TPTP Problem File: ITP235^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP235^2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_InsertCorrectness 01015_065053
% 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 : 0067_VEBT_InsertCorrectness_01015_065053 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 9762 (2704 unt; 741 typ; 0 def)
% Number of atoms : 31396 (8943 equ; 0 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 178902 (2487 ~; 360 |;2570 &;158711 @)
% ( 0 <=>;14774 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 8 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 5455 (5455 >; 0 *; 0 +; 0 <<)
% Number of symbols : 732 ( 729 usr; 15 con; 0-9 aty)
% Number of variables : 31661 (2864 ^;27070 !; 985 ?;31661 :)
% ( 742 !>; 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 21:53:14.167
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_t_VEBT__Definitions_OVEBT,type,
vEBT_VEBT: $tType ).
thf(ty_t_Code__Numeral_Ointeger,type,
code_integer: $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Complex_Ocomplex,type,
complex: $tType ).
thf(ty_t_String_Oliteral,type,
literal: $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Rat_Orat,type,
rat: $tType ).
thf(ty_t_Num_Onum,type,
num: $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
% Explicit typings (722)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oabs__if,type,
abs_if:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom,type,
semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__Gcd,type,
semiring_Gcd:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd,type,
semiring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_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_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__lattice,type,
finite_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_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_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__modulo,type,
semidom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oalgebraic__semidom,type,
algebraic_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__cancel,type,
semiring_1_cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Odistrib__lattice,type,
distrib_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Obanach,type,
real_Vector_banach:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring,type,
linordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Partial__Order_Occpo,type,
comple9053668089753744459l_ccpo:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot1__space,type,
topological_t1_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot2__space,type,
topological_t2_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot3__space,type,
topological_t3_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot4__space,type,
topological_t4_space:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord4710134922213307826strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1802427076303600483id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__monoid__add,type,
topolo6943815403480290642id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__field,type,
real_V7773925162809079976_field:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
real_V4867850818363320053vector:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim2362893244070406136eiling:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere166539214618696060dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere6658533253407199908up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__ab__group__add,type,
topolo1287966508704411220up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V7819770556892013058_space:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Operfect__space,type,
topolo8386298272705272623_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ouniform__space,type,
topolo7287701948861334536_space:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__semigroup__mult,type,
topolo4211221413907600880p_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ocomplete__space,type,
real_V8037385150606011577_space:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__algebra__1,type,
real_V2191834092415804123ebra_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
semiri2026040879449505780visors:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Oorder__topology,type,
topolo2564578578187576103pology:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
bit_ri3973907225187159222ations:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple6319245703460814977attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__comm__monoid__add,type,
topolo5987344860129210374id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ounbounded__dense__linorder,type,
unboun7993243217541854897norder:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ouniformity__dist,type,
real_V768167426530841204y_dist:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim462609752435547400_field:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__div__algebra,type,
real_V5047593784448816457lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
bounde4346867609351753570nf_top:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde4967611905675639751up_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__field,type,
real_V3459762299906320749_field:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Odiscrete__topology,type,
topolo8865339358273720382pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Olinorder__topology,type,
topolo1944317154257567458pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo4958980785337419405_space:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V822414075346904944vector:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord2810124833399127020strict:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
bit_se359711467146920520ations:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere2412721322843649153imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere580206878836729694up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere1170586879665033532d_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict9044650504122735259up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oordered__real__vector,type,
real_V5355595471888546746vector:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra,type,
real_V4412858255891104859lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri6575147826004484403cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid8851590272496341667cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra__1,type,
real_V2822296259951069270ebra_1:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
comple592849572758109894attice:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__div__algebra,type,
real_V8999393235501362500lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
topolo3112930676232923870pology:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
euclid4440199948858584721cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
euclid3128863361964157862miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Olinear__continuum__topology,type,
topolo8458572112393995274pology:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere1937475149494474687imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit5016429287641298734tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid5411537665997757685th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Countable__Complete__Lattices_Ocountable__complete__lattice,type,
counta3822494911875563373attice:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
thf(sy_cl_Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,type,
counta4013691401010221786attice:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit6923001295902523014norder:
!>[A: $tType] : $o ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofrac,type,
archimedean_frac:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocexp,type,
bNF_Cardinal_cexp:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
bNF_Ca4139267488887388095finite:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocsum,type,
bNF_Cardinal_csum:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Oczero,type,
bNF_Cardinal_czero:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OcardSuc,type,
bNF_Ca8387033319878233205ardSuc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of,type,
bNF_Ca6860139660246222851ard_of:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on,type,
bNF_Ca8970107618336181345der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
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_OregularCard,type,
bNF_Ca7133664381575040944arCard:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
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__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2p,type,
bNF_Greatest_image2p:
!>[C: $tType,A: $tType,D: $tType,B: $tType] : ( ( C > A ) > ( D > B ) > ( C > D > $o ) > A > B > $o ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
bNF_Gr4221423524335903396lImage:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( B > A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
bNF_Gr7122648621184425601vImage:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
bNF_Gr1419584066657907630d_pred:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > $o ) ).
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__Constructions_OofilterIncl,type,
bNF_We413866401316099525erIncl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OordIso,type,
bNF_Wellorder_ordIso:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
bNF_Wellorder_ordLeq:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OordLess,type,
bNF_We4044943003108391690rdLess:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Embedding_Oembed,type,
bNF_Wellorder_embed:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_OembedS,type,
bNF_Wellorder_embedS:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_Oiso,type,
bNF_Wellorder_iso:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
bNF_Wellorder_wo_rel:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
bNF_We4791949203932849705sMinim:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
bNF_We1388413361240627857o_max2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > A > A ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
bNF_We6954850376910717587_minim:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc,type,
bNF_Wellorder_wo_suc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A ) ).
thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
basic_BNF_size_prod:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
thf(sy_c_Basic__BNFs_Ofsts,type,
basic_fsts:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_Basic__BNFs_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
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_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_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
bit_un2480387367778600638or_num: num > num > ( option @ num ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff__axioms,type,
boolea5476839437570043046axioms:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > ( A > A > A ) > $o ) ).
thf(sy_c_Code__Numeral_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_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
comple1908693960933563346ssible:
!>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > $o ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp,type,
comple7512665784863727008ratesp:
!>[A: $tType] : ( ( A > A ) > A > $o ) ).
thf(sy_c_Complete__Partial__Order_Ochain,type,
comple1602240252501008431_chain:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Complex_OArg,type,
arg: complex > real ).
thf(sy_c_Complex_Ocis,type,
cis: real > complex ).
thf(sy_c_Complex_Ocnj,type,
cnj: complex > complex ).
thf(sy_c_Complex_Ocomplex_OComplex,type,
complex2: real > real > complex ).
thf(sy_c_Complex_Ocomplex_OIm,type,
im: complex > real ).
thf(sy_c_Complex_Ocomplex_ORe,type,
re: complex > real ).
thf(sy_c_Complex_Ocsqrt,type,
csqrt: complex > complex ).
thf(sy_c_Complex_Oimaginary__unit,type,
imaginary_unit: complex ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
condit941137186595557371_above:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
condit1013018076250108175_below:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Countable__Set_Ocountable,type,
countable_countable:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Countable__Set_Ofrom__nat__into,type,
counta4804993851260445106t_into:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Countable__Set_Oto__nat__on,type,
countable_to_nat_on:
!>[A: $tType] : ( ( set @ A ) > A > nat ) ).
thf(sy_c_Deriv_Odifferentiable,type,
differentiable:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__derivative,type,
has_derivative:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__field__derivative,type,
has_field_derivative:
!>[A: $tType] : ( ( A > A ) > A > ( filter @ A ) > $o ) ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_Equiv__Relations_Ocongruent,type,
equiv_congruent:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Equiv__Relations_Ocongruent2,type,
equiv_congruent2:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B > C ) > $o ) ).
thf(sy_c_Equiv__Relations_Oequiv,type,
equiv_equiv:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Equiv__Relations_Oproj,type,
equiv_proj:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
thf(sy_c_Equiv__Relations_Oquotient,type,
equiv_quotient:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
euclid6346220572633701492n_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
euclid7384307370059645450egment:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Extended__Nat_OeSuc,type,
extended_eSuc: extended_enat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
extended_case_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
extend4730790105801354508finity:
!>[A: $tType] : A ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Ocofinite,type,
cofinite:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofilter_OAbs__filter,type,
abs_filter:
!>[A: $tType] : ( ( ( A > $o ) > $o ) > ( filter @ A ) ) ).
