: set @ A,D6: set @ B] :
( ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 )
@ ( product_Sigma @ A @ B @ C5
@ ^ [Uu3: A] : D6 ) )
= ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A3 @ C5 )
@ ^ [Uu3: A] : ( inf_inf @ ( set @ B ) @ B3 @ D6 ) ) ) ).
% Times_Int_Times
thf(fact_7868_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_7869_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_7870_Sigma__cong,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C5: A > ( set @ B ),D6: A > ( set @ B )] :
( ( A3 = B3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B3 )
=> ( ( C5 @ X5 )
= ( D6 @ X5 ) ) )
=> ( ( product_Sigma @ A @ B @ A3 @ C5 )
= ( product_Sigma @ A @ B @ B3 @ D6 ) ) ) ) ).
% Sigma_cong
thf(fact_7871_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X3: A,C5: set @ A,A3: set @ B,B3: set @ B] :
( ( member @ A @ X3 @ C5 )
=> ( ( ( product_Sigma @ B @ A @ A3
@ ^ [Uu3: B] : C5 )
= ( product_Sigma @ B @ A @ B3
@ ^ [Uu3: B] : C5 ) )
= ( A3 = B3 ) ) ) ).
% Times_eq_cancel2
thf(fact_7872_Times__Diff__distrib1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A3 @ B3 )
@ ^ [Uu3: A] : C5 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B3
@ ^ [Uu3: A] : C5 ) ) ) ).
% Times_Diff_distrib1
thf(fact_7873_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B ),B3: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I6
@ ^ [I3: A] : ( minus_minus @ ( set @ B ) @ ( A3 @ I3 ) @ ( B3 @ I3 ) ) )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I6 @ A3 ) @ ( product_Sigma @ A @ B @ I6 @ B3 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_7874_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X: A,Y2: B] :
( ( P @ X )
& ( Q @ X @ Y2 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [X: A] : ( collect @ B @ ( Q @ X ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_7875_swap__product,axiom,
! [B: $tType,A: $tType,A3: set @ B,B3: set @ A] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I3: B,J3: A] : ( product_Pair @ A @ B @ J3 @ I3 ) )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu3: B] : B3 ) )
= ( product_Sigma @ A @ B @ B3
@ ^ [Uu3: A] : A3 ) ) ).
% swap_product
thf(fact_7876_times__subset__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,C5: set @ B,B3: set @ A,D6: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B3
@ ^ [Uu3: A] : D6 ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( C5
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
& ( ord_less_eq @ ( set @ B ) @ C5 @ D6 ) ) ) ) ).
% times_subset_iff
thf(fact_7877_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X3: product_prod @ A @ B,A3: set @ A,B3: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X3
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 ) )
= ( ( member @ A @ ( product_fst @ A @ B @ X3 ) @ A3 )
& ( member @ B @ ( product_snd @ A @ B @ X3 ) @ B3 ) ) ) ).
% mem_Times_iff
thf(fact_7878_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F2: C > A,G: D > B,A3: set @ C,B3: set @ D] :
( ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X: C,Y2: D] : ( product_Pair @ A @ B @ ( F2 @ X ) @ ( G @ Y2 ) ) )
@ ( product_Sigma @ C @ D @ A3
@ ^ [Uu3: C] : B3 ) )
= ( product_Sigma @ A @ B @ ( image @ C @ A @ F2 @ A3 )
@ ^ [Uu3: A] : ( image @ D @ B @ G @ B3 ) ) ) ).
% image_paired_Times
thf(fact_7879_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A3 )
& ( ( B3 @ X )
!= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A3 )
=> ( finite_finite2 @ B @ ( B3 @ A5 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) ) ) ).
% finite_SigmaI2
thf(fact_7880_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 ) )
=> ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_cartesian_productD1
thf(fact_7881_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A3: set @ A,B3: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( finite_finite2 @ B @ B3 ) ) ) ).
% finite_cartesian_productD2
thf(fact_7882_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,B3: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B3
= ( bot_bot @ ( set @ B ) ) )
| ( ( finite_finite2 @ A @ A3 )
& ( finite_finite2 @ B @ B3 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_7883_UN__Times__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,E5: C > ( set @ A ),F5: D > ( set @ B ),A3: set @ C,B3: set @ D] :
( ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ C @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A4: C,B4: D] :
( product_Sigma @ A @ B @ ( E5 @ A4 )
@ ^ [Uu3: A] : ( F5 @ B4 ) ) )
@ ( product_Sigma @ C @ D @ A3
@ ^ [Uu3: C] : B3 ) ) )
= ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ C @ ( set @ A ) @ E5 @ A3 ) )
@ ^ [Uu3: A] : ( complete_Sup_Sup @ ( set @ B ) @ ( image @ D @ ( set @ B ) @ F5 @ B3 ) ) ) ) ).
% UN_Times_distrib
thf(fact_7884_snd__image__Sigma,axiom,
! [A: $tType,B: $tType,A3: set @ B,B3: B > ( set @ A )] :
( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( product_Sigma @ B @ A @ A3 @ B3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% snd_image_Sigma
thf(fact_7885_subset__fst__imageI,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,S3: set @ ( product_prod @ A @ B ),Y: B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 )
@ S3 )
=> ( ( member @ B @ Y @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S3 ) ) ) ) ).
% subset_fst_imageI
thf(fact_7886_subset__snd__imageI,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,S3: set @ ( product_prod @ A @ B ),X3: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 )
@ S3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ B3 @ ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S3 ) ) ) ) ).
% subset_snd_imageI
thf(fact_7887_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3
@ ( product_Sigma @ A @ B @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 )
@ ^ [Uu3: A] : ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) ) ) ).
% subset_fst_snd
thf(fact_7888_card__cartesian__product__singleton,axiom,
! [A: $tType,B: $tType,X3: A,A3: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu3: A] : A3 ) )
= ( finite_card @ B @ A3 ) ) ).
% card_cartesian_product_singleton
thf(fact_7889_Sigma__interval__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( order @ A )
=> ! [A3: set @ B,V2: B > A,W2: A] :
( ( inf_inf @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A3
@ ^ [I3: B] : ( set_ord_atMost @ A @ ( V2 @ I3 ) ) )
@ ( product_Sigma @ B @ A @ A3
@ ^ [I3: B] : ( set_or3652927894154168847AtMost @ A @ ( V2 @ I3 ) @ W2 ) ) )
= ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) ) ).
% Sigma_interval_disjoint
thf(fact_7890_sum_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B3: B > ( set @ C ),G: B > C > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( finite_finite2 @ C @ ( B3 @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ X ) @ ( B3 @ X ) )
@ A3 )
= ( groups7311177749621191930dd_sum @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G ) @ ( product_Sigma @ B @ C @ A3 @ B3 ) ) ) ) ) ) ).
% sum.Sigma
thf(fact_7891_prod_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B3: B > ( set @ C ),G: B > C > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( finite_finite2 @ C @ ( B3 @ X5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ X ) @ ( B3 @ X ) )
@ A3 )
= ( groups7121269368397514597t_prod @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G ) @ ( product_Sigma @ B @ C @ A3 @ B3 ) ) ) ) ) ) ).
% prod.Sigma
thf(fact_7892_Sigma__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B )
= ( ^ [A6: set @ A,B6: A > ( set @ B )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X: A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y2: B] : ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
@ ( B6 @ X ) ) )
@ A6 ) ) ) ) ).
% Sigma_def
thf(fact_7893_product__fold,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X: A,Z6: set @ ( product_prod @ A @ B )] :
( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y2: B] : ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) )
@ Z6
@ B3 )
@ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
@ A3 ) ) ) ) ).
% product_fold
thf(fact_7894_uniformly__continuous__on__uniformity,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo7287701948861334536_space @ A )
& ( topolo7287701948861334536_space @ B ) )
=> ( ( topolo6026614971017936543ous_on @ A @ B )
= ( ^ [S8: set @ A,F3: A > B] :
( filterlim @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B )
@ ( product_case_prod @ A @ A @ ( product_prod @ B @ B )
@ ^ [X: A,Y2: A] : ( product_Pair @ B @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
@ ( 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_7895_sum__mult__sum__if__inj,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_0 @ B )
=> ! [F2: A > B,G: C > B,A3: set @ A,B3: set @ C] :
( ( inj_on @ ( product_prod @ A @ C ) @ B
@ ( product_case_prod @ A @ C @ B
@ ^ [A4: A,B4: C] : ( times_times @ B @ ( F2 @ A4 ) @ ( G @ B4 ) ) )
@ ( product_Sigma @ A @ C @ A3
@ ^ [Uu3: A] : B3 ) )
=> ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B3 ) )
= ( groups7311177749621191930dd_sum @ B @ B @ ( id @ B )
@ ( collect @ B
@ ^ [Uu3: B] :
? [A4: A,B4: C] :
( ( Uu3
= ( times_times @ B @ ( F2 @ A4 ) @ ( G @ B4 ) ) )
& ( member @ A @ A4 @ A3 )
& ( member @ C @ B4 @ B3 ) ) ) ) ) ) ) ).
% sum_mult_sum_if_inj
thf(fact_7896_infinite__cartesian__product,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ~ ( finite_finite2 @ B @ B3 )
=> ~ ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_7897_Gr__incl,axiom,
! [A: $tType,B: $tType,A3: set @ A,F2: A > B,B3: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A3 @ F2 )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B3 ) )
= ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B3 ) ) ).
% Gr_incl
thf(fact_7898_GrD2,axiom,
! [A: $tType,B: $tType,X3: A,Fx: B,A3: set @ A,F2: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Fx ) @ ( bNF_Gr @ A @ B @ A3 @ F2 ) )
=> ( ( F2 @ X3 )
= Fx ) ) ).
% GrD2
thf(fact_7899_GrD1,axiom,
! [B: $tType,A: $tType,X3: A,Fx: B,A3: set @ A,F2: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Fx ) @ ( bNF_Gr @ A @ B @ A3 @ F2 ) )
=> ( member @ A @ X3 @ A3 ) ) ).
% GrD1
thf(fact_7900_Gr__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr @ A @ B )
= ( ^ [A6: set @ A,F3: A > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A4: A] :
( ( Uu3
= ( product_Pair @ A @ B @ A4 @ ( F3 @ A4 ) ) )
& ( member @ A @ A4 @ A6 ) ) ) ) ) ).
% Gr_def
thf(fact_7901_pairs__le__eq__Sigma,axiom,
! [M: nat] :
( ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ M ) ) )
= ( product_Sigma @ nat @ nat @ ( set_ord_atMost @ nat @ M )
@ ^ [R5: nat] : ( set_ord_atMost @ nat @ ( minus_minus @ nat @ M @ R5 ) ) ) ) ).
% pairs_le_eq_Sigma
thf(fact_7902_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ A ),As4: A > B] :
! [I3: A,J3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I3 @ J3 ) @ R5 )
=> ( ord_less_eq @ B @ ( As4 @ I3 ) @ ( As4 @ J3 ) ) ) ) ) ) ).
% relChain_def
thf(fact_7903_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% natLess_def
thf(fact_7904_rotate__drop__take,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N: nat,Xs2: list @ A] : ( append @ A @ ( drop @ A @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ Xs2 ) @ ( take @ A @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ Xs2 ) ) ) ) ).
% rotate_drop_take
thf(fact_7905_image__split__eq__Sigma,axiom,
! [C: $tType,B: $tType,A: $tType,F2: C > A,G: C > B,A3: set @ C] :
( ( image @ C @ ( product_prod @ A @ B )
@ ^ [X: C] : ( product_Pair @ A @ B @ ( F2 @ X ) @ ( G @ X ) )
@ A3 )
= ( product_Sigma @ A @ B @ ( image @ C @ A @ F2 @ A3 )
@ ^ [X: A] : ( image @ C @ B @ G @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_7906_vimage__eq,axiom,
! [A: $tType,B: $tType,A2: A,F2: A > B,B3: set @ B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F2 @ B3 ) )
= ( member @ B @ ( F2 @ A2 ) @ B3 ) ) ).