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_Ofrequently,type,
frequently:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( filter @ B ) > ( filter @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Filter_Orel__filter,type,
rel_filter:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( filter @ A ) > ( filter @ B ) > $o ) ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
finite4664212375090638736ute_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
finite673082921795544331dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__on__axioms,type,
finite4980608107308702382axioms:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Finite__Set_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B > $o ) ).
thf(sy_c_Finite__Set_Ofolding,type,
finite_folding:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem,type,
finite_folding_idem:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem__axioms,type,
finite7837460588564673216axioms:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem__on,type,
finite1890593828518410140dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem__on__axioms,type,
finite6916993218817215295axioms:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on,type,
finite_folding_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on_OF,type,
finite_folding_F:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
thf(sy_c_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
thf(sy_c_Fun_Ostrict__mono__on,type,
strict_mono_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Fun__Def_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_Obounded__quasi__semilattice__set,type,
bounde6485984586167503788ce_set:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
thf(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
bounde2362111253966948842tice_F:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( set @ A ) > A ) ).
thf(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
semiri4206861660011772517g_char:
!>[A: $tType] : ( ( itself @ A ) > nat ) ).
thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
semiring_gcd_Gcd_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_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_OUniq,type,
uniq:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
complete_lattice_gfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini527867602293511546merate:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Int_OAbs__Integ,type,
abs_Integ: ( product_prod @ nat @ nat ) > int ).
thf(sy_c_Int_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_Ointrel,type,
intrel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Opcr__int,type,
pcr_int: ( product_prod @ nat @ nat ) > int > $o ).
thf(sy_c_Int_Opower__int,type,
power_int:
!>[A: $tType] : ( A > int > A ) ).
thf(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_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,type,
lattices_ord_arg_min:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
lattic501386751177426532rg_min:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
lattic4895041142388067077er_set:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__set,type,
lattic149705377957585745ce_set:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF,type,
lattic1715443433743089157tice_F:
!>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Limits_OBfun,type,
bfun:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Limits_OZfun,type,
zfun:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Limits_Oat__infinity,type,
at_infinity:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
thf(sy_c_List_Oarg__min__list__rel,type,
arg_min_list_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( list @ A ) ) > ( product_prod @ ( A > B ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocoset,type,
coset:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
thf(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
thf(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olinorder_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__insert__key,type,
linord329482645794927042rt_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Olist__all2,type,
list_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Orec__list,type,
rec_list:
!>[C: $tType,A: $tType] : ( C > ( A > ( list @ A ) > C > C ) > ( list @ A ) > C ) ).
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_Olists,type,
lists:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Onths,type,
nths:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_List_Onull,type,
null:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Opartition,type,
partition:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OremoveAll,type,
removeAll:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_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_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__add,type,
map_add:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > A > ( option @ 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_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: ( set @ nat ) > nat ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_NthRoot_Osqrt,type,
sqrt: real > real ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( num > num > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Ocase__num,type,
case_num:
!>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
thf(sy_c_Num_Onum_Orec__num,type,
rec_num:
!>[A: $tType] : ( A > ( num > A > A ) > ( num > A > A ) > num > A ) ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Num_Oring__1__class_Oiszero,type,
ring_1_iszero:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Osqr,type,
sqr: num > num ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
thf(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Order__Continuity_Ocountable__complete__lattice__class_Occlfp,type,
order_532582986084564980_cclfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Order__Continuity_Oinf__continuous,type,
order_inf_continuous:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Order__Continuity_Osup__continuous,type,
order_sup_continuous:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Order__Relation_OAboveS,type,
order_AboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ 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_Oofilter,type,
order_ofilter:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Order__Relation_Opartial__order__on,type,
order_7125193373082350890der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Ounder,type,
order_under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord_Omax,type,
max:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord_Omin,type,
min:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Partial__Function_Oflat__lub,type,
partial_flat_lub:
!>[A: $tType] : ( A > ( set @ A ) > A ) ).
thf(sy_c_Power_Opower_Opower,type,
power2:
!>[A: $tType] : ( A > ( A > A > A ) > A > nat > A ) ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Rat_OAbs__Rat,type,
abs_Rat: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_OFract,type,
fract: int > int > rat ).
thf(sy_c_Rat_ORep__Rat,type,
rep_Rat: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
field_char_0_Rats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : ( rat > A ) ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Opcr__rat,type,
pcr_rat: ( product_prod @ int @ int ) > rat > $o ).
thf(sy_c_Rat_Opositive,type,
positive: rat > $o ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oratrel,type,
ratrel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Real_OReal,type,
real2: ( nat > rat ) > real ).
thf(sy_c_Real_Ocauchy,type,
cauchy: ( nat > rat ) > $o ).
thf(sy_c_Real_Opcr__real,type,
pcr_real: ( nat > rat ) > real > $o ).
thf(sy_c_Real_Opositive,type,
positive2: real > $o ).
thf(sy_c_Real_Orealrel,type,
realrel: ( nat > rat ) > ( nat > rat ) > $o ).
thf(sy_c_Real_Orep__real,type,
rep_real: real > nat > rat ).
thf(sy_c_Real_Ovanishes,type,
vanishes: ( nat > rat ) > $o ).
thf(sy_c_Real__Vector__Spaces_OReals,type,
real_Vector_Reals:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__bilinear,type,
real_V2442710119149674383linear:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear,type,
real_V3181309239436604168linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear__axioms,type,
real_V4916620083959148203axioms:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Oconstruct,type,
real_V4425403222259421789struct:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B ) ).
thf(sy_c_Real__Vector__Spaces_Odependent,type,
real_V358717886546972837endent:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Odim,type,
real_Vector_dim:
!>[A: $tType] : ( ( set @ A ) > nat ) ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
real_V557655796197034286t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oextend__basis,type,
real_V4986007116245087402_basis:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Real__Vector__Spaces_Olinear,type,
real_Vector_linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
real_V7770717601297561774m_norm:
!>[A: $tType] : ( A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oof__real,type,
real_Vector_of_real:
!>[A: $tType] : ( real > A ) ).
thf(sy_c_Real__Vector__Spaces_Orepresentation,type,
real_V7696804695334737415tation:
!>[A: $tType] : ( ( set @ A ) > A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V8093663219630862766scaleR:
!>[A: $tType] : ( real > A > A ) ).
thf(sy_c_Real__Vector__Spaces_Ospan,type,
real_Vector_span:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Real__Vector__Spaces_Osubspace,type,
real_Vector_subspace:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OField,type,
field2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OId,type,
id2:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_OImage,type,
image:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Relation_ORange,type,
range:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Relation_ORangep,type,
rangep:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > $o ) ).
thf(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oantisymp,type,
antisymp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Oconverse,type,
converse:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oirreflp,type,
irreflp:
!>[A: $tType] : ( ( A > A > $o ) > $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_Osingle__valued,type,
single_valued:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
thf(sy_c_Relation_Osingle__valuedp,type,
single_valuedp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Relation_Osym,type,
sym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
algebr8660921524188924756oprime:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_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_Odisjnt,type,
disjnt:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Ofilter,type,
filter3:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image2:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert2:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_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_Ofold__atLeastAtMost__nat__rel,type,
set_fo1817059534552279752at_rel:
!>[A: $tType] : ( ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > $o ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or1337092689740270186AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
set_or7035219750837199246ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or3652927894154168847AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or5935395276787703475ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_String_OCode_Oabort,type,
abort:
!>[A: $tType] : ( literal > ( product_unit > A ) > A ) ).
thf(sy_c_String_OLiteral,type,
literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : ( char > A ) ).
thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : ( A > char ) ).
thf(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
thf(sy_c_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_Oconnected,type,
topolo1966860045006549960nected:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent,type,
topolo6863149650580417670ergent:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
topolo7230453075368039082e_nhds:
!>[A: $tType] : ( A > ( filter @ A ) ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type,
topolo3814608138187158403Cauchy:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_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_Opowr__real,type,
powr_real: real > real > real ).
thf(sy_c_Transcendental_Osin,type,
sin:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_Transcendental_Osinh,type,
sinh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otan,type,
tan:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otanh,type,
tanh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Obi__unique,type,
bi_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transitive__Closure_Ontrancl,type,
transitive_ntrancl:
!>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
vEBT_Node: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
vEBT_V312737461966249ad_rel: ( product_prod @ vEBT_VEBT @ extended_enat ) > ( product_prod @ vEBT_VEBT @ extended_enat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
vEBT_V5917875025757280293ildren: nat > ( list @ vEBT_VEBT ) > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
vEBT_V4351362008482014158ma_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
vEBT_V5765760719290551771er_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
vEBT_VEBT_valid_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_Oset__vebt,type,
vEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(sy_c_VEBT__Insert_Ovebt__insert,type,
vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
vEBT_vebt_insert_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Member_Ovebt__member,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_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_Zorn_OChains,type,
chains:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Zorn_Ochain__subset,type,
chain_subset:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > $o ) ).
thf(sy_c_Zorn_Ochains,type,
chains2:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ ( set @ A ) ) ) ) ).
thf(sy_c_Zorn_Oinit__seg__of,type,
init_seg_of:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
thf(sy_c_Zorn_Opred__on_Ochain,type,
pred_chain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Zorn_Opred__on_Omaxchain,type,
pred_maxchain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Zorn_Opred__on_Osuc,type,
pred_suc:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_t,type,
t: vEBT_VEBT ).
thf(sy_v_x,type,
x: nat ).
thf(sy_v_y____,type,
y: nat ).