% vimage_eq
thf(fact_7907_vimageI,axiom,
! [B: $tType,A: $tType,F2: B > A,A2: B,B2: A,B3: set @ A] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member @ A @ B2 @ B3 )
=> ( member @ B @ A2 @ ( vimage @ B @ A @ F2 @ B3 ) ) ) ) ).
% vimageI
thf(fact_7908_vimage__Collect__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,P: B > $o] :
( ( vimage @ A @ B @ F2 @ ( collect @ B @ P ) )
= ( collect @ A
@ ^ [Y2: A] : ( P @ ( F2 @ Y2 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_7909_vimage__ident,axiom,
! [A: $tType,Y7: set @ A] :
( ( vimage @ A @ A
@ ^ [X: A] : X
@ Y7 )
= Y7 ) ).
% vimage_ident
thf(fact_7910_vimage__UNIV,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( vimage @ A @ B @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% vimage_UNIV
thf(fact_7911_vimage__empty,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( vimage @ A @ B @ F2 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% vimage_empty
thf(fact_7912_vimage__Int,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ B,B3: set @ B] :
( ( vimage @ A @ B @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A3 ) @ ( vimage @ A @ B @ F2 @ B3 ) ) ) ).
% vimage_Int
thf(fact_7913_vimage__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ B,B3: set @ B] :
( ( vimage @ A @ B @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A3 ) @ ( vimage @ A @ B @ F2 @ B3 ) ) ) ).
% vimage_Un
thf(fact_7914_rotate__is__Nil__conv,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( rotate @ A @ N2 @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate_is_Nil_conv
thf(fact_7915_set__rotate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( set2 @ A @ ( rotate @ A @ N2 @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rotate
thf(fact_7916_length__rotate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate @ A @ N2 @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate
thf(fact_7917_distinct__rotate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( distinct @ A @ ( rotate @ A @ N2 @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_rotate
thf(fact_7918_vimage__const,axiom,
! [B: $tType,A: $tType,C2: B,A3: set @ B] :
( ( ( member @ B @ C2 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X: A] : C2
@ A3 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ C2 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X: A] : C2
@ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% vimage_const
thf(fact_7919_rotate__Suc,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( rotate @ A @ ( suc @ N2 ) @ Xs )
= ( rotate1 @ A @ ( rotate @ A @ N2 @ Xs ) ) ) ).
% rotate_Suc
thf(fact_7920_rotate0,axiom,
! [A: $tType] :
( ( rotate @ A @ ( zero_zero @ nat ) )
= ( id @ ( list @ A ) ) ) ).
% rotate0
thf(fact_7921_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ A] :
( ( image @ B @ A @ F2 @ ( vimage @ B @ A @ F2 @ A3 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% image_vimage_eq
thf(fact_7922_rotate__length01,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate @ A @ N2 @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_7923_rotate__id,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
= ( zero_zero @ nat ) )
=> ( ( rotate @ A @ N2 @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_7924_vimage__if,axiom,
! [B: $tType,A: $tType,C2: B,A3: set @ B,D2: B,B3: set @ A] :
( ( ( member @ B @ C2 @ A3 )
=> ( ( ( member @ B @ D2 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ B3 ) @ C2 @ D2 )
@ A3 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ D2 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ B3 ) @ C2 @ D2 )
@ A3 )
= B3 ) ) ) )
& ( ~ ( member @ B @ C2 @ A3 )
=> ( ( ( member @ B @ D2 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ B3 ) @ C2 @ D2 )
@ A3 )
= ( uminus_uminus @ ( set @ A ) @ B3 ) ) )
& ( ~ ( member @ B @ D2 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ B3 ) @ C2 @ D2 )
@ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% vimage_if
thf(fact_7925_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X3: B,A3: set @ B,F2: B > ( set @ A )] :
( ( ( member @ B @ X3 @ A3 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 ) @ ( product_Sigma @ B @ A @ A3 @ F2 ) )
= ( F2 @ X3 ) ) )
& ( ~ ( member @ B @ X3 @ A3 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 ) @ ( product_Sigma @ B @ A @ A3 @ F2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pair_vimage_Sigma
thf(fact_7926_vimage__snd,axiom,
! [B: $tType,A: $tType,A3: set @ B] :
( ( vimage @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : A3 ) ) ).
% vimage_snd
thf(fact_7927_vimage__fst,axiom,
! [B: $tType,A: $tType,A3: set @ A] :
( ( vimage @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 )
= ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) ) ).
% vimage_fst
thf(fact_7928_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > ( product_prod @ B @ C ),A3: set @ B,B3: set @ C] :
( ( vimage @ A @ ( product_prod @ B @ C ) @ F2
@ ( product_Sigma @ B @ C @ A3
@ ^ [Uu3: B] : B3 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F2 ) @ A3 ) @ ( vimage @ A @ C @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F2 ) @ B3 ) ) ) ).
% vimage_Times
thf(fact_7929_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A2: A,F2: A > B,B2: B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F2 @ ( insert @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( ( F2 @ A2 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_7930_vimage__insert,axiom,
! [A: $tType,B: $tType,F2: A > B,A2: B,B3: set @ B] :
( ( vimage @ A @ B @ F2 @ ( insert @ B @ A2 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( vimage @ A @ B @ F2 @ B3 ) ) ) ).
% vimage_insert
thf(fact_7931_vimage__Suc__insert__Suc,axiom,
! [N2: nat,A3: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( suc @ N2 ) @ A3 ) )
= ( insert @ nat @ N2 @ ( vimage @ nat @ nat @ suc @ A3 ) ) ) ).
% vimage_Suc_insert_Suc
thf(fact_7932_vimage__Suc__insert__0,axiom,
! [A3: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( zero_zero @ nat ) @ A3 ) )
= ( vimage @ nat @ nat @ suc @ A3 ) ) ).
% vimage_Suc_insert_0
thf(fact_7933_rotate__add,axiom,
! [A: $tType,M: nat,N2: nat] :
( ( rotate @ A @ ( plus_plus @ nat @ M @ N2 ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( rotate @ A @ M ) @ ( rotate @ A @ N2 ) ) ) ).
% rotate_add
thf(fact_7934_rotate__map,axiom,
! [A: $tType,B: $tType,N2: nat,F2: B > A,Xs: list @ B] :
( ( rotate @ A @ N2 @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( rotate @ B @ N2 @ Xs ) ) ) ).
% rotate_map
thf(fact_7935_rotate__def,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N: nat] : ( compow @ ( ( list @ A ) > ( list @ A ) ) @ N @ ( rotate1 @ A ) ) ) ) ).
% rotate_def
thf(fact_7936_rotate__rotate,axiom,
! [A: $tType,M: nat,N2: nat,Xs: list @ A] :
( ( rotate @ A @ M @ ( rotate @ A @ N2 @ Xs ) )
= ( rotate @ A @ ( plus_plus @ nat @ M @ N2 ) @ Xs ) ) ).
% rotate_rotate
thf(fact_7937_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F3: A > B,B6: set @ B] :
( collect @ A
@ ^ [X: A] : ( member @ B @ ( F3 @ X ) @ B6 ) ) ) ) ).
% vimage_def
thf(fact_7938_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F2: A > B,Q: A > $o] :
( ! [X5: A] :
( ( P @ ( F2 @ X5 ) )
= ( Q @ X5 ) )
=> ( ( vimage @ A @ B @ F2 @ ( collect @ B @ P ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_7939_vimageI2,axiom,
! [B: $tType,A: $tType,F2: B > A,A2: B,A3: set @ A] :
( ( member @ A @ ( F2 @ A2 ) @ A3 )
=> ( member @ B @ A2 @ ( vimage @ B @ A @ F2 @ A3 ) ) ) ).
% vimageI2
thf(fact_7940_vimageE,axiom,
! [A: $tType,B: $tType,A2: A,F2: A > B,B3: set @ B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F2 @ B3 ) )
=> ( member @ B @ ( F2 @ A2 ) @ B3 ) ) ).
% vimageE
thf(fact_7941_vimageD,axiom,
! [A: $tType,B: $tType,A2: A,F2: A > B,A3: set @ B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F2 @ A3 ) )
=> ( member @ B @ ( F2 @ A2 ) @ A3 ) ) ).
% vimageD
thf(fact_7942_rotate1__rotate__swap,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( rotate1 @ A @ ( rotate @ A @ N2 @ Xs ) )
= ( rotate @ A @ N2 @ ( rotate1 @ A @ Xs ) ) ) ).
% rotate1_rotate_swap
thf(fact_7943_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_7944_continuous__imp__open__vimage,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [S: set @ A,F2: A > B,B3: set @ B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( topolo1002775350975398744n_open @ B @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ B3 ) @ S )
=> ( topolo1002775350975398744n_open @ A @ ( vimage @ A @ B @ F2 @ B3 ) ) ) ) ) ) ) ).
% continuous_imp_open_vimage
thf(fact_7945_vimage__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,F2: B > A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ A3 ) @ ( vimage @ B @ A @ F2 @ B3 ) ) ) ).
% vimage_mono
thf(fact_7946_subset__vimage__iff,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B,B3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( vimage @ A @ B @ F2 @ B3 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( member @ B @ ( F2 @ X ) @ B3 ) ) ) ) ).
% subset_vimage_iff
thf(fact_7947_rotate__conv__mod,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N: nat,Xs2: list @ A] : ( rotate @ A @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ Xs2 ) ) ) ).
% rotate_conv_mod
thf(fact_7948_rotate__append,axiom,
! [A: $tType,L: list @ A,Q4: list @ A] :
( ( rotate @ A @ ( size_size @ ( list @ A ) @ L ) @ ( append @ A @ L @ Q4 ) )
= ( append @ A @ Q4 @ L ) ) ).
% rotate_append
thf(fact_7949_finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F5: set @ A,H2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ F5 )
=> ( ( inj_on @ B @ A @ H2 @ A3 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ F5 ) @ A3 ) ) ) ) ).
% finite_vimage_IntI
thf(fact_7950_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_7951_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B,G: A > B,Y: set @ B] :
( ! [W: A] :
( ( member @ A @ W @ S3 )
=> ( ( F2 @ W )
= ( G @ W ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ Y ) @ S3 )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G @ Y ) @ S3 ) ) ) ).
% vimage_inter_cong
thf(fact_7952_vimage__Diff,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ B,B3: set @ B] :
( ( vimage @ A @ B @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A3 ) @ ( vimage @ A @ B @ F2 @ B3 ) ) ) ).
% vimage_Diff
thf(fact_7953_vimage__Compl,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ B] :
( ( vimage @ A @ B @ F2 @ ( uminus_uminus @ ( set @ B ) @ A3 ) )
= ( uminus_uminus @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A3 ) ) ) ).
% vimage_Compl
thf(fact_7954_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( vimage @ B @ A @ F2 @ A3 )
= ( bot_bot @ ( set @ B ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% surj_vimage_empty
thf(fact_7955_image__subset__iff__subset__vimage,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B3 )
= ( ord_less_eq @ ( set @ B ) @ A3 @ ( vimage @ B @ A @ F2 @ B3 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_7956_image__vimage__subset,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( vimage @ B @ A @ F2 @ A3 ) ) @ A3 ) ).