% Relevant facts (8186)
thf(fact_0__092_060open_062y_A_092_060in_062_Aset__vebt_H_A_Ivebt__insert_At_Ax_J_092_060close_062,axiom,
member @ nat @ y @ ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_insert @ t @ x ) ) ).
% \<open>y \<in> set_vebt' (vebt_insert t x)\<close>
thf(fact_1_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% buildup_gives_empty
thf(fact_2_Un__insert__left,axiom,
! [A: $tType,A3: A,B2: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A3 @ B2 ) @ C2 )
= ( insert2 @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_3_Un__insert__right,axiom,
! [A: $tType,A4: set @ A,A3: A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) )
= ( insert2 @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_4_Un__empty,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_5_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_6_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_7_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_8_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_9_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_10_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B3: A] :
( ( ( sup_sup @ A @ A3 @ B3 )
= ( bot_bot @ A ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B3
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_11_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
= A3 ) ) ).
% sup_bot.left_neutral
thf(fact_12_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B3: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A3 @ B3 ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B3
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_13_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_14_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_15_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_16_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_17_insert__absorb2,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( insert2 @ A @ X @ ( insert2 @ A @ X @ A4 ) )
= ( insert2 @ A @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_18_insert__iff,axiom,
! [A: $tType,A3: A,B3: A,A4: set @ A] :
( ( member @ A @ A3 @ ( insert2 @ A @ B3 @ A4 ) )
= ( ( A3 = B3 )
| ( member @ A @ A3 @ A4 ) ) ) ).
% insert_iff
thf(fact_19_insertCI,axiom,
! [A: $tType,A3: A,B2: set @ A,B3: A] :
( ( ~ ( member @ A @ A3 @ B2 )
=> ( A3 = B3 ) )
=> ( member @ A @ A3 @ ( insert2 @ A @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_20_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_21_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ B3 )
= ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.right_idem
thf(fact_22_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_23_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) )
= ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.left_idem
thf(fact_24_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_25_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ A3 )
= A3 ) ) ).
% sup.idem
thf(fact_26_Un__iff,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ( member @ A @ C3 @ A4 )
| ( member @ A @ C3 @ B2 ) ) ) ).
% Un_iff
thf(fact_27_UnCI,axiom,
! [A: $tType,C3: A,B2: set @ A,A4: set @ A] :
( ( ~ ( member @ A @ C3 @ B2 )
=> ( member @ A @ C3 @ A4 ) )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% UnCI
thf(fact_28_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
= A3 ) ) ).
% sup_bot.right_neutral
thf(fact_29_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_30_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_31_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_32_equals0D,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A4 ) ) ).
% equals0D
thf(fact_33_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_34_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ? [B4: set @ A] :
( ( A4
= ( insert2 @ A @ A3 @ B4 ) )
& ~ ( member @ A @ A3 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_35_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( insert2 @ A @ X @ ( insert2 @ A @ Y @ A4 ) )
= ( insert2 @ A @ Y @ ( insert2 @ A @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_36_insert__eq__iff,axiom,
! [A: $tType,A3: A,A4: set @ A,B3: A,B2: set @ A] :
( ~ ( member @ A @ A3 @ A4 )
=> ( ~ ( member @ A @ B3 @ B2 )
=> ( ( ( insert2 @ A @ A3 @ A4 )
= ( insert2 @ A @ B3 @ B2 ) )
= ( ( ( A3 = B3 )
=> ( A4 = B2 ) )
& ( ( A3 != B3 )
=> ? [C4: set @ A] :
( ( A4
= ( insert2 @ A @ B3 @ C4 ) )
& ~ ( member @ A @ B3 @ C4 )
& ( B2
= ( insert2 @ A @ A3 @ C4 ) )
& ~ ( member @ A @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_37_insert__absorb,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( insert2 @ A @ A3 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_38_insert__ident,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ~ ( member @ A @ X @ B2 )
=> ( ( ( insert2 @ A @ X @ A4 )
= ( insert2 @ A @ X @ B2 ) )
= ( A4 = B2 ) ) ) ) ).
% insert_ident
thf(fact_39_Set_Oset__insert,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ~ ! [B4: set @ A] :
( ( A4
= ( insert2 @ A @ X @ B4 ) )
=> ( member @ A @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_40_insertI2,axiom,
! [A: $tType,A3: A,B2: set @ A,B3: A] :
( ( member @ A @ A3 @ B2 )
=> ( member @ A @ A3 @ ( insert2 @ A @ B3 @ B2 ) ) ) ).
% insertI2
thf(fact_41_insertI1,axiom,
! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert2 @ A @ A3 @ B2 ) ) ).
% insertI1
thf(fact_42_insertE,axiom,
! [A: $tType,A3: A,B3: A,A4: set @ A] :
( ( member @ A @ A3 @ ( insert2 @ A @ B3 @ A4 ) )
=> ( ( A3 != B3 )
=> ( member @ A @ A3 @ A4 ) ) ) ).
% insertE
thf(fact_43_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_44_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G2: A > B] :
( ! [X3: A] :
( ( F3 @ X3 )
= ( G2 @ X3 ) )
=> ( F3 = G2 ) ) ).
% ext
thf(fact_49_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A3 @ C3 ) )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C3 ) ) ) ) ).
% sup.left_commute
thf(fact_50_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% sup_commute
thf(fact_51_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A5: A,B5: A] : ( sup_sup @ A @ B5 @ A5 ) ) ) ) ).
% sup.commute
thf(fact_52_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% sup_assoc
thf(fact_53_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ C3 )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C3 ) ) ) ) ).
% sup.assoc
thf(fact_54_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_55_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_56_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_57_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_58_Un__left__commute,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_59_Un__left__absorb,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ).
% Un_left_absorb
thf(fact_60_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_61_Un__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_62_Un__assoc,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C2 )
= ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_63_ball__Un,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_64_bex__Un,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( P @ X2 ) )
| ? [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_65_UnI2,axiom,
! [A: $tType,C3: A,B2: set @ A,A4: set @ A] :
( ( member @ A @ C3 @ B2 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% UnI2
thf(fact_66_UnI1,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% UnI1
thf(fact_67_UnE,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ( ~ ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% UnE
thf(fact_68_singleton__inject,axiom,
! [A: $tType,A3: A,B3: A] :
( ( ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_69_insert__not__empty,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( insert2 @ A @ A3 @ A4 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_70_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B3: A,C3: A,D2: A] :
( ( ( insert2 @ A @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ C3 @ ( insert2 @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C3 )
& ( B3 = D2 ) )
| ( ( A3 = D2 )
& ( B3 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_71_singleton__iff,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_72_singletonD,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_73_Un__empty__right,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Un_empty_right
thf(fact_74_Un__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_75_singleton__Un__iff,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ( ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_76_Un__singleton__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_77_insert__is__Un,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A5: A] : ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_78__092_060open_062set__vebt_H_At_A_092_060union_062_A_123x_125_A_092_060subseteq_062_Aset__vebt_H_A_Ivebt__insert_At_Ax_J_092_060close_062,axiom,
ord_less_eq @ ( set @ nat ) @ ( sup_sup @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ t ) @ ( insert2 @ nat @ x @ ( bot_bot @ ( set @ nat ) ) ) ) @ ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_insert @ t @ x ) ) ).
% \<open>set_vebt' t \<union> {x} \<subseteq> set_vebt' (vebt_insert t x)\<close>
thf(fact_79_buildup__nothing__in__leaf,axiom,
! [N: nat,X: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_leaf
thf(fact_80_assms_I1_J,axiom,
vEBT_invar_vebt @ t @ n ).