% image_vimage_subset
thf(fact_7957_vimage__subsetD,axiom,
! [A: $tType,B: $tType,F2: B > A,B3: set @ A,A3: set @ B] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ B3 ) @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ).
% vimage_subsetD
thf(fact_7958_finite__vimageD,axiom,
! [A: $tType,B: $tType,H2: A > B,F5: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ H2 @ F5 ) )
=> ( ( ( image @ A @ B @ H2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B @ F5 ) ) ) ).
% finite_vimageD
thf(fact_7959_finite__vimageD_H,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ F2 @ A3 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) )
=> ( finite_finite2 @ B @ A3 ) ) ) ).
% finite_vimageD'
thf(fact_7960_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ~ ( finite_finite2 @ B @ A3 )
=> ~ ! [Y4: A] :
( ( member @ A @ Y4 @ ( image @ B @ A @ F2 @ A3 ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ F2 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_7961_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ~ ( finite_finite2 @ B @ A3 )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F2 @ A3 ) )
& ~ ( finite_finite2 @ B @ ( vimage @ B @ A @ F2 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_7962_vimage__subsetI,axiom,
! [B: $tType,A: $tType,F2: A > B,B3: set @ B,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ ( image @ A @ B @ F2 @ A3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ B3 ) @ A3 ) ) ) ).
% vimage_subsetI
thf(fact_7963_finite__finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F5: set @ A,H2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ F5 )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ F5 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ F5 ) @ A3 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_7964_vimage__eq__UN,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F3: A > B,B6: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y2: B] : ( vimage @ A @ B @ F3 @ ( insert @ B @ Y2 @ ( bot_bot @ ( set @ B ) ) ) )
@ B6 ) ) ) ) ).
% vimage_eq_UN
thf(fact_7965_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ~ ( finite_finite2 @ B @ A3 )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F2 @ A3 ) )
& ~ ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_7966_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ~ ( finite_finite2 @ B @ A3 )
=> ~ ! [Y4: A] :
( ( member @ A @ Y4 @ ( image @ B @ A @ F2 @ A3 ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_7967_card__vimage__inj,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( finite_card @ A @ ( vimage @ A @ B @ F2 @ A3 ) )
= ( finite_card @ B @ A3 ) ) ) ) ).
% card_vimage_inj
thf(fact_7968_card__vimage__inj__on__le,axiom,
! [A: $tType,B: $tType,F2: A > B,D6: set @ A,A3: set @ B] :
( ( inj_on @ A @ B @ F2 @ D6 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A3 ) @ D6 ) ) @ ( finite_card @ B @ A3 ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_7969_rotate__rev,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( rotate @ A @ N2 @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( rotate @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_7970_set__decode__div__2,axiom,
! [X3: nat] :
( ( nat_set_decode @ ( divide_divide @ nat @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( vimage @ nat @ nat @ suc @ ( nat_set_decode @ X3 ) ) ) ).
% set_decode_div_2
thf(fact_7971_set__encode__vimage__Suc,axiom,
! [A3: set @ nat] :
( ( nat_set_encode @ ( vimage @ nat @ nat @ suc @ A3 ) )
= ( divide_divide @ nat @ ( nat_set_encode @ A3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% set_encode_vimage_Suc
thf(fact_7972_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) )
@ ( insert @ A
@ ( the @ A
@ ^ [X: A] :
( ( F2 @ X )
= A2 ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_vimage_singleton
thf(fact_7973_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,A2: B] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) @ A3 )
@ ( insert @ A
@ ( the @ A
@ ^ [X: A] :
( ( member @ A @ X @ A3 )
& ( ( F2 @ X )
= A2 ) ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_on_vimage_singleton
thf(fact_7974_inv__image__partition,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ ( set2 @ A @ Ys ) )
=> ~ ( P @ Y4 ) )
=> ( ( vimage @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( partition @ A @ P ) @ ( 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_7975_nth__rotate,axiom,
! [A: $tType,N2: nat,Xs: list @ A,M: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate @ A @ M @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_7976_hd__rotate__conv__nth,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( rotate @ A @ N2 @ Xs ) )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_7977_dual__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( min @ A
@ ^ [X: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X ) )
= ( ord_max @ A ) ) ) ).
% dual_min
thf(fact_7978_listrel1__subset__listrel,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R3 )
=> ( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R2 ) @ ( listrel @ A @ A @ R3 ) ) ) ) ).
% listrel1_subset_listrel
thf(fact_7979_refl__onD2,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( member @ A @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_7980_refl__onD1,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( member @ A @ X3 @ A3 ) ) ) ).
% refl_onD1
thf(fact_7981_refl__onD,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( member @ A @ A2 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_7982_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_7983_ord_Omin__def,axiom,
! [A: $tType] :
( ( min @ A )
= ( ^ [Less_eq: A > A > $o,A4: A,B4: A] : ( if @ A @ ( Less_eq @ A4 @ B4 ) @ A4 @ B4 ) ) ) ).
% ord.min_def
thf(fact_7984_ord_Omin_Ocong,axiom,
! [A: $tType] :
( ( min @ A )
= ( min @ A ) ) ).
% ord.min.cong
thf(fact_7985_refl__on__def,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A6: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : A6 ) )
& ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R5 ) ) ) ) ) ).
% refl_on_def
thf(fact_7986_refl__onI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 ) )
=> ( refl_on @ A @ A3 @ R2 ) ) ) ).
% refl_onI
thf(fact_7987_refl__on__def_H,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A6: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ! [X: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y2: A,Z6: A] :
( ( member @ A @ Y2 @ A6 )
& ( member @ A @ Z6 @ A6 ) )
@ X ) )
& ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R5 ) ) ) ) ) ).
% refl_on_def'
thf(fact_7988_refl__on__singleton,axiom,
! [A: $tType,X3: A] : ( refl_on @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% refl_on_singleton
thf(fact_7989_refl__on__domain,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( ( member @ A @ A2 @ A3 )
& ( member @ A @ B2 @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_7990_Restr__natLeq,axiom,
! [N2: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat
@ ( collect @ nat
@ ^ [X: nat] : ( ord_less @ nat @ X @ N2 ) )
@ ^ [Uu3: nat] :
( collect @ nat
@ ^ [X: nat] : ( ord_less @ nat @ X @ N2 ) ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X: nat,Y2: nat] :
( ( ord_less @ nat @ X @ N2 )
& ( ord_less @ nat @ Y2 @ N2 )
& ( ord_less_eq @ nat @ X @ Y2 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_7991_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less_eq @ nat ) ) ) ) ).
% natLeq_def
thf(fact_7992_Restr__natLeq2,axiom,
! [N2: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat @ ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N2 )
@ ^ [Uu3: nat] : ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N2 ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X: nat,Y2: nat] :
( ( ord_less @ nat @ X @ N2 )
& ( ord_less @ nat @ Y2 @ N2 )
& ( ord_less_eq @ nat @ X @ Y2 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_7993_linear__order__on__singleton,axiom,
! [A: $tType,X3: A] : ( order_679001287576687338der_on @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% linear_order_on_singleton
thf(fact_7994_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_7995_underS__def,axiom,
! [A: $tType] :
( ( order_underS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A4: A] :
( collect @ A
@ ^ [B4: A] :
( ( B4 != A4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ R5 ) ) ) ) ) ).
% underS_def
thf(fact_7996_natLeq__underS__less,axiom,
! [N2: nat] :
( ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N2 )
= ( collect @ nat
@ ^ [X: nat] : ( ord_less @ nat @ X @ N2 ) ) ) ).
% natLeq_underS_less
thf(fact_7997_underS__I,axiom,
! [A: $tType,I: A,J: A,R: set @ ( product_prod @ A @ A )] :
( ( I != J )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
=> ( member @ A @ I @ ( order_underS @ A @ R @ J ) ) ) ) ).
% underS_I
thf(fact_7998_underS__E,axiom,
! [A: $tType,I: A,R: set @ ( product_prod @ A @ A ),J: A] :
( ( member @ A @ I @ ( order_underS @ A @ R @ J ) )
=> ( ( I != J )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R ) ) ) ).
% underS_E
thf(fact_7999_total__on__singleton,axiom,
! [A: $tType,X3: A] : ( total_on @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_8000_Arg__bounded,axiom,
! [Z2: complex] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z2 ) )
& ( ord_less_eq @ real @ ( arg @ Z2 ) @ pi ) ) ).
% Arg_bounded
thf(fact_8001_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_8002_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_8003_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_8004_total__onI,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( member @ A @ Y4 @ A3 )
=> ( ( X5 != Y4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X5 ) @ R2 ) ) ) ) )
=> ( total_on @ A @ A3 @ R2 ) ) ).
% total_onI
thf(fact_8005_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A6: set @ A,R5: set @ ( product_prod @ A @ A )] :
! [X: A] :
( ( member @ A @ X @ A6 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ A6 )
=> ( ( X != Y2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ R5 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_8006_Arg__correct,axiom,
! [Z2: complex] :
( ( Z2
!= ( zero_zero @ complex ) )
=> ( ( ( sgn_sgn @ complex @ Z2 )
= ( cis @ ( arg @ Z2 ) ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z2 ) )
& ( ord_less_eq @ real @ ( arg @ Z2 ) @ pi ) ) ) ).
% Arg_correct
thf(fact_8007_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 ) ) ) )
| ? [A5: A] :
( R2
= ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% Total_subset_Id
thf(fact_8008_Field__empty,axiom,
! [A: $tType] :
( ( field2 @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Field_empty
thf(fact_8009_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_8010_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_8011_Order__Relation_OunderS__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A] : ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A2 ) @ ( field2 @ A @ R2 ) ) ).
% Order_Relation.underS_Field
thf(fact_8012_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_8013_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_8014_FieldI2,axiom,
! [A: $tType,I: A,J: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
=> ( member @ A @ J @ ( field2 @ A @ R ) ) ) ).
% FieldI2
thf(fact_8015_FieldI1,axiom,
! [A: $tType,I: A,J: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
=> ( member @ A @ I @ ( field2 @ A @ R ) ) ) ).
% FieldI1
thf(fact_8016_mono__Field,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 @ A ) @ ( field2 @ A @ R2 ) @ ( field2 @ A @ S ) ) ) ).
% mono_Field
thf(fact_8017_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_8018_Field__Restr__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ord_less_eq @ ( set @ A )
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) ) )
@ A3 ) ).
% Field_Restr_subset
thf(fact_8019_Field__natLeq__on,axiom,
! [N2: nat] :
( ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X: nat,Y2: nat] :
( ( ord_less @ nat @ X @ N2 )
& ( ord_less @ nat @ Y2 @ N2 )
& ( ord_less_eq @ nat @ X @ Y2 ) ) ) ) )
= ( collect @ nat
@ ^ [X: nat] : ( ord_less @ nat @ X @ N2 ) ) ) ).
% Field_natLeq_on
thf(fact_8020_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_8021_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_8022_Refl__Field__Restr2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) ) )
= A3 ) ) ) ).
% Refl_Field_Restr2
thf(fact_8023_cis__Arg__unique,axiom,
! [Z2: complex,X3: real] :
( ( ( sgn_sgn @ complex @ Z2 )
= ( cis @ X3 ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ pi )
=> ( ( arg @ Z2 )
= X3 ) ) ) ) ).
% cis_Arg_unique
thf(fact_8024_Under__def,axiom,
! [A: $tType] :
( ( order_Under @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A6: set @ A] :
( collect @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ ( field2 @ A @ R5 ) )
& ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ X ) @ R5 ) ) ) ) ) ) ).
% Under_def
thf(fact_8025_UnderS__def,axiom,
! [A: $tType] :
( ( order_UnderS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A6: set @ A] :
( collect @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ ( field2 @ A @ R5 ) )
& ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( ( B4 != X )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ X ) @ R5 ) ) ) ) ) ) ) ).