% assms(1)
thf(fact_81_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_82_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X2: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_83_buildup__nothing__in__min__max,axiom,
! [N: nat,X: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_min_max
thf(fact_84_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_85_boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% boolean_algebra.disj_zero_right
thf(fact_86_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_87_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
? [X2: A] :
( A6
= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_88_is__singletonE,axiom,
! [A: $tType,A4: set @ A] :
( ( is_singleton @ A @ A4 )
=> ~ ! [X3: A] :
( A4
!= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_89_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,K: A,B3: A,A3: A] :
( ( B2
= ( sup_sup @ A @ K @ B3 ) )
=> ( ( sup_sup @ A @ A3 @ B2 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_90_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_91_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_92_subsetI,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% subsetI
thf(fact_93_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% subset_antisym
thf(fact_94_set__vebt__set__vebt_H__valid,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_set_vebt @ T2 )
= ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).
% set_vebt_set_vebt'_valid
thf(fact_95_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X @ Z )
& ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_96_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C3 ) @ A3 )
= ( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% sup.bounded_iff
thf(fact_97_subset__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_98_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_99_insert__subset,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ B2 )
= ( ( member @ A @ X @ B2 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% insert_subset
thf(fact_100_Un__subset__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C2 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
& ( ord_less_eq @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_101_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A3: A,A4: set @ A] :
( ( ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ A @ A3 @ A4 ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_102_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A4: set @ A,B3: A] :
( ( ( insert2 @ A @ A3 @ A4 )
= ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_103_in__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_104_subsetD,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_105_equalityE,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ) ).
% equalityE
thf(fact_106_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( member @ A @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_107_equalityD1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% equalityD1
thf(fact_108_equalityD2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ).
% equalityD2
thf(fact_109_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_110_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_111_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_112_subset__trans,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_113_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z2: set @ A] : Y4 = Z2 )
= ( ^ [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_114_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_115_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% order_antisym_conv
thf(fact_116_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_le_cases
thf(fact_117_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F3: A > B,C3: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F3 @ B3 )
= C3 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_118_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F3: B > A,B3: B,C3: B] :
( ( A3
= ( F3 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_119_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_linear
thf(fact_120_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_eq_refl
thf(fact_121_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F3: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F3 @ B3 ) @ C3 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_122_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_123_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_124_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_125_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G2: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G2 ) ) ) ).
% le_funI
thf(fact_126_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funE
thf(fact_127_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funD
thf(fact_128_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% antisym
thf(fact_129_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_130_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_131_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_132_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A7: A,B7: A] :
( ( ord_less_eq @ A @ A7 @ B7 )
=> ( P @ A7 @ B7 ) )
=> ( ! [A7: A,B7: A] :
( ( P @ B7 @ A7 )
=> ( P @ A7 @ B7 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_133_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_134_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% order.trans
thf(fact_135_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% order_antisym
thf(fact_136_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_137_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_138_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_139_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_140_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( ord_less_eq @ A @ A3 @ B3 ) )
= ( ( ord_less_eq @ A @ B3 @ A3 )
& ( B3 != A3 ) ) ) ) ).
% nle_le
thf(fact_141_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_142_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_143_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_144_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_145_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_146_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A3 @ X )
=> ~ ( ord_less_eq @ A @ B3 @ X ) ) ) ) ).
% le_supE
thf(fact_147_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,X: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ( ord_less_eq @ A @ B3 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X ) ) ) ) ).
% le_supI
thf(fact_148_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge1
thf(fact_149_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge2
thf(fact_150_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% le_supI1
thf(fact_151_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B3: A,A3: A] :
( ( ord_less_eq @ A @ X @ B3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% le_supI2
thf(fact_152_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A3: A,D2: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ( ord_less_eq @ A @ D2 @ B3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C3 @ D2 ) @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% sup.mono
thf(fact_153_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,C3: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ( ord_less_eq @ A @ B3 @ D2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ ( sup_sup @ A @ C3 @ D2 ) ) ) ) ) ).
% sup_mono
thf(fact_154_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z ) @ X ) ) ) ) ).
% sup_least
thf(fact_155_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y3: A] :
( ( sup_sup @ A @ X2 @ Y3 )
= Y3 ) ) ) ) ).
% le_iff_sup
thf(fact_156_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3
= ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.orderE
thf(fact_157_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( sup_sup @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% sup.orderI
thf(fact_158_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F3: A > A > A,X: A,Y: A] :
( ! [X3: A,Y2: A] : ( ord_less_eq @ A @ X3 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ Y2 @ X3 )
=> ( ( ord_less_eq @ A @ Z3 @ X3 )
=> ( ord_less_eq @ A @ ( F3 @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_159_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb1
thf(fact_160_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( sup_sup @ A @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb2
thf(fact_161_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( sup_sup @ A @ X @ Y )
= X ) ) ) ).
% sup_absorb1
thf(fact_162_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( sup_sup @ A @ X @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_163_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C3 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B3 @ A3 )
=> ~ ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% sup.boundedE
thf(fact_164_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C3 ) @ A3 ) ) ) ) ).
% sup.boundedI
thf(fact_165_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B5 ) ) ) ) ) ).
% sup.order_iff
thf(fact_166_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.cobounded1
thf(fact_167_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.cobounded2
thf(fact_168_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_169_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= B5 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_170_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.coboundedI1
thf(fact_171_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.coboundedI2
thf(fact_172_insert__mono,axiom,
! [A: $tType,C2: set @ A,D3: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ C2 ) @ ( insert2 @ A @ A3 @ D3 ) ) ) ).
% insert_mono
thf(fact_173_subset__insert,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% subset_insert
thf(fact_174_subset__insertI,axiom,
! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert2 @ A @ A3 @ B2 ) ) ).
% subset_insertI
thf(fact_175_subset__insertI2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ B3 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_176_Un__mono,axiom,
! [A: $tType,A4: set @ A,C2: set @ A,B2: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C2 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_177_Un__least,axiom,
! [A: $tType,A4: set @ A,C2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_178_Un__upper1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ).
% Un_upper1
thf(fact_179_Un__upper2,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ).
% Un_upper2
thf(fact_180_Un__absorb1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_181_Un__absorb2,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= A4 ) ) ).
% Un_absorb2
thf(fact_182_subset__UnE,axiom,
! [A: $tType,C2: set @ A,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ~ ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ A4 )
=> ! [B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B8 @ B2 )
=> ( C2
!= ( sup_sup @ ( set @ A ) @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_183_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_184_subset__singletonD,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( A4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_185_subset__singleton__iff,axiom,
! [A: $tType,X4: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X4
= ( bot_bot @ ( set @ A ) ) )
| ( X4
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_186_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
( A6
= ( insert2 @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_187_is__singletonI_H,axiom,
! [A: $tType,A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( X3 = Y2 ) ) )
=> ( is_singleton @ A @ A4 ) ) ) ).
% is_singletonI'
thf(fact_188_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_189_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,K: A,A3: A,B3: A] :
( ( A4
= ( sup_sup @ A @ K @ A3 ) )
=> ( ( sup_sup @ A @ A4 @ B3 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_190_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ Tree @ N )
=> ( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X )
| ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% member_valid_both_member_options
thf(fact_191_valid__eq2,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_VEBT_valid @ T2 @ D2 )
=> ( vEBT_invar_vebt @ T2 @ D2 ) ) ).
% valid_eq2
thf(fact_192_valid__eq1,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_invar_vebt @ T2 @ D2 )
=> ( vEBT_VEBT_valid @ T2 @ D2 ) ) ).
% valid_eq1
thf(fact_193_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_194_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_195_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_196_set__vebt__finite,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( finite_finite2 @ nat @ ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).
% set_vebt_finite
thf(fact_197_insert__subsetI,axiom,
! [A: $tType,X: A,A4: set @ A,X4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ X4 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_198_subset__emptyI,axiom,
! [A: $tType,A4: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_199_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T3: vEBT_VEBT,X2: nat] :
( ( vEBT_V5719532721284313246member @ T3 @ X2 )
| ( vEBT_VEBT_membermima @ T3 @ X2 ) ) ) ) ).
% both_member_options_def
thf(fact_200_valid__tree__deg__neq__0,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_tree_deg_neq_0
thf(fact_201_valid__0__not,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_0_not
thf(fact_202_both__member__options__equiv__member,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
= ( vEBT_vebt_member @ T2 @ X ) ) ) ).
% both_member_options_equiv_member
thf(fact_203_valid__member__both__member__options,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
=> ( vEBT_vebt_member @ T2 @ X ) ) ) ).
% valid_member_both_member_options
thf(fact_204_member__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_vebt_member @ T2 @ X )
= ( member @ nat @ X @ ( vEBT_set_vebt @ T2 ) ) ) ) ).