% UnderS_def
thf(fact_8026_Above__def,axiom,
! [A: $tType] :
( ( order_Above @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A6: set @ A] :
( collect @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ ( field2 @ A @ R5 ) )
& ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ B4 ) @ R5 ) ) ) ) ) ) ).
% Above_def
thf(fact_8027_bij__betw__roots__unity,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( 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 @ N2 ) ) )
@ ( set_ord_lessThan @ nat @ N2 )
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= ( one_one @ complex ) ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_8028_bij__betw__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A3: set @ A,B3: set @ A] :
( ( bij_betw @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 @ B3 )
= ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 )
= B3 ) ) ) ).
% bij_betw_add
thf(fact_8029_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B3: set @ B,C5: set @ A,D6: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A3 @ B3 )
=> ( ( bij_betw @ A @ B @ F2 @ C5 @ D6 )
=> ( ( ( inf_inf @ ( set @ B ) @ B3 @ D6 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ C5 ) @ ( sup_sup @ ( set @ B ) @ B3 @ D6 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_8030_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,C5: set @ A,B3: set @ B,D6: set @ B] :
( ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ C5 ) @ ( sup_sup @ ( set @ B ) @ B3 @ D6 ) )
=> ( ( bij_betw @ A @ B @ F2 @ C5 @ D6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ C5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ B3 @ D6 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F2 @ A3 @ B3 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_8031_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,C5: set @ B,G: A > B,B3: set @ A,D6: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A3 @ C5 )
=> ( ( bij_betw @ A @ B @ G @ B3 @ D6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C5 @ D6 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ A3 ) @ ( F2 @ X ) @ ( G @ X ) )
@ ( sup_sup @ ( set @ A ) @ A3 @ B3 )
@ ( sup_sup @ ( set @ B ) @ C5 @ D6 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_8032_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,A16: set @ B,B3: set @ A,B15: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A3 @ A16 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( ( image @ A @ B @ F2 @ B3 )
= B15 )
=> ( bij_betw @ A @ B @ F2 @ B3 @ B15 ) ) ) ) ).
% bij_betw_subset
thf(fact_8033_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A3: set @ A,F11: B > A,F2: A > B,A16: set @ B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( F11 @ ( F2 @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A16 )
=> ( ( F2 @ ( F11 @ X5 ) )
= X5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ A16 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F11 @ A16 ) @ A3 )
=> ( bij_betw @ A @ B @ F2 @ A3 @ A16 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_8034_bij__betw__UNION__chain,axiom,
! [B: $tType,C: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B ),F2: B > C,A16: A > ( set @ C )] :
( ! [I4: A,J2: A] :
( ( member @ A @ I4 @ I6 )
=> ( ( member @ A @ J2 @ I6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A3 @ I4 ) @ ( A3 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A3 @ J2 ) @ ( A3 @ I4 ) ) ) ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( bij_betw @ B @ C @ F2 @ ( A3 @ I4 ) @ ( A16 @ I4 ) ) )
=> ( bij_betw @ B @ C @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ A @ ( set @ C ) @ A16 @ I6 ) ) ) ) ) ).
% bij_betw_UNION_chain
thf(fact_8035_bij__betw__comp__iff2,axiom,
! [C: $tType,A: $tType,B: $tType,F11: A > B,A16: set @ A,A19: set @ B,F2: C > A,A3: set @ C] :
( ( bij_betw @ A @ B @ F11 @ A16 @ A19 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ A3 ) @ A16 )
=> ( ( bij_betw @ C @ A @ F2 @ A3 @ A16 )
= ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F11 @ F2 ) @ A3 @ A19 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_8036_finite__same__card__bij,axiom,
! [A: $tType,B: $tType,A3: set @ A,B3: set @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( ( finite_card @ A @ A3 )
= ( finite_card @ B @ B3 ) )
=> ? [H6: A > B] : ( bij_betw @ A @ B @ H6 @ A3 @ B3 ) ) ) ) ).
% finite_same_card_bij
thf(fact_8037_bij__betw__iff__card,axiom,
! [A: $tType,B: $tType,A3: set @ A,B3: set @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( ? [F3: A > B] : ( bij_betw @ A @ B @ F3 @ A3 @ B3 ) )
= ( ( finite_card @ A @ A3 )
= ( finite_card @ B @ B3 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_8038_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A] :
( ( bij_betw @ A @ B @ F2 @ A3 @ ( bot_bot @ ( set @ B ) ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% bij_betw_empty2
thf(fact_8039_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ B] :
( ( bij_betw @ A @ B @ F2 @ ( bot_bot @ ( set @ A ) ) @ A3 )
=> ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ).
% bij_betw_empty1
thf(fact_8040_bij__betw__finite,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B3: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A3 @ B3 )
=> ( ( finite_finite2 @ A @ A3 )
= ( finite_finite2 @ B @ B3 ) ) ) ).
% bij_betw_finite
thf(fact_8041_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B2: A,A3: set @ A,F2: A > B,A16: set @ B] :
( ~ ( member @ A @ B2 @ A3 )
=> ( ~ ( member @ B @ ( F2 @ B2 ) @ A16 )
=> ( ( bij_betw @ A @ B @ F2 @ A3 @ A16 )
=> ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A16 @ ( insert @ B @ ( F2 @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_8042_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B2: A,A3: set @ A,F2: A > B,A16: set @ B] :
( ~ ( member @ A @ B2 @ A3 )
=> ( ~ ( member @ B @ ( F2 @ B2 ) @ A16 )
=> ( ( bij_betw @ A @ B @ F2 @ A3 @ A16 )
= ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A16 @ ( insert @ B @ ( F2 @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_8043_bij__betw__funpow,axiom,
! [A: $tType,F2: A > A,S3: set @ A,N2: nat] :
( ( bij_betw @ A @ A @ F2 @ S3 @ S3 )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ S3 @ S3 ) ) ).
% bij_betw_funpow
thf(fact_8044_bij__fn,axiom,
! [A: $tType,F2: A > A,N2: nat] :
( ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_fn
thf(fact_8045_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_8046_infinite__imp__bij__betw2,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [H6: A > A] : ( bij_betw @ A @ A @ H6 @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_8047_infinite__imp__bij__betw,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [H6: A > A] : ( bij_betw @ A @ A @ H6 @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_8048_vimage__subset__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,B3: set @ B,A3: set @ A] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ B3 ) @ A3 )
= ( ord_less_eq @ ( set @ B ) @ B3 @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ).
% vimage_subset_eq
thf(fact_8049_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_8050_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_8051_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S6: set @ B,T6: set @ C,H2: B > C,S3: set @ B,T4: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S6 )
=> ( ( G @ ( H2 @ A5 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T6 )
=> ( ( G @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( G @ ( H2 @ X ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T4 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_8052_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S6: set @ B,T6: set @ C,H2: B > C,S3: set @ B,T4: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S6 )
=> ( ( G @ ( H2 @ A5 ) )
= ( one_one @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T6 )
=> ( ( G @ B5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( G @ ( H2 @ X ) )
@ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T4 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_8053_bij__betw__nth,axiom,
! [A: $tType,Xs: list @ A,A3: set @ nat,B3: set @ A] :
( ( distinct @ A @ Xs )
=> ( ( A3
= ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( ( B3
= ( set2 @ A @ Xs ) )
=> ( bij_betw @ nat @ A @ ( nth @ A @ Xs ) @ A3 @ B3 ) ) ) ) ).
% bij_betw_nth
thf(fact_8054_ex__bij__betw__strict__mono__card,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ~ ! [H6: nat > A] :
( ( bij_betw @ nat @ A @ H6 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ M7 ) ) @ M7 )
=> ~ ( strict_mono_on @ nat @ A @ H6 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ M7 ) ) ) ) ) ) ).
% ex_bij_betw_strict_mono_card
thf(fact_8055_Schroeder__Bernstein,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B3: set @ B,G: B > A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B3 )
=> ( ( inj_on @ B @ A @ G @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ B3 ) @ A3 )
=> ? [H6: A > B] : ( bij_betw @ A @ B @ H6 @ A3 @ B3 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_8056_bij__betw__nth__root__unity,axiom,
! [C2: complex,N2: nat] :
( ( C2
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( bij_betw @ complex @ complex @ ( times_times @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( root @ N2 @ ( real_V7770717601297561774m_norm @ complex @ C2 ) ) ) @ ( cis @ ( divide_divide @ real @ ( arg @ C2 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) )
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= ( one_one @ complex ) ) )
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= C2 ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_8057_bij__betw__Suc,axiom,
! [M7: set @ nat,N7: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M7 @ N7 )
= ( ( image @ nat @ nat @ suc @ M7 )
= N7 ) ) ).
% bij_betw_Suc
thf(fact_8058_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X6: $o > A,Y9: $o > A] :
( ( ord_less_eq @ A @ ( X6 @ $false ) @ ( Y9 @ $false ) )
& ( ord_less_eq @ A @ ( X6 @ $true ) @ ( Y9 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_8059_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H6: A > nat] : ( bij_betw @ A @ nat @ H6 @ M7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_8060_finite__Linear__order__induct,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,P: A > $o] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
=> ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R2 ) )
=> ( ! [Y5: A] :
( ( member @ A @ Y5 @ ( order_aboveS @ A @ R2 @ X5 ) )
=> ( P @ Y5 ) )
=> ( P @ X5 ) ) )
=> ( P @ X3 ) ) ) ) ) ).
% finite_Linear_order_induct
thf(fact_8061_Arg__def,axiom,
( arg
= ( ^ [Z6: complex] :
( if @ real
@ ( Z6
= ( zero_zero @ complex ) )
@ ( zero_zero @ real )
@ ( fChoice @ real
@ ^ [A4: real] :
( ( ( sgn_sgn @ complex @ Z6 )
= ( cis @ A4 ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ A4 )
& ( ord_less_eq @ real @ A4 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_8062_some__in__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X: A] : ( member @ A @ X @ A3 ) )
@ A3 )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_8063_exE__some,axiom,
! [A: $tType,P: A > $o,C2: A] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ( C2
= ( fChoice @ A @ P ) )
=> ( P @ C2 ) ) ) ).
% exE_some
thf(fact_8064_aboveS__def,axiom,
! [A: $tType] :
( ( order_aboveS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A4: A] :
( collect @ A
@ ^ [B4: A] :
( ( B4 != A4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R5 ) ) ) ) ) ).
% aboveS_def
thf(fact_8065_arg__min__SOME__Min,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( lattic7623131987881927897min_on @ A @ B @ F2 @ S3 )
= ( fChoice @ A
@ ^ [Y2: A] :
( ( member @ A @ Y2 @ S3 )
& ( ( F2 @ Y2 )
= ( lattic643756798350308766er_Min @ B @ ( image @ A @ B @ F2 @ S3 ) ) ) ) ) ) ) ) ).
% arg_min_SOME_Min
thf(fact_8066_cofinal__def,axiom,
! [A: $tType] :
( ( bNF_Ca7293521722713021262ofinal @ A )
= ( ^ [A6: set @ A,R5: set @ ( product_prod @ A @ A )] :
! [X: A] :
( ( member @ A @ X @ ( field2 @ A @ R5 ) )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ A6 )
& ( X != Y2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R5 ) ) ) ) ) ).
% cofinal_def
thf(fact_8067_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 )
@ ^ [A1: A,A22: A] :
( product_case_prod @ A @ A @ $o
@ ^ [B16: A,B23: A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A1 @ ( insert @ A @ A22 @ ( insert @ A @ B16 @ ( insert @ A @ B23 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R5 ) )
& ( ( ( A1 = B16 )
& ( A22 = B23 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A1 @ A22 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B16 @ B23 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A1 @ A22 )
= ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B16 @ B23 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B16 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A1 @ A22 )
= ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B16 @ B23 ) )
& ( A1 = B16 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A22 @ B23 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_8068_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X3: A,Y: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y6: B] :
( ( X3 = X9 )
& ( Y = Y6 ) ) ) )
= ( product_Pair @ A @ B @ X3 @ Y ) ) ).
% Eps_case_prod_eq
thf(fact_8069_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P3: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A4: A,B4: B] : ( P3 @ ( product_Pair @ A @ B @ A4 @ B4 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_8070_to__nat__on__def,axiom,
! [A: $tType] :
( ( countable_to_nat_on @ A )
= ( ^ [S7: set @ A] :
( fChoice @ ( A > nat )
@ ^ [F3: A > nat] :
( ( ( finite_finite2 @ A @ S7 )
=> ( bij_betw @ A @ nat @ F3 @ S7 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S7 ) ) ) )
& ( ~ ( finite_finite2 @ A @ S7 )
=> ( bij_betw @ A @ nat @ F3 @ S7 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% to_nat_on_def
thf(fact_8071_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_8072_under__def,axiom,
! [A: $tType] :
( ( order_under @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A4: A] :
( collect @ A
@ ^ [B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ R5 ) ) ) ) ).
% under_def
thf(fact_8073_underS__subset__under,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A] : ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A2 ) @ ( order_under @ A @ R2 @ A2 ) ) ).
% underS_subset_under
thf(fact_8074_under__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A] : ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R2 @ A2 ) @ ( field2 @ A @ R2 ) ) ).
% under_Field
thf(fact_8075_to__nat__on__finite,axiom,
! [A: $tType,S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( bij_betw @ A @ nat @ ( countable_to_nat_on @ A @ S3 ) @ S3 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S3 ) ) ) ) ).
% to_nat_on_finite
thf(fact_8076_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 ) ) )
= ( ! [A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R2 ) )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X: A] :
( ( member @ A @ X @ A6 )
& ! [Y2: A] :
( ( member @ A @ Y2 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_8077_slice__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat,M: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) ) )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ M @ N2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ) ).
% slice_eq_mask
thf(fact_8078_wf__empty,axiom,
! [A: $tType] : ( wf @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% wf_empty
thf(fact_8079_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_8080_wf__lex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( wf @ ( list @ A ) @ ( lex @ A @ R2 ) ) ) ).
% wf_lex
thf(fact_8081_wf__lenlex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( wf @ ( list @ A ) @ ( lenlex @ A @ R2 ) ) ) ).
% wf_lenlex
thf(fact_8082_not__negative__int__iff,axiom,
! [K: int] :
( ( ord_less @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% not_negative_int_iff
thf(fact_8083_not__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ K ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% not_nonnegative_int_iff
thf(fact_8084_bit_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X3 ) @ X3 )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_left
thf(fact_8085_bit_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se5824344872417868541ns_and @ A @ X3 @ ( bit_ri4277139882892585799ns_not @ A @ X3 ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_right
thf(fact_8086_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_8087_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_8088_wf__insert,axiom,
! [A: $tType,Y: A,X3: A,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ R2 ) )
= ( ( wf @ A @ R2 )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% wf_insert
thf(fact_8089_reduction__pairI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S3 ) @ R )
=> ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ S3 ) ) ) ) ).
% reduction_pairI
thf(fact_8090_wf__bounded__measure,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > nat,F2: A > nat] :
( ! [A5: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A5 ) @ R2 )
=> ( ( ord_less_eq @ nat @ ( Ub @ B5 ) @ ( Ub @ A5 ) )
& ( ord_less_eq @ nat @ ( F2 @ B5 ) @ ( Ub @ A5 ) )
& ( ord_less @ nat @ ( F2 @ A5 ) @ ( F2 @ B5 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wf_bounded_measure
thf(fact_8091_wfI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : B3 ) )
=> ( ! [X5: A,P8: A > $o] :
( ! [Xa: A] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa ) @ R2 )
=> ( P8 @ Y4 ) )
=> ( P8 @ Xa ) )
=> ( ( member @ A @ X5 @ A3 )
=> ( ( member @ A @ X5 @ B3 )
=> ( P8 @ X5 ) ) ) )
=> ( wf @ A @ R2 ) ) ) ).
% wfI
thf(fact_8092_wf__lexn,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),N2: nat] :
( ( wf @ A @ R2 )
=> ( wf @ ( list @ A ) @ ( lexn @ A @ R2 @ N2 ) ) ) ).
% wf_lexn
thf(fact_8093_wf__int__ge__less__than2,axiom,
! [D2: int] : ( wf @ int @ ( int_ge_less_than2 @ D2 ) ) ).
% wf_int_ge_less_than2
thf(fact_8094_wf__int__ge__less__than,axiom,
! [D2: int] : ( wf @ int @ ( int_ge_less_than @ D2 ) ) ).
% wf_int_ge_less_than
thf(fact_8095_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_8096_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 )
=> ~ ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ R )
=> ~ ( member @ A @ Y5 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_8097_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P: B > $o,K: B,M: B > A] :
( ( wf @ A @ R2 )
=> ( ! [X5: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ ( transitive_trancl @ A @ R2 ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X5 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) )
=> ( ( P @ K )
=> ? [X5: B] :
( ( P @ X5 )
& ! [Y5: B] :
( ( P @ Y5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M @ X5 ) @ ( M @ Y5 ) ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_8098_wf__def,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [P3: A > $o] :
( ! [X: A] :
( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ R5 )
=> ( P3 @ Y2 ) )
=> ( P3 @ X ) )
=> ! [X6: A] : ( P3 @ X6 ) ) ) ) ).
% wf_def
thf(fact_8099_wfE__min,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X3: A,Q: set @ A] :
( ( wf @ A @ R )
=> ( ( member @ A @ X3 @ Q )
=> ~ ! [Z3: A] :
( ( member @ A @ Z3 @ Q )
=> ~ ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ R )
=> ~ ( member @ A @ Y5 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_8100_wfI__min,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X5: A,Q8: set @ A] :
( ( member @ A @ X5 @ Q8 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Q8 )
& ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa ) @ R )
=> ~ ( member @ A @ Y4 @ Q8 ) ) ) )
=> ( wf @ A @ R ) ) ).
% wfI_min
thf(fact_8101_wfUNIVI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [P8: A > $o,X5: A] :
( ! [Xa: A] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa ) @ R2 )
=> ( P8 @ Y4 ) )
=> ( P8 @ Xa ) )
=> ( P8 @ X5 ) )
=> ( wf @ A @ R2 ) ) ).
% wfUNIVI
thf(fact_8102_wf__asym,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,X3: A] :
( ( wf @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A2 ) @ R2 ) ) ) ).
% wf_asym
thf(fact_8103_wf__induct,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A2: A] :
( ( wf @ A @ R2 )
=> ( ! [X5: A] :
( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X5 ) @ R2 )
=> ( P @ Y5 ) )
=> ( P @ X5 ) )
=> ( P @ A2 ) ) ) ).
% wf_induct
thf(fact_8104_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_8105_wf__not__sym,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,X3: A] :
( ( wf @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A2 ) @ R2 ) ) ) ).
% wf_not_sym
thf(fact_8106_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_8107_wf__eq__minimal,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [Q7: set @ A] :
( ? [X: A] : ( member @ A @ X @ Q7 )
=> ? [X: A] :
( ( member @ A @ X @ Q7 )
& ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ R5 )
=> ~ ( member @ A @ Y2 @ Q7 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_8108_wf__induct__rule,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A2: A] :
( ( wf @ A @ R2 )
=> ( ! [X5: A] :
( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X5 ) @ R2 )
=> ( P @ Y5 ) )
=> ( P @ X5 ) )
=> ( P @ A2 ) ) ) ).
% wf_induct_rule
thf(fact_8109_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
~ ? [F3: nat > A] :
! [I3: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F3 @ ( suc @ I3 ) ) @ ( F3 @ I3 ) ) @ R5 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_8110_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),F2: nat > A] :
( ( wf @ A @ R2 )
=> ~ ! [K2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ ( suc @ K2 ) ) @ ( F2 @ K2 ) ) @ R2 ) ) ).
% wf_no_infinite_down_chainE
thf(fact_8111_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A4: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A4 ) @ ( one_one @ A ) ) ) ) ) ).
% minus_eq_not_plus_1
thf(fact_8112_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_8113_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_8114_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_8115_wf__if__measure,axiom,
! [A: $tType,P: A > $o,F2: A > nat,G: A > A] :
( ! [X5: A] :
( ( P @ X5 )
=> ( ord_less @ nat @ ( F2 @ ( G @ X5 ) ) @ ( F2 @ X5 ) ) )
=> ( wf @ A
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [Y2: A,X: A] :
( ( P @ X )
& ( Y2
= ( G @ X ) ) ) ) ) ) ) ).
% wf_if_measure
thf(fact_8116_wf__less,axiom,
wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ).
% wf_less
thf(fact_8117_wf__eq__minimal2,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [A6: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R5 ) )
& ( A6
!= ( bot_bot @ ( set @ A ) ) ) )
=> ? [X: A] :
( ( member @ A @ X @ A6 )
& ! [Y2: A] :
( ( member @ A @ Y2 @ A6 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ R5 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_8118_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F2: A > ( set @ B )] :
( ! [A5: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A5 ) @ R2 )
=> ( ( finite_finite2 @ B @ ( Ub @ A5 ) )
& ( ord_less_eq @ ( set @ B ) @ ( Ub @ B5 ) @ ( Ub @ A5 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F2 @ B5 ) @ ( Ub @ A5 ) )
& ( ord_less @ ( set @ B ) @ ( F2 @ A5 ) @ ( F2 @ B5 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wf_bounded_set
thf(fact_8119_take__bit__not__mask__eq__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_not_mask_eq_0
thf(fact_8120_push__bit__mask__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N2: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ N2 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) ) ) ) ) ).
% push_bit_mask_eq
thf(fact_8121_finite__subset__wf,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( wf @ ( set @ A )
@ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X6: set @ A,Y9: set @ A] :
( ( ord_less @ ( set @ A ) @ X6 @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ Y9 @ A3 ) ) ) ) ) ) ).
% finite_subset_wf
thf(fact_8122_bit_Ocompl__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ X3 @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ X3 @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( bit_ri4277139882892585799ns_not @ A @ X3 )
= Y ) ) ) ) ).
% bit.compl_unique
thf(fact_8123_bit__not__iff__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ N2 )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
!= ( zero_zero @ A ) )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ) ).
% bit_not_iff_eq
thf(fact_8124_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_8125_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P: ( A > B ) > A > B > $o] :
( ( wf @ A @ R )
=> ( ! [F4: A > B,G4: A > B,X5: A,R4: B] :
( ! [Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X5 ) @ R )
=> ( ( F4 @ Z4 )
= ( G4 @ Z4 ) ) )
=> ( ( P @ F4 @ X5 @ R4 )
= ( P @ G4 @ X5 @ R4 ) ) )
=> ( ! [X5: A,F4: A > B] :
( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X5 ) @ R )
=> ( P @ F4 @ Y5 @ ( F4 @ Y5 ) ) )
=> ? [X_12: B] : ( P @ F4 @ X5 @ X_12 ) )
=> ? [F4: A > B] :
! [X2: A] : ( P @ F4 @ X2 @ ( F4 @ X2 ) ) ) ) ) ).