% member_correct
thf(fact_205_finite__Un,axiom,
! [A: $tType,F4: set @ A,G3: set @ A] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F4 @ G3 ) )
= ( ( finite_finite2 @ A @ F4 )
& ( finite_finite2 @ A @ G3 ) ) ) ).
% finite_Un
thf(fact_206_finite__insert,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( finite_finite2 @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( finite_finite2 @ A @ A4 ) ) ).
% finite_insert
thf(fact_207_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_208_min__Null__member,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_minNull @ T2 )
=> ~ ( vEBT_vebt_member @ T2 @ X ) ) ).
% min_Null_member
thf(fact_209_finite__subset__induct_H,axiom,
! [A: $tType,F4: set @ A,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F4 @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A7: A,F5: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( member @ A @ A7 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A4 )
=> ( ~ ( member @ A @ A7 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert2 @ A @ A7 @ F5 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_210_finite__subset__induct,axiom,
! [A: $tType,F4: set @ A,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F4 @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A7: A,F5: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( member @ A @ A7 @ A4 )
=> ( ~ ( member @ A @ A7 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert2 @ A @ A7 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_211_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S: set @ B,P: ( set @ B ) > $o,F3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B,S2: set @ B] :
( ( finite_finite2 @ B @ S2 )
=> ( ! [Y5: B] :
( ( member @ B @ Y5 @ S2 )
=> ( ord_less_eq @ A @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert2 @ B @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_212_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_213_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A4: set @ A] :
( ! [A9: set @ A] :
( ~ ( finite_finite2 @ A @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F5: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ~ ( member @ A @ X3 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert2 @ A @ X3 @ F5 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_214_finite__ne__induct,axiom,
! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( F4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] : ( P @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A,F5: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( F5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X3 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert2 @ A @ X3 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_215_finite__induct,axiom,
! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F5: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ~ ( member @ A @ X3 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert2 @ A @ X3 @ F5 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_216_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_217_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_218_finite__set__choice,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ? [X_12: B] : ( P @ X3 @ X_12 ) )
=> ? [F6: A > B] :
! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( P @ X5 @ ( F6 @ X5 ) ) ) ) ) ).
% finite_set_choice
thf(fact_219_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A4: set @ A] : ( finite_finite2 @ A @ A4 ) ) ).
% finite
thf(fact_220_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_221_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ord_less_eq @ A @ X3 @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A4 )
=> ( ( ord_less_eq @ A @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_222_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ord_less_eq @ A @ A3 @ X3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A4 )
=> ( ( ord_less_eq @ A @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_223_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_224_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_225_rev__finite__subset,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_226_infinite__super,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T4 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% infinite_super
thf(fact_227_finite__subset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% finite_subset
thf(fact_228_finite_OinsertI,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ A @ ( insert2 @ A @ A3 @ A4 ) ) ) ).
% finite.insertI
thf(fact_229_finite__UnI,axiom,
! [A: $tType,F4: set @ A,G3: set @ A] :
( ( finite_finite2 @ A @ F4 )
=> ( ( finite_finite2 @ A @ G3 )
=> ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F4 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_230_Un__infinite,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T4 ) ) ) ).
% Un_infinite
thf(fact_231_infinite__Un,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T4 ) ) )
= ( ~ ( finite_finite2 @ A @ S )
| ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% infinite_Un
thf(fact_232_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A4 )
=> ( ( ord_less_eq @ A @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_233_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A4 )
=> ( ( ord_less_eq @ A @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_234_finite_Ocases,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A9: set @ A] :
( ? [A7: A] :
( A3
= ( insert2 @ A @ A7 @ A9 ) )
=> ~ ( finite_finite2 @ A @ A9 ) ) ) ) ).
% finite.cases
thf(fact_235_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A5: set @ A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ? [A6: set @ A,B5: A] :
( ( A5
= ( insert2 @ A @ B5 @ A6 ) )
& ( finite_finite2 @ A @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_236_buildup__gives__valid,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% buildup_gives_valid
thf(fact_237_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A3 ) ).
% bot_nat_0.extremum
thf(fact_238_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_239_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S: set @ A,Y: A,F3: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y @ S )
=> ( ord_less_eq @ B @ ( F3 @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ) ).
% arg_min_least
thf(fact_240_finite__transitivity__chain,axiom,
! [A: $tType,A4: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X3: A] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: A,Y2: A,Z3: A] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A4 )
& ( R @ X3 @ Y5 ) ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_241_deg__not__0,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% deg_not_0
thf(fact_242_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_243_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_244_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S ) @ S ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_245_Leaf__0__not,axiom,
! [A3: $o,B3: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A3 @ B3 ) @ ( zero_zero @ nat ) ) ).
% Leaf_0_not
thf(fact_246_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ B2 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_247_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_248_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_249_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_250_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_251_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Sup_fin.singleton
thf(fact_252_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F3: A > nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less @ nat @ ( F3 @ Y2 ) @ B3 ) )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_253_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y5 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_254_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_imp_not_less
thf(fact_255_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% order_less_imp_not_eq2
thf(fact_256_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% order_less_imp_not_eq
thf(fact_257_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_less_linear
thf(fact_258_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% order_less_imp_triv
thf(fact_259_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_not_sym
thf(fact_260_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F3: A > C,C3: C] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ C @ ( F3 @ B3 ) @ C3 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ C @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% order_less_subst2
thf(fact_261_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B3: B,C3: B] :
( ( ord_less @ A @ A3 @ ( F3 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C3 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_262_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_263_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F3: A > B,C3: B] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ( F3 @ B3 )
= C3 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ B @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_264_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F3: B > A,B3: B,C3: B] :
( ( A3
= ( F3 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C3 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_265_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% order_less_trans
thf(fact_266_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% order_less_asym'
thf(fact_267_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neq_iff
thf(fact_268_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_asym
thf(fact_269_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE
thf(fact_270_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( A3 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_271_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( A3 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_272_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_273_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_274_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans
thf(fact_275_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A7: A,B7: A] :
( ( ord_less @ A @ A7 @ B7 )
=> ( P @ A7 @ B7 ) )
=> ( ! [A7: A] : ( P @ A7 @ A7 )
=> ( ! [A7: A,B7: A] :
( ( P @ B7 @ A7 )
=> ( P @ A7 @ B7 ) )
=> ( P @ A3 @ B3 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_276_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X6: A] : ( P2 @ X6 ) )
= ( ^ [P3: A > $o] :
? [N2: A] :
( ( P3 @ N2 )
& ! [M2: A] :
( ( ord_less @ A @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_277_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_278_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_279_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_280_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_281_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_282_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_283_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_284_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% order.asym
thf(fact_285_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_286_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_287_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_288_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% lt_ex
thf(fact_289_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_290_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_291_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_292_less__not__refl3,axiom,
! [S3: nat,T2: nat] :
( ( ord_less @ nat @ S3 @ T2 )
=> ( S3 != T2 ) ) ).
% less_not_refl3
thf(fact_293_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F3 @ Y5 ) @ ( F3 @ X3 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct
thf(fact_294_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_295_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_296_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_297_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less @ nat @ X @ Y )
=> ( ord_less @ nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_298_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F3 @ Y5 ) @ ( F3 @ X3 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct_rule
thf(fact_299_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X3 ) )
& ~ ( P @ Y5 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_300_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_301_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_302_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_303_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_304_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_305_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
| ( ord_less_eq @ nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_306_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_307_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_308_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less @ nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_309_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ B3 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_310_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( M != N )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_311_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F3 @ I2 ) @ ( F3 @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F3 @ I ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_312_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_313_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_314_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_315_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_316_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_317_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_318_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_319_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_320_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_321_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X: A] :
( ! [X3: A] :
( ( ( V @ X3 )
= ( zero_zero @ nat ) )
=> ( P @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
=> ( ~ ( P @ X3 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X3 ) )
& ~ ( P @ Y5 ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_322_unbounded__k__infinite,axiom,
! [K: nat,S: set @ nat] :
( ! [M4: nat] :
( ( ord_less @ nat @ K @ M4 )
=> ? [N4: nat] :
( ( ord_less @ nat @ M4 @ N4 )
& ( member @ nat @ N4 @ S ) ) )
=> ~ ( finite_finite2 @ nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_323_infinite__nat__iff__unbounded,axiom,
! [S: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S ) )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ( member @ nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_324_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_325_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_326_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( ord_less @ A @ A3 @ B3 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% nless_le
thf(fact_327_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_328_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_329_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,Y: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_330_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_331_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_332_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_333_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less @ A @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_334_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_335_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans1
thf(fact_336_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans2
thf(fact_337_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ~ ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_338_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_339_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_340_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less @ A @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_341_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_342_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_343_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_344_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ~ ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_345_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_346_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_347_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ) ).