% dependent_wf_choice
thf(fact_8126_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A )
=> ! [P: ( A > B ) > A > B > $o] :
( ! [R4: B,F4: A > B,G4: A > B,X5: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X5 )
=> ( ( F4 @ Y5 )
= ( G4 @ Y5 ) ) )
=> ( ( P @ F4 @ X5 @ R4 )
= ( P @ G4 @ X5 @ R4 ) ) )
=> ( ! [X5: A,F4: A > B] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X5 )
=> ( P @ F4 @ Y5 @ ( F4 @ Y5 ) ) )
=> ? [X_12: B] : ( P @ F4 @ X5 @ X_12 ) )
=> ? [F4: A > B] :
! [X2: A] : ( P @ F4 @ X2 @ ( F4 @ X2 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_8127_chains__extend,axiom,
! [A: $tType,C2: set @ ( set @ A ),S3: set @ ( set @ A ),Z2: set @ A] :
( ( member @ ( set @ ( set @ A ) ) @ C2 @ ( chains @ A @ S3 ) )
=> ( ( member @ ( set @ A ) @ Z2 @ S3 )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ X5 @ Z2 ) )
=> ( member @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ Z2 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C2 ) @ ( chains @ A @ S3 ) ) ) ) ) ).
% chains_extend
thf(fact_8128_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_8129_chainsD,axiom,
! [A: $tType,C2: set @ ( set @ A ),S3: set @ ( set @ A ),X3: set @ A,Y: set @ A] :
( ( member @ ( set @ ( set @ A ) ) @ C2 @ ( chains @ A @ S3 ) )
=> ( ( member @ ( set @ A ) @ X3 @ C2 )
=> ( ( member @ ( set @ A ) @ Y @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Y )
| ( ord_less_eq @ ( set @ A ) @ Y @ X3 ) ) ) ) ) ).
% chainsD
thf(fact_8130_Zorn__Lemma2,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ! [X5: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X5 @ ( chains @ A @ A3 ) )
=> ? [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A3 )
& ! [Xb2: set @ A] :
( ( member @ ( set @ A ) @ Xb2 @ X5 )
=> ( ord_less_eq @ ( set @ A ) @ Xb2 @ Xa ) ) ) )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A3 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ Xa )
=> ( Xa = X5 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_8131_Zorn__Lemma,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ! [X5: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X5 @ ( chains @ A @ A3 ) )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ X5 ) @ A3 ) )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A3 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ Xa )
=> ( Xa = X5 ) ) ) ) ) ).
% Zorn_Lemma
thf(fact_8132_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_8133_and__not__num__eq__None__iff,axiom,
! [M: num,N2: num] :
( ( ( bit_and_not_num @ M @ N2 )
= ( none @ num ) )
= ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( zero_zero @ int ) ) ) ).
% and_not_num_eq_None_iff
thf(fact_8134_partial__order__on__well__order__on,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( order_7125193373082350890der_on @ A @ A3 @ R2 )
=> ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ).
% partial_order_on_well_order_on
thf(fact_8135_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_8136_and__not__num_Osimps_I1_J,axiom,
( ( bit_and_not_num @ one2 @ one2 )
= ( none @ num ) ) ).
% and_not_num.simps(1)
thf(fact_8137_and__not__num_Osimps_I3_J,axiom,
! [N2: num] :
( ( bit_and_not_num @ one2 @ ( bit1 @ N2 ) )
= ( none @ num ) ) ).
% and_not_num.simps(3)
thf(fact_8138_finite__Partial__order__induct,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,P: A > $o] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
=> ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R2 ) )
=> ( ! [Y5: A] :
( ( member @ A @ Y5 @ ( order_aboveS @ A @ R2 @ X5 ) )
=> ( P @ Y5 ) )
=> ( P @ X5 ) ) )
=> ( P @ X3 ) ) ) ) ) ).
% finite_Partial_order_induct
thf(fact_8139_and__not__num_Oelims,axiom,
! [X3: num,Xa2: num,Y: option @ num] :
( ( ( bit_and_not_num @ X3 @ Xa2 )
= Y )
=> ( ( ( X3 = one2 )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X3 = one2 )
=> ( ? [N3: num] :
( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X3 = one2 )
=> ( ? [N3: num] :
( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ! [M3: num] :
( ( X3
= ( bit0 @ M3 ) )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X3
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X3
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X3
= ( bit1 @ M3 ) )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X3
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
=> ~ ! [M3: num] :
( ( X3
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
thf(fact_8140_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
@ ^ [X6: set @ A,Y9: set @ A] :
( ( finite_finite2 @ A @ X6 )
& ( finite_finite2 @ A @ Y9 )
& ( Y9
!= ( bot_bot @ ( set @ A ) ) )
& ! [X: A] :
( ( member @ A @ X @ X6 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ Y9 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R6 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_8141_bex__empty,axiom,
! [A: $tType,P: A > $o] :
~ ? [X2: A] :
( ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
& ( P @ X2 ) ) ).
% bex_empty
thf(fact_8142_map__option__eq__Some,axiom,
! [A: $tType,B: $tType,F2: B > A,Xo: option @ B,Y: A] :
( ( ( map_option @ B @ A @ F2 @ Xo )
= ( some @ A @ Y ) )
= ( ? [Z6: B] :
( ( Xo
= ( some @ B @ Z6 ) )
& ( ( F2 @ Z6 )
= Y ) ) ) ) ).
% map_option_eq_Some
thf(fact_8143_option_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: option @ A] :
( ( ( map_option @ A @ B @ F2 @ A2 )
= ( none @ B ) )
= ( A2
= ( none @ A ) ) ) ).
% option.map_disc_iff
thf(fact_8144_map__option__is__None,axiom,
! [A: $tType,B: $tType,F2: B > A,Opt: option @ B] :
( ( ( map_option @ B @ A @ F2 @ Opt )
= ( none @ A ) )
= ( Opt
= ( none @ B ) ) ) ).
% map_option_is_None
thf(fact_8145_None__eq__map__option__iff,axiom,
! [A: $tType,B: $tType,F2: B > A,X3: option @ B] :
( ( ( none @ A )
= ( map_option @ B @ A @ F2 @ X3 ) )
= ( X3
= ( none @ B ) ) ) ).
% None_eq_map_option_iff
thf(fact_8146_finite__Collect__bex,axiom,
! [B: $tType,A: $tType,A3: set @ A,Q: B > A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
? [Y2: A] :
( ( member @ A @ Y2 @ A3 )
& ( Q @ X @ Y2 ) ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Y2: B] : ( Q @ Y2 @ X ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_8147_map__option__o__empty,axiom,
! [C: $tType,B: $tType,A: $tType,F2: C > B] :
( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F2 )
@ ^ [X: A] : ( none @ C ) )
= ( ^ [X: A] : ( none @ B ) ) ) ).
% map_option_o_empty
thf(fact_8148_bex__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( top_top @ ( set @ A ) ) )
& ( P @ X ) ) )
= ( ? [X6: A] : ( P @ X6 ) ) ) ).
% bex_UNIV
thf(fact_8149_case__map__option,axiom,
! [B: $tType,A: $tType,C: $tType,G: A,H2: B > A,F2: C > B,X3: option @ C] :
( ( case_option @ A @ B @ G @ H2 @ ( map_option @ C @ B @ F2 @ X3 ) )
= ( case_option @ A @ C @ G @ ( comp @ B @ A @ C @ H2 @ F2 ) @ X3 ) ) ).
% case_map_option
thf(fact_8150_map__option__o__map__upd,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > B,M: A > ( option @ C ),A2: A,B2: C] :
( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F2 ) @ ( fun_upd @ A @ ( option @ C ) @ M @ A2 @ ( some @ C @ B2 ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F2 ) @ M ) @ A2 @ ( some @ B @ ( F2 @ B2 ) ) ) ) ).
% map_option_o_map_upd
thf(fact_8151_nths__nths,axiom,
! [A: $tType,Xs: list @ A,A3: set @ nat,B3: set @ nat] :
( ( nths @ A @ ( nths @ A @ Xs @ A3 ) @ B3 )
= ( nths @ A @ Xs
@ ( collect @ nat
@ ^ [I3: nat] :
( ( member @ nat @ I3 @ A3 )
& ( member @ nat
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [I10: nat] :
( ( member @ nat @ I10 @ A3 )
& ( ord_less @ nat @ I10 @ I3 ) ) ) )
@ B3 ) ) ) ) ) ).
% nths_nths
thf(fact_8152_vimage__image__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A] :
( ( vimage @ A @ B @ F2 @ ( image @ A @ B @ F2 @ A3 ) )
= ( collect @ A
@ ^ [Y2: A] :
? [X: A] :
( ( member @ A @ X @ A3 )
& ( ( F2 @ X )
= ( F2 @ Y2 ) ) ) ) ) ).
% vimage_image_eq
thf(fact_8153_option_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F2: B > nat,G: A > B] :
( ( comp @ ( option @ B ) @ nat @ ( option @ A ) @ ( size_option @ B @ F2 ) @ ( map_option @ A @ B @ G ) )
= ( size_option @ A @ ( comp @ B @ nat @ A @ F2 @ G ) ) ) ).
% option.size_gen_o_map
thf(fact_8154_map__option_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F2: B > C,G: A > B] :
( ( comp @ ( option @ B ) @ ( option @ C ) @ ( option @ A ) @ ( map_option @ B @ C @ F2 ) @ ( map_option @ A @ B @ G ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) ) ) ).
% map_option.comp
thf(fact_8155_option_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F2: A > B,V2: option @ A] :
( ( map_option @ B @ C @ G @ ( map_option @ A @ B @ F2 @ V2 ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ G @ F2 ) @ V2 ) ) ).
% option.map_comp
thf(fact_8156_map__option_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F2: B > C,G: A > B,Option: option @ A] :
( ( map_option @ B @ C @ F2 @ ( map_option @ A @ B @ G @ Option ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) @ Option ) ) ).
% map_option.compositionality
thf(fact_8157_option_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( map_option @ A @ B @ F2 @ ( none @ A ) )
= ( none @ B ) ) ).
% option.simps(8)
thf(fact_8158_option_Omap__sel,axiom,
! [B: $tType,A: $tType,A2: option @ A,F2: A > B] :
( ( A2
!= ( none @ A ) )
=> ( ( the2 @ B @ ( map_option @ A @ B @ F2 @ A2 ) )
= ( F2 @ ( the2 @ A @ A2 ) ) ) ) ).
% option.map_sel
thf(fact_8159_option_Omap__id,axiom,
! [A: $tType,T2: option @ A] :
( ( map_option @ A @ A @ ( id @ A ) @ T2 )
= T2 ) ).
% option.map_id
thf(fact_8160_option_Omap__id0,axiom,
! [A: $tType] :
( ( map_option @ A @ A @ ( id @ A ) )
= ( id @ ( option @ A ) ) ) ).
% option.map_id0
thf(fact_8161_map__option_Oidentity,axiom,
! [A: $tType] :
( ( map_option @ A @ A
@ ^ [X: A] : X )
= ( id @ ( option @ A ) ) ) ).
% map_option.identity
thf(fact_8162_option_Omap__ident,axiom,
! [A: $tType,T2: option @ A] :
( ( map_option @ A @ A
@ ^ [X: A] : X
@ T2 )
= T2 ) ).
% option.map_ident
thf(fact_8163_Bex__def,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A6: set @ A,P3: A > $o] :
? [X: A] :
( ( member @ A @ X @ A6 )
& ( P3 @ X ) ) ) ) ).
% Bex_def
thf(fact_8164_Bex__def__raw,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A6: set @ A,P3: A > $o] :
? [X: A] :
( ( member @ A @ X @ A6 )
& ( P3 @ X ) ) ) ) ).
% Bex_def_raw
thf(fact_8165_map__option__cong,axiom,
! [B: $tType,A: $tType,X3: option @ A,Y: option @ A,F2: A > B,G: A > B] :
( ( X3 = Y )
=> ( ! [A5: A] :
( ( Y
= ( some @ A @ A5 ) )
=> ( ( F2 @ A5 )
= ( G @ A5 ) ) )
=> ( ( map_option @ A @ B @ F2 @ X3 )
= ( map_option @ A @ B @ G @ Y ) ) ) ) ).
% map_option_cong
thf(fact_8166_option_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F2: A > B,X22: A] :
( ( map_option @ A @ B @ F2 @ ( some @ A @ X22 ) )
= ( some @ B @ ( F2 @ X22 ) ) ) ).
% option.simps(9)
thf(fact_8167_Bex__fold,axiom,
! [A: $tType,A3: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ? [X: A] :
( ( member @ A @ X @ A3 )
& ( P @ X ) ) )
= ( finite_fold @ A @ $o
@ ^ [K3: A,S8: $o] :
( S8
| ( P @ K3 ) )
@ $false
@ A3 ) ) ) ).
% Bex_fold
thf(fact_8168_option_Oinj__map,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( option @ A ) @ ( option @ B ) @ ( map_option @ A @ B @ F2 ) @ ( top_top @ ( set @ ( option @ A ) ) ) ) ) ).
% option.inj_map
thf(fact_8169_image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [F3: A > B,A6: set @ A] :
( collect @ B
@ ^ [Y2: B] :
? [X: A] :
( ( member @ A @ X @ A6 )
& ( Y2
= ( F3 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_8170_map__option__case,axiom,
! [A: $tType,B: $tType] :
( ( map_option @ B @ A )
= ( ^ [F3: B > A] :
( case_option @ ( option @ A ) @ B @ ( none @ A )
@ ^ [X: B] : ( some @ A @ ( F3 @ X ) ) ) ) ) ).
% map_option_case
thf(fact_8171_max__extp_Ocases,axiom,
! [A: $tType,R: A > A > $o,A12: set @ A,A23: set @ A] :
( ( max_extp @ A @ R @ A12 @ A23 )
=> ~ ( ( finite_finite2 @ A @ A12 )
=> ( ( finite_finite2 @ A @ A23 )
=> ( ( A23
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ A12 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A23 )
& ( R @ X2 @ Xa3 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_8172_max__extp_Osimps,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R6: A > A > $o,A1: set @ A,A22: set @ A] :
( ( finite_finite2 @ A @ A1 )
& ( finite_finite2 @ A @ A22 )
& ( A22
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
& ! [X: A] :
( ( member @ A @ X @ A1 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ A22 )
& ( R6 @ X @ Y2 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_8173_max__extp_Omax__extI,axiom,
! [A: $tType,X8: set @ A,Y7: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ X8 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( Y7
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Y7 )
& ( R @ X5 @ Xa ) ) )
=> ( max_extp @ A @ R @ X8 @ Y7 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_8174_map__of__map,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,Xs: list @ ( product_prod @ A @ C )] :
( ( map_of @ A @ B
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [K3: A,V5: C] : ( product_Pair @ A @ B @ K3 @ ( F2 @ V5 ) ) )
@ Xs ) )
= ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F2 ) @ ( map_of @ A @ C @ Xs ) ) ) ).
% map_of_map
thf(fact_8175_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 )] :
? [X6: set @ A,Y9: set @ A] :
( ( Uu3
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X6 @ Y9 ) )
& ( X6
!= ( bot_bot @ ( set @ A ) ) )
& ! [X: A] :
( ( member @ A @ X @ Y9 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ X6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ R5 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_8176_map__project__def,axiom,
! [B: $tType,A: $tType] :
( ( map_project @ A @ B )
= ( ^ [F3: A > ( option @ B ),A6: set @ A] :
( collect @ B
@ ^ [B4: B] :
? [X: A] :
( ( member @ A @ X @ A6 )
& ( ( F3 @ X )
= ( some @ B @ B4 ) ) ) ) ) ) ).
% map_project_def
thf(fact_8177_and__num_Oelims,axiom,
! [X3: num,Xa2: num,Y: option @ num] :
( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
= Y )
=> ( ( ( X3 = one2 )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X3 = one2 )
=> ( ? [N3: num] :
( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X3 = one2 )
=> ( ? [N3: num] :
( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ? [M3: num] :
( X3
= ( bit0 @ M3 ) )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ! [M3: num] :
( ( X3
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X3
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
=> ( ( ? [M3: num] :
( X3
= ( bit1 @ M3 ) )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ! [M3: num] :
( ( X3
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
=> ~ ! [M3: num] :
( ( X3
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
thf(fact_8178_and__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one2 )
= ( none @ num ) ) ).
% and_num.simps(4)
thf(fact_8179_and__num_Osimps_I2_J,axiom,
! [N2: num] :
( ( bit_un7362597486090784418nd_num @ one2 @ ( bit0 @ N2 ) )
= ( none @ num ) ) ).
% and_num.simps(2)
thf(fact_8180_and__num__eq__None__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N2: num] :
( ( ( bit_un7362597486090784418nd_num @ M @ N2 )
= ( none @ num ) )
= ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
= ( zero_zero @ A ) ) ) ) ).
% and_num_eq_None_iff
thf(fact_8181_numeral__and__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ) ).
% numeral_and_num
thf(fact_8182_xor__num_Oelims,axiom,
! [X3: num,Xa2: num,Y: option @ num] :
( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
= Y )
=> ( ( ( X3 = one2 )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( some @ num @ ( bit1 @ N3 ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( some @ num @ ( bit0 @ N3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X3
= ( bit0 @ M3 ) )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( some @ num @ ( bit1 @ M3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X3
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X3
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X3
= ( bit1 @ M3 ) )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X3
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X3
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.elims
thf(fact_8183_option_Orec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,G: C,Ga: B > C,F2: A > B] :
( ( comp @ ( option @ B ) @ C @ ( option @ A ) @ ( rec_option @ C @ B @ G @ Ga ) @ ( map_option @ A @ B @ F2 ) )
= ( rec_option @ C @ A @ G
@ ^ [X: A] : ( Ga @ ( F2 @ X ) ) ) ) ).
% option.rec_o_map
thf(fact_8184_option_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F22: A > C] :
( ( rec_option @ C @ A @ F1 @ F22 @ ( none @ A ) )
= F1 ) ).
% option.simps(6)
thf(fact_8185_xor__num_Osimps_I1_J,axiom,
( ( bit_un2480387367778600638or_num @ one2 @ one2 )
= ( none @ num ) ) ).
% xor_num.simps(1)
% Type constructors (813)
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A10: $tType,A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( condit1219197933456340205attice @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
! [A10: $tType,A20: $tType] :
( ( counta3822494911875563373attice @ A20 )
=> ( counta3822494911875563373attice @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A10: $tType,A20: $tType] :
( ( comple592849572758109894attice @ A20 )
=> ( comple592849572758109894attice @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A10: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounde4967611905675639751up_bot @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A10: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounde4346867609351753570nf_top @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A10: $tType,A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple6319245703460814977attice @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A10: $tType,A20: $tType] :
( ( boolea8198339166811842893lgebra @ A20 )
=> ( boolea8198339166811842893lgebra @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A10: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounded_lattice_bot @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A10: $tType,A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple9053668089753744459l_ccpo @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A10: $tType,A20: $tType] :
( ( semilattice_sup @ A20 )
=> ( semilattice_sup @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A10: $tType,A20: $tType] :
( ( semilattice_inf @ A20 )
=> ( semilattice_inf @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A10: $tType,A20: $tType] :
( ( distrib_lattice @ A20 )
=> ( distrib_lattice @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A10: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounded_lattice @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A10: $tType,A20: $tType] :
( ( order_top @ A20 )
=> ( order_top @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A10: $tType,A20: $tType] :
( ( order_bot @ A20 )
=> ( order_bot @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A10: $tType,A20: $tType] :
( ( preorder @ A20 )
=> ( preorder @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A10: $tType,A20: $tType] :
( ( ( finite_finite @ A10 )
& ( finite_finite @ A20 ) )
=> ( finite_finite @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A10: $tType,A20: $tType] :
( ( lattice @ A20 )
=> ( lattice @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A10: $tType,A20: $tType] :
( ( order @ A20 )
=> ( order @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A10: $tType,A20: $tType] :
( ( top @ A20 )
=> ( top @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A10: $tType,A20: $tType] :
( ( ord @ A20 )
=> ( ord @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A10: $tType,A20: $tType] :
( ( bot @ A20 )
=> ( bot @ ( A10 > A20 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A10: $tType,A20: $tType] :
( ( uminus @ A20 )
=> ( uminus @ ( A10 > A20 ) ) ) ).
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___GCD_Osemiring__gcd__mult__normalize,axiom,
semiri6843258321239162965malize @ int ).
thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add @ int ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors @ int ).
thf(tcon_Int_Oint___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___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_5,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_6,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_7,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_8,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_9,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
abs_if @ int ).
thf(tcon_Int_Oint___GCD_Oring__gcd,axiom,
ring_gcd @ int ).
thf(tcon_Int_Oint___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Groups_Oplus,axiom,
plus @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_10,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_12,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_13,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_14,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_15,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_16,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_17,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_18,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_19,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_20,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_21,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_22,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_23,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_24,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_25,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_26,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_27,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_28,axiom,
topolo4958980785337419405_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_29,axiom,
topolo1944317154257567458pology @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_30,axiom,
topolo8865339358273720382pology @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_31,axiom,
topolo5987344860129210374id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_32,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_33,axiom,
topolo2564578578187576103pology @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_34,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_35,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_36,axiom,
topolo4211221413907600880p_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_37,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_38,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_39,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_40,axiom,
semiri6843258321239162965malize @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_41,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_42,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_43,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_44,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_45,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_46,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_47,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space_48,axiom,
topological_t1_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_49,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_50,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_51,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_52,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_53,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_54,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_55,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_56,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_57,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_58,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_59,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_60,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_61,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_62,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_63,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_64,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_65,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_66,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_67,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_68,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_69,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_70,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_71,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_72,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_73,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_74,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_75,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_76,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_77,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_78,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_79,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_80,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_81,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_82,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_83,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_84,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_85,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_86,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_87,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_88,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_89,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_90,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom_91,axiom,
semidom @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_92,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_93,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Power_Opower_94,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_95,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_96,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_97,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_98,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_99,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_100,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_101,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_102,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_103,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Oplus_104,axiom,
plus @ num ).