% order_le_less
thf(fact_348_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ) ).
% order_less_le
thf(fact_349_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_not_le
thf(fact_350_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_not_less
thf(fact_351_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_less_imp_le
thf(fact_352_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% order_le_neq_trans
thf(fact_353_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( A3 != B3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% order_neq_le_trans
thf(fact_354_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% order_le_less_trans
thf(fact_355_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% order_less_le_trans
thf(fact_356_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F3 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C3 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_357_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F3: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ C @ ( F3 @ B3 ) @ C3 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ C @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_358_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B3: B,C3: B] :
( ( ord_less @ A @ A3 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_359_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F3: A > C,C3: C] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F3 @ B3 ) @ C3 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ C @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_360_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_le_less_linear
thf(fact_361_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_362_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_363_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_364_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_365_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_366_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( A3
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% bot.not_eq_extremum
thf(fact_367_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_368_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_369_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ C3 @ A3 )
=> ( ord_less @ A @ C3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_370_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( A5
= ( sup_sup @ A @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_371_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B3 @ C3 ) @ A3 )
=> ~ ( ( ord_less @ A @ B3 @ A3 )
=> ~ ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_372_sup_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( sup_sup @ A @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb4
thf(fact_373_sup_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb3
thf(fact_374_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B3: A,A3: A] :
( ( ord_less @ A @ X @ B3 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% less_supI2
thf(fact_375_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B3: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% less_supI1
thf(fact_376_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X5: A] :
( ( member @ A @ X5 @ S )
& ( ord_less @ B @ ( F3 @ X5 ) @ ( F3 @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_377_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: set @ A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ X4 )
& ( ord_less @ A @ X3 @ Xa ) ) )
=> ~ ( finite_finite2 @ A @ X4 ) ) ) ) ).
% infinite_growing
thf(fact_378_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ S )
& ~ ? [Xa: A] :
( ( member @ A @ Xa @ S )
& ( ord_less @ A @ Xa @ X3 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_379_Sup__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ord_less_eq @ A @ A3 @ ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_380_Sup__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_381_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B7: A,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A9 )
=> ( ord_less @ A @ B7 @ X5 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert2 @ A @ B7 @ A9 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_382_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B7: A,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A9 )
=> ( ord_less @ A @ X5 @ B7 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert2 @ A @ B7 @ A9 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_383_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_384_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_385_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
=> ( A3
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_386_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
= ( A3
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_387_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_388_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_389_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A7: A] :
( ( member @ A @ A7 @ A4 )
=> ( ord_less_eq @ A @ A7 @ X ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ X ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_390_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ X )
=> ! [A10: A] :
( ( member @ A @ A10 @ A4 )
=> ( ord_less_eq @ A @ A10 @ X ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_391_infinite__nat__iff__unbounded__le,axiom,
! [S: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S ) )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
& ( member @ nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_392_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ B2 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_393_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B2 ) @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_394_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_395_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] : ( member @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_396_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_397_VEBT_Oinject_I2_J,axiom,
! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
( ( ( vEBT_Leaf @ X21 @ X22 )
= ( vEBT_Leaf @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% VEBT.inject(2)
thf(fact_398_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% VEBT_internal.minNull.simps(3)
thf(fact_399_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% VEBT_internal.minNull.simps(2)
thf(fact_400_VEBT__internal_OminNull_Osimps_I1_J,axiom,
vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% VEBT_internal.minNull.simps(1)
thf(fact_401_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_402_deg1Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
= ( ? [A5: $o,B5: $o] :
( T2
= ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ).
% deg1Leaf
thf(fact_403_deg__1__Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
=> ? [A7: $o,B7: $o] :
( T2
= ( vEBT_Leaf @ A7 @ B7 ) ) ) ).
% deg_1_Leaf
thf(fact_404_deg__1__Leafy,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( N
= ( one_one @ nat ) )
=> ? [A7: $o,B7: $o] :
( T2
= ( vEBT_Leaf @ A7 @ B7 ) ) ) ) ).
% deg_1_Leafy
thf(fact_405_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N5: set @ nat] :
? [M2: nat] :
! [X2: nat] :
( ( member @ nat @ X2 @ N5 )
=> ( ord_less_eq @ nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_406_bounded__nat__set__is__finite,axiom,
! [N6: set @ nat,N: nat] :
( ! [X3: nat] :
( ( member @ nat @ X3 @ N6 )
=> ( ord_less @ nat @ X3 @ N ) )
=> ( finite_finite2 @ nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_407_psubsetI,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( A4 != B2 )
=> ( ord_less @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% psubsetI
thf(fact_408_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_409_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_410_not__psubset__empty,axiom,
! [A: $tType,A4: set @ A] :
~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_411_finite__psubset__induct,axiom,
! [A: $tType,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [B9: set @ A] :
( ( ord_less @ ( set @ A ) @ B9 @ A9 )
=> ( P @ B9 ) )
=> ( P @ A9 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_412_psubsetE,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ) ).
% psubsetE
thf(fact_413_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_414_psubset__imp__subset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_415_psubset__subset__trans,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_416_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_417_subset__psubset__trans,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_418_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_419_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_420_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_421_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
& ~ ( ord_less_eq @ ( A > B ) @ G @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_422_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_423_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_424_vebt__member_Osimps_I1_J,axiom,
! [A3: $o,B3: $o,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A3 @ B3 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A3 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B3 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_425_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A3: $o,B3: $o,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A3 @ B3 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A3 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B3 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_426_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq @ nat @ X3 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq @ nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_427_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N5: set @ nat] :
? [M2: nat] :
! [X2: nat] :
( ( member @ nat @ X2 @ N5 )
=> ( ord_less @ nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_428_vebt__insert_Osimps_I1_J,axiom,
! [X: nat,A3: $o,B3: $o] :
( ( ( X
= ( zero_zero @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ X )
= ( vEBT_Leaf @ $true @ B3 ) ) )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ X )
= ( vEBT_Leaf @ A3 @ $true ) ) )
& ( ( X
!= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ X )
= ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) ).
% vebt_insert.simps(1)
thf(fact_429_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_430_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_431_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_432_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_433_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_434_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less @ nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_435_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_436_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_437_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C5: A] :
( ( ord_less_eq @ A @ A3 @ C5 )
& ( ord_less_eq @ A @ C5 @ B3 )
& ! [X5: A] :
( ( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less @ A @ X5 @ C5 ) )
=> ( P @ X5 ) )
& ! [D4: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less @ A @ X3 @ D4 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D4 @ C5 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_438_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B10: B,A11: B] :
( ( ~ ( ord_less_eq @ B @ B10 @ A11 ) )
= ( ord_less @ B @ A11 @ B10 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_439_psubset__trans,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% psubset_trans
thf(fact_440_psubsetD,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C3: A] :
( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% psubsetD
thf(fact_441_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(2)
thf(fact_442_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( A3 = B3 )
| ~ ( ord_less_eq @ A @ A3 @ B3 )
| ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% verit_la_disequality
thf(fact_443_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(1)
thf(fact_444_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_445_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit5016429287641298734tinuum @ A )
=> ! [A3: A] :
? [B7: A] :
( ( ord_less @ A @ A3 @ B7 )
| ( ord_less @ A @ B7 @ A3 ) ) ) ).
% ex_gt_or_lt
thf(fact_446_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% minf(8)
thf(fact_447_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% minf(6)
thf(fact_448_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% pinf(8)
thf(fact_449_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% pinf(6)
thf(fact_450_dbl__inc__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_inc_simps(2)
thf(fact_451_VEBT_Osize__gen_I2_J,axiom,
! [X21: $o,X22: $o] :
( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size_gen(2)
thf(fact_452_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_453_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_454_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_455_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_456_VEBT_Osize_I4_J,axiom,
! [X21: $o,X22: $o] :
( ( size_size @ vEBT_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size(4)
thf(fact_457_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A6: A > B,B6: A > B,X2: A] : ( minus_minus @ B @ ( A6 @ X2 ) @ ( B6 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_458_DiffI,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ A4 )
=> ( ~ ( member @ A @ C3 @ B2 )
=> ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% DiffI
thf(fact_459_Diff__iff,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( ( member @ A @ C3 @ A4 )
& ~ ( member @ A @ C3 @ B2 ) ) ) ).
% Diff_iff
thf(fact_460_Diff__idemp,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ).
% Diff_idemp
thf(fact_461_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_462_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_zero
thf(fact_463_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_464_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_0_right
thf(fact_465_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_466_Diff__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Diff_empty
thf(fact_467_empty__Diff,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_468_Diff__cancel,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_469_finite__Diff2,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( finite_finite2 @ A @ A4 ) ) ) ).
% finite_Diff2
thf(fact_470_finite__Diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% finite_Diff
thf(fact_471_Diff__insert0,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ B2 ) )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_472_insert__Diff1,axiom,
! [A: $tType,X: A,B2: set @ A,A4: set @ A] :
( ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_473_Un__Diff__cancel,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_474_Un__Diff__cancel2,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) @ A4 )
= ( sup_sup @ ( set @ A ) @ B2 @ A4 ) ) ).
% Un_Diff_cancel2
thf(fact_475_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_476_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_477_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_478_Diff__eq__empty__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_479_insert__Diff__single,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( insert2 @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert2 @ A @ A3 @ A4 ) ) ).
% insert_Diff_single
thf(fact_480_finite__Diff__insert,axiom,
! [A: $tType,A4: set @ A,A3: A,B2: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_481_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Inf_fin.singleton
thf(fact_482_sup__Inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ A3 )
= A3 ) ) ) ) ).
% sup_Inf_absorb
thf(fact_483_DiffE,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
=> ~ ( ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% DiffE
thf(fact_484_DiffD1,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
=> ( member @ A @ C3 @ A4 ) ) ).
% DiffD1
thf(fact_485_DiffD2,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
=> ~ ( member @ A @ C3 @ B2 ) ) ).
% DiffD2
thf(fact_486_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A6: A > B,B6: A > B,X2: A] : ( minus_minus @ B @ ( A6 @ X2 ) @ ( B6 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_487_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C3 @ D2 ) )
=> ( ( A3 = B3 )
= ( C3 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_488_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C3 ) @ B3 )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% diff_right_commute
thf(fact_489_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X: A,Y: A] :
( ( ( size_size @ A @ X )
!= ( size_size @ A @ Y ) )
=> ( X != Y ) ) ) ).
% size_neq_size_imp_neq
thf(fact_490_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C3 @ D2 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
= ( ord_less_eq @ A @ C3 @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_491_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B3 @ C3 ) ) ) ) ).
% diff_right_mono
thf(fact_492_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C3 @ A3 ) @ ( minus_minus @ A @ C3 @ B3 ) ) ) ) ).
% diff_left_mono
thf(fact_493_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,D2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ D2 @ C3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_494_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( minus_minus @ A @ A5 @ B5 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_495_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,D2: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ D2 @ C3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_496_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C3 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
= ( ord_less @ A @ C3 @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_497_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ord_less @ A @ ( minus_minus @ A @ C3 @ A3 ) @ ( minus_minus @ A @ C3 @ B3 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_498_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B3 @ C3 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_499_Diff__infinite__finite,axiom,
! [A: $tType,T4: set @ A,S: set @ A] :
( ( finite_finite2 @ A @ T4 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ T4 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_500_Diff__mono,axiom,
! [A: $tType,A4: set @ A,C2: set @ A,D3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ D3 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ ( minus_minus @ ( set @ A ) @ C2 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_501_Diff__subset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ A4 ) ).
% Diff_subset
thf(fact_502_double__diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ( minus_minus @ ( set @ A ) @ B2 @ ( minus_minus @ ( set @ A ) @ C2 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_503_insert__Diff__if,axiom,
! [A: $tType,X: A,B2: set @ A,A4: set @ A] :
( ( ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) )
& ( ~ ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ B2 )
= ( insert2 @ A @ X @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_504_Un__Diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C2 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ C2 ) @ ( minus_minus @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_505_psubset__imp__ex__mem,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ? [B7: A] : ( member @ A @ B7 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_506_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_507_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] : ( ord_less @ A @ ( minus_minus @ A @ A5 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_508_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( minus_minus @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% diff_shunt_var
thf(fact_509_Diff__insert,axiom,
! [A: $tType,A4: set @ A,A3: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_510_insert__Diff,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( insert2 @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_511_Diff__insert2,axiom,
! [A: $tType,A4: set @ A,A3: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_512_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_513_subset__Diff__insert,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,X: A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ ( insert2 @ A @ X @ C2 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ C2 ) )
& ~ ( member @ A @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_514_Diff__subset__conv,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ C2 )
= ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_515_Diff__partition,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_516_Inf__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ A3 ) ) ) ) ).
% Inf_fin.coboundedI
thf(fact_517_infinite__remove,axiom,
! [A: $tType,S: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_518_infinite__coinduct,axiom,
! [A: $tType,X4: ( set @ A ) > $o,A4: set @ A] :
( ( X4 @ A4 )
=> ( ! [A9: set @ A] :
( ( X4 @ A9 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A9 )
& ( ( X4 @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_519_finite__empty__induct,axiom,
! [A: $tType,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( P @ A4 )
=> ( ! [A7: A,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ( member @ A @ A7 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert2 @ A @ A7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_520_subset__insert__iff,axiom,
! [A: $tType,A4: set @ A,X: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ B2 ) )
= ( ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_521_Diff__single__insert,axiom,
! [A: $tType,A4: set @ A,X: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_522_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_523_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_524_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T2 ) ) ) ).
% pinf(3)
thf(fact_525_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T2 ) ) ) ).
% pinf(4)
thf(fact_526_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less @ A @ X5 @ T2 ) ) ) ).
% pinf(5)
thf(fact_527_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less @ A @ T2 @ X5 ) ) ) ).
% pinf(7)
thf(fact_528_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F4: D] :
? [Z3: C] :
! [X5: C] :
( ( ord_less @ C @ Z3 @ X5 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_529_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_530_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_531_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T2 ) ) ) ).
% minf(3)
thf(fact_532_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T2 ) ) ) ).
% minf(4)
thf(fact_533_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less @ A @ X5 @ T2 ) ) ) ).
% minf(5)
thf(fact_534_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less @ A @ T2 @ X5 ) ) ) ).
% minf(7)
thf(fact_535_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F4: D] :
? [Z3: C] :
! [X5: C] :
( ( ord_less @ C @ X5 @ Z3 )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_536_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
=> ! [A10: A] :
( ( member @ A @ A10 @ A4 )
=> ( ord_less_eq @ A @ X @ A10 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_537_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A7: A] :
( ( member @ A @ A7 @ A4 )
=> ( ord_less_eq @ A @ X @ A7 ) )
=> ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_538_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_539_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B2: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B2 )
=> ( P @ B2 ) )
=> ( ! [A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A9 @ B2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_540_finite__remove__induct,axiom,
! [A: $tType,B2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A9 @ B2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_541_finite__induct__select,axiom,
! [A: $tType,S: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T5: set @ A] :
( ( ord_less @ ( set @ A ) @ T5 @ S )
=> ( ( P @ T5 )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( minus_minus @ ( set @ A ) @ S @ T5 ) )
& ( P @ ( insert2 @ A @ X5 @ T5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_542_psubset__insert__iff,axiom,
! [A: $tType,A4: set @ A,X: A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ B2 ) )
= ( ( ( member @ A @ X @ B2 )
=> ( ord_less @ ( set @ A ) @ A4 @ B2 ) )
& ( ~ ( member @ A @ X @ B2 )
=> ( ( ( member @ A @ X @ A4 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_543_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B2 ) @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_544_artanh__0,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% artanh_0
thf(fact_545_arsinh__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% arsinh_0
thf(fact_546_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_547_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X2: A,A6: set @ A] : ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_548_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_549_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_550_card__Diff1__less__iff,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) )
= ( ( finite_finite2 @ A @ A4 )
& ( member @ A @ X @ A4 ) ) ) ).
% card_Diff1_less_iff
thf(fact_551_card__Diff2__less,axiom,
! [A: $tType,A4: set @ A,X: A,Y: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_552_card__Diff1__less,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) ) ) ) ).
% card_Diff1_less
thf(fact_553_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_554_of__nat__0__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% of_nat_0_less_iff
thf(fact_555_finite__Int,axiom,
! [A: $tType,F4: set @ A,G3: set @ A] :
( ( ( finite_finite2 @ A @ F4 )
| ( finite_finite2 @ A @ G3 ) )
=> ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F4 @ G3 ) ) ) ).
% finite_Int
thf(fact_556_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_557_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_558_Int__subset__iff,axiom,
! [A: $tType,C2: set @ A,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C2 @ A4 )
& ( ord_less_eq @ ( set @ A ) @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_559_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% inf_apply
thf(fact_560_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_right_idem
thf(fact_561_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B3 ) @ B3 )
= ( inf_inf @ A @ A3 @ B3 ) ) ) ).