thf(tcon_Num_Onum___Nat_Osize_105,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_106,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_107,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_108,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_109,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_110,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_111,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_112,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_113,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_114,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_115,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_116,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_117,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_118,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_119,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_120,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_121,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_122,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_123,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_124,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_125,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_126,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_127,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_128,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_129,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_130,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_131,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_132,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_133,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_134,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_135,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_136,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_137,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_138,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_139,axiom,
distrib_lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_140,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_141,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_142,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_143,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_144,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_145,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_146,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_147,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_148,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_149,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_150,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_151,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_152,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_153,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_154,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_155,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_156,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_157,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_158,axiom,
ab_group_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__neq__one_159,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_160,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_161,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_162,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_163,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_164,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_165,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_166,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_167,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_168,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_169,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_170,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_171,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_172,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_173,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_174,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_175,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_176,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_177,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_178,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_179,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom_180,axiom,
semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_181,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_182,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_183,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_184,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_185,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_186,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_187,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_188,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_189,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_190,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_191,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_192,axiom,
dvd @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_193,axiom,
! [A10: $tType] : ( condit1219197933456340205attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_194,axiom,
! [A10: $tType] : ( counta3822494911875563373attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_195,axiom,
! [A10: $tType] : ( comple592849572758109894attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_196,axiom,
! [A10: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_197,axiom,
! [A10: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_198,axiom,
! [A10: $tType] : ( comple6319245703460814977attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_199,axiom,
! [A10: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_200,axiom,
! [A10: $tType] : ( bounded_lattice_bot @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_201,axiom,
! [A10: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_202,axiom,
! [A10: $tType] : ( semilattice_sup @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_203,axiom,
! [A10: $tType] : ( semilattice_inf @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_204,axiom,
! [A10: $tType] : ( distrib_lattice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_205,axiom,
! [A10: $tType] : ( bounded_lattice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_206,axiom,
! [A10: $tType] : ( order_top @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_207,axiom,
! [A10: $tType] : ( order_bot @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_208,axiom,
! [A10: $tType] : ( preorder @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_209,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( finite_finite @ ( set @ A10 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_210,axiom,
! [A10: $tType] : ( lattice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_211,axiom,
! [A10: $tType] : ( order @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_212,axiom,
! [A10: $tType] : ( top @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_213,axiom,
! [A10: $tType] : ( ord @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_214,axiom,
! [A10: $tType] : ( bot @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_215,axiom,
! [A10: $tType] : ( uminus @ ( set @ A10 ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_216,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_217,axiom,
counta3822494911875563373attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_218,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_219,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_220,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_221,axiom,
topolo8865339358273720382pology @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_222,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_223,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_224,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_225,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_226,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_227,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_228,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_229,axiom,
topological_t1_space @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_230,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_231,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_232,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_233,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_234,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_235,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_236,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_237,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_238,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_239,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_240,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_241,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_242,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_243,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_244,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_245,axiom,
uminus @ $o ).
thf(tcon_List_Olist___Nat_Osize_246,axiom,
! [A10: $tType] : ( size @ ( list @ A10 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_247,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_248,axiom,
condit1219197933456340205attice @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_249,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_250,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_251,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_252,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_253,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_254,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_255,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_256,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_257,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_258,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_259,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_260,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_261,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_262,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_263,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_264,axiom,
linord4140545234300271783up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_265,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_266,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_267,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_268,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_269,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_270,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_271,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_272,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_273,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_274,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_275,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_276,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_277,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_278,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_279,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_280,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_281,axiom,
comm_s4317794764714335236cancel @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
topolo1633459387980952147up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_282,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot1__space_283,axiom,
topological_t1_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_284,axiom,
ordere2520102378445227354miring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_285,axiom,
linord6961819062388156250ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_286,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_287,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring_288,axiom,
linordered_semiring @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Obanach,axiom,
real_Vector_banach @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring__0_289,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_290,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder_291,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_292,axiom,
semilattice_sup @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__inf_293,axiom,
semilattice_inf @ real ).
thf(tcon_Real_Oreal___Lattices_Odistrib__lattice_294,axiom,
distrib_lattice @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_295,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_296,axiom,
semiring_1_cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_297,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_298,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field_299,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_300,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_301,axiom,
ordered_ring_abs @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_302,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_303,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_304,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_305,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_306,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order_307,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_308,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_309,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_310,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_311,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__abs__sgn_312,axiom,
field_abs_sgn @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring_313,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_314,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_315,axiom,
comm_semiring @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_316,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_317,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0_318,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_319,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_320,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_321,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_322,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_323,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_324,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__divide_325,axiom,
idom_divide @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_326,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_327,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_328,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_329,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_330,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_331,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_332,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_333,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_334,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_335,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_336,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_337,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_338,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_339,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse_340,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom_341,axiom,
semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_342,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_343,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_344,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if_345,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Fields_Ofield_346,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_347,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_348,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_349,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_350,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oring_351,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_352,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_353,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_354,axiom,
dvd @ real ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_355,axiom,
! [A10: $tType,A20: $tType] :
( ( ( finite_finite @ A10 )
& ( finite_finite @ A20 ) )
=> ( finite_finite @ ( sum_sum @ A10 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_356,axiom,
! [A10: $tType,A20: $tType] : ( size @ ( sum_sum @ A10 @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_357,axiom,
! [A10: $tType] : ( condit1219197933456340205attice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_358,axiom,
! [A10: $tType] : ( counta3822494911875563373attice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_359,axiom,
! [A10: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_360,axiom,
! [A10: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_361,axiom,
! [A10: $tType] : ( comple6319245703460814977attice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_362,axiom,
! [A10: $tType] : ( bounded_lattice_bot @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_363,axiom,
! [A10: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_364,axiom,
! [A10: $tType] : ( semilattice_sup @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_365,axiom,
! [A10: $tType] : ( semilattice_inf @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_366,axiom,
! [A10: $tType] : ( distrib_lattice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_367,axiom,
! [A10: $tType] : ( bounded_lattice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_368,axiom,
! [A10: $tType] : ( order_top @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_369,axiom,
! [A10: $tType] : ( order_bot @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_370,axiom,
! [A10: $tType] : ( preorder @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_371,axiom,
! [A10: $tType] : ( lattice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_372,axiom,
! [A10: $tType] : ( order @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_373,axiom,
! [A10: $tType] : ( top @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_374,axiom,
! [A10: $tType] : ( ord @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_375,axiom,
! [A10: $tType] : ( bot @ ( filter @ A10 ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_376,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( finite_finite @ ( option @ A10 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_377,axiom,
! [A10: $tType] : ( size @ ( option @ A10 ) ) ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_378,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_379,axiom,
topolo3112930676232923870pology @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_380,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_381,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_382,axiom,
semiri6575147826004484403cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_383,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_384,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_385,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_386,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_387,axiom,
real_V5047593784448816457lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_388,axiom,
real_V768167426530841204y_dist @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_389,axiom,
topolo5987344860129210374id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_390,axiom,
semiri2026040879449505780visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_391,axiom,
real_V2191834092415804123ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_392,axiom,
real_V8037385150606011577_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_393,axiom,
topolo4211221413907600880p_mult @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_394,axiom,
topolo7287701948861334536_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_395,axiom,
topolo8386298272705272623_space @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_396,axiom,
semiri3467727345109120633visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_397,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_398,axiom,
topolo1287966508704411220up_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_399,axiom,
real_V4867850818363320053vector @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_400,axiom,
cancel2418104881723323429up_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_401,axiom,
ring_15535105094025558882visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_402,axiom,
real_V7773925162809079976_field @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_403,axiom,
topolo6943815403480290642id_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_404,axiom,
cancel1802427076303600483id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_405,axiom,
comm_s4317794764714335236cancel @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__group__add_406,axiom,
topolo1633459387980952147up_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_407,axiom,
topological_t2_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_408,axiom,
topological_t1_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_409,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_410,axiom,
real_Vector_banach @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_411,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_412,axiom,
semiring_1_cancel @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_413,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_414,axiom,
ab_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_415,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_416,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_417,axiom,
comm_semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_418,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_419,axiom,
semidom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_420,axiom,
semiring_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_421,axiom,
semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_422,axiom,
field_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_423,axiom,
division_ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_424,axiom,
comm_semiring @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_425,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_426,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_427,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_428,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_429,axiom,
idom_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_430,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_431,axiom,
idom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_432,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_433,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_434,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_435,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_436,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_437,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_438,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_439,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_440,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring_441,axiom,
semiring @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_442,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom_443,axiom,
semidom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_444,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_445,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_446,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_447,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_448,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_449,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oplus_450,axiom,
plus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring_451,axiom,
ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_452,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_453,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_454,axiom,
dvd @ complex ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_455,axiom,
condit6923001295902523014norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_456,axiom,
condit1219197933456340205attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_457,axiom,
counta3822494911875563373attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_458,axiom,
comple592849572758109894attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_459,axiom,
strict9044650504122735259up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_460,axiom,
strict7427464778891057005id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_461,axiom,
canoni5634975068530333245id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_462,axiom,
bounde4967611905675639751up_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_463,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_464,axiom,
linord4140545234300271783up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_465,axiom,
comple6319245703460814977attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_466,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_467,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_468,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_469,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_470,axiom,
bounded_lattice_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_471,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_472,axiom,
comple9053668089753744459l_ccpo @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_473,axiom,
semilattice_sup @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_474,axiom,
semilattice_inf @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_475,axiom,
distrib_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_476,axiom,
bounded_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_477,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_478,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_479,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_480,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_481,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_482,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_483,axiom,
comm_semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_484,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_485,axiom,
semiring_numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_486,axiom,
semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_487,axiom,
zero_less_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_488,axiom,
comm_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_489,axiom,
wellorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_490,axiom,
order_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_491,axiom,
order_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_492,axiom,
semiring_char_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_493,axiom,
zero_neq_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_494,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_495,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_496,axiom,
monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_497,axiom,
monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_498,axiom,
semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_499,axiom,
semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_500,axiom,
lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_501,axiom,
mult_zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_502,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_503,axiom,
semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Otop_504,axiom,
top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_505,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Obot_506,axiom,
bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Power_Opower_507,axiom,
power @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_508,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_509,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oplus_510,axiom,
plus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_511,axiom,
one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Odvd_512,axiom,
dvd @ extended_enat ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_513,axiom,
! [A10: $tType,A20: $tType] :
( ( ( topolo4958980785337419405_space @ A10 )
& ( topolo4958980785337419405_space @ A20 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A10 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_514,axiom,
! [A10: $tType,A20: $tType] :
( ( ( topological_t2_space @ A10 )
& ( topological_t2_space @ A20 ) )
=> ( topological_t2_space @ ( product_prod @ A10 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_515,axiom,
! [A10: $tType,A20: $tType] :
( ( ( topological_t1_space @ A10 )
& ( topological_t1_space @ A20 ) )
=> ( topological_t1_space @ ( product_prod @ A10 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_516,axiom,
! [A10: $tType,A20: $tType] :
( ( ( finite_finite @ A10 )
& ( finite_finite @ A20 ) )
=> ( finite_finite @ ( product_prod @ A10 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_517,axiom,
! [A10: $tType,A20: $tType] : ( size @ ( product_prod @ A10 @ A20 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_518,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_519,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_520,axiom,
counta3822494911875563373attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_521,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_522,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_523,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_524,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_525,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_526,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_527,axiom,
bounded_lattice_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_528,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_529,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_530,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_531,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_532,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_533,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_534,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_535,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_536,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_537,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_538,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_539,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_540,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_541,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_542,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_543,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_544,axiom,
uminus @ product_unit ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_545,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_546,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_547,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_548,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_549,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_550,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_551,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_552,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_553,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_554,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_555,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_556,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_557,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_558,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_559,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_560,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_561,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_562,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_563,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_564,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_565,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_566,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_567,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_568,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_569,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_570,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_571,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_572,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_573,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_574,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_575,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_576,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_577,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_578,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_579,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_580,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_581,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_582,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_583,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_584,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_585,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_586,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_587,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_588,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_589,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_590,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_591,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_592,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_593,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_594,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_595,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_596,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_597,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_598,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_599,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_600,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_601,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_602,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_603,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_604,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_605,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_606,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_607,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_608,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_609,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_610,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_611,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_612,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_613,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_614,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_615,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_616,axiom,
idom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_617,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_618,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_619,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_620,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_621,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_622,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_623,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_624,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_625,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_626,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_627,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_628,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_629,axiom,
semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_630,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_631,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_632,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_633,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_634,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_635,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_636,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_637,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_638,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_639,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_640,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_641,axiom,
dvd @ code_integer ).
thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_642,axiom,
size @ vEBT_VEBT ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $true @ X3 @ Y )
= X3 ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X6: A] : ( P @ X6 ) ) ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [X2: nat] :
( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ maxs ) @ X2 )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------