% inf.right_idem
thf(fact_562_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_left_idem
thf(fact_563_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( inf_inf @ A @ A3 @ ( inf_inf @ A @ A3 @ B3 ) )
= ( inf_inf @ A @ A3 @ B3 ) ) ) ).
% inf.left_idem
thf(fact_564_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ X )
= X ) ) ).
% inf_idem
thf(fact_565_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ A3 @ A3 )
= A3 ) ) ).
% inf.idem
thf(fact_566_Int__insert__right__if1,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_567_Int__insert__right__if0,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ~ ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_568_insert__inter__insert,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ A4 ) @ ( insert2 @ A @ A3 @ B2 ) )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_569_Int__insert__left__if1,axiom,
! [A: $tType,A3: A,C2: set @ A,B2: set @ A] :
( ( member @ A @ A3 @ C2 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B2 ) @ C2 )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_570_Int__insert__left__if0,axiom,
! [A: $tType,A3: A,C2: set @ A,B2: set @ A] :
( ~ ( member @ A @ A3 @ C2 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B2 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_571_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( semiring_1_of_nat @ A @ N ) )
= ( M = N ) ) ) ).
% of_nat_eq_iff
thf(fact_572_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_573_Un__Int__eq_I1_J,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S @ T4 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_574_Un__Int__eq_I2_J,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S @ T4 ) @ T4 )
= T4 ) ).
% Un_Int_eq(2)
thf(fact_575_Un__Int__eq_I3_J,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ S @ ( sup_sup @ ( set @ A ) @ S @ T4 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_576_Un__Int__eq_I4_J,axiom,
! [A: $tType,T4: set @ A,S: set @ A] :
( ( inf_inf @ ( set @ A ) @ T4 @ ( sup_sup @ ( set @ A ) @ S @ T4 ) )
= T4 ) ).
% Un_Int_eq(4)
thf(fact_577_Int__Un__eq_I1_J,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S @ T4 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_578_Int__Un__eq_I2_J,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S @ T4 ) @ T4 )
= T4 ) ).
% Int_Un_eq(2)
thf(fact_579_Int__Un__eq_I3_J,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ S @ ( inf_inf @ ( set @ A ) @ S @ T4 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_580_Int__Un__eq_I4_J,axiom,
! [A: $tType,T4: set @ A,S: set @ A] :
( ( sup_sup @ ( set @ A ) @ T4 @ ( inf_inf @ ( set @ A ) @ S @ T4 ) )
= T4 ) ).
% Int_Un_eq(4)
thf(fact_581_member__remove,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( member @ A @ X @ ( remove @ A @ Y @ A4 ) )
= ( ( member @ A @ X @ A4 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_582_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C3 ) )
= ( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% inf.bounded_iff
thf(fact_583_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% le_inf_iff
thf(fact_584_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) )
= ( ord_less @ nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_585_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_586_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_587_boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_right
thf(fact_588_boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_left
thf(fact_589_insert__disjoint_I1_J,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ A4 ) @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B2 )
& ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_590_insert__disjoint_I2_J,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ A4 ) @ B2 ) )
= ( ~ ( member @ A @ A3 @ B2 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_591_disjoint__insert_I1_J,axiom,
! [A: $tType,B2: set @ A,A3: A,A4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B2 @ ( insert2 @ A @ A3 @ A4 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B2 )
& ( ( inf_inf @ ( set @ A ) @ B2 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_592_disjoint__insert_I2_J,axiom,
! [A: $tType,A4: set @ A,B3: A,B2: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A4 @ ( insert2 @ A @ B3 @ B2 ) ) )
= ( ~ ( member @ A @ B3 @ A4 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_593_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% diff_is_0_eq'
thf(fact_594_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_595_sup__inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= X ) ) ).
% sup_inf_absorb
thf(fact_596_inf__sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= X ) ) ).
% inf_sup_absorb
thf(fact_597_Diff__disjoint,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_598_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_599_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( zero_zero @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( ( zero_zero @ nat )
= N ) ) ) ).
% of_nat_0_eq_iff
thf(fact_600_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_601_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ).
% of_nat_less_iff
thf(fact_602_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% of_nat_le_iff
thf(fact_603_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_604_card_Oinfinite,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ A @ A4 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_605_of__nat__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ A @ N )
= ( one_one @ A ) )
= ( N
= ( one_one @ nat ) ) ) ) ).
% of_nat_eq_1_iff
thf(fact_606_of__nat__1__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( one_one @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( N
= ( one_one @ nat ) ) ) ) ).
% of_nat_1_eq_iff
thf(fact_607_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_608_inf__Sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ A @ A3 @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= A3 ) ) ) ) ).
% inf_Sup_absorb
thf(fact_609_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_610_card__0__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( finite_card @ A @ A4 )
= ( zero_zero @ nat ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_611_card__Diff__insert,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ~ ( member @ A @ A3 @ B2 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_612_boolean__algebra__cancel_Oinf2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,K: A,B3: A,A3: A] :
( ( B2
= ( inf_inf @ A @ K @ B3 ) )
=> ( ( inf_inf @ A @ A3 @ B2 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_613_boolean__algebra__cancel_Oinf1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,K: A,A3: A,B3: A] :
( ( A4
= ( inf_inf @ A @ K @ A3 ) )
=> ( ( inf_inf @ A @ A4 @ B3 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_614_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% inf_fun_def
thf(fact_615_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% inf_left_commute
thf(fact_616_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( inf_inf @ A @ B3 @ ( inf_inf @ A @ A3 @ C3 ) )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B3 @ C3 ) ) ) ) ).
% inf.left_commute
thf(fact_617_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X2: A,Y3: A] : ( inf_inf @ A @ Y3 @ X2 ) ) ) ) ).
% inf_commute
thf(fact_618_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A5: A,B5: A] : ( inf_inf @ A @ B5 @ A5 ) ) ) ) ).
% inf.commute
thf(fact_619_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% inf_assoc
thf(fact_620_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C3 )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B3 @ C3 ) ) ) ) ).
% inf.assoc
thf(fact_621_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( inf_inf @ A )
= ( ^ [X2: A,Y3: A] : ( inf_inf @ A @ Y3 @ X2 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_622_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_623_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_624_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_sup_aci(4)
thf(fact_625_card__Diff__subset__Int,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_626_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y ) ) )
= ( ( ( ord_less_eq @ nat @ Y @ X )
=> ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y ) ) ) )
& ( ( ord_less @ nat @ X @ Y )
=> ( P @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_627_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% inf.coboundedI2
thf(fact_628_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% inf.coboundedI1
thf(fact_629_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( inf_inf @ A @ A5 @ B5 )
= B5 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_630_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( inf_inf @ A @ A5 @ B5 )
= A5 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_631_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ B3 ) ) ).
% inf.cobounded2
thf(fact_632_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ A3 ) ) ).
% inf.cobounded1
thf(fact_633_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( A5
= ( inf_inf @ A @ A5 @ B5 ) ) ) ) ) ).
% inf.order_iff
thf(fact_634_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Z )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ) ).
% inf_greatest
thf(fact_635_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C3 ) ) ) ) ) ).
% inf.boundedI
thf(fact_636_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C3 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B3 )
=> ~ ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% inf.boundedE
thf(fact_637_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( inf_inf @ A @ X @ Y )
= Y ) ) ) ).
% inf_absorb2
thf(fact_638_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( inf_inf @ A @ X @ Y )
= X ) ) ) ).
% inf_absorb1
thf(fact_639_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb2
thf(fact_640_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( inf_inf @ A @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb1
thf(fact_641_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y3: A] :
( ( inf_inf @ A @ X2 @ Y3 )
= X2 ) ) ) ) ).
% le_iff_inf
thf(fact_642_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F3: A > A > A,X: A,Y: A] :
( ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( F3 @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( F3 @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ X3 @ Z3 )
=> ( ord_less_eq @ A @ X3 @ ( F3 @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf @ A @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ) ).
% inf_unique
thf(fact_643_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( inf_inf @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% inf.orderI
thf(fact_644_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3
= ( inf_inf @ A @ A3 @ B3 ) ) ) ) ).
% inf.orderE
thf(fact_645_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,X: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X ) ) ) ).
% le_infI2
thf(fact_646_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X ) ) ) ).
% le_infI1
thf(fact_647_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C3: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ( ord_less_eq @ A @ B3 @ D2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ ( inf_inf @ A @ C3 @ D2 ) ) ) ) ) ).
% inf_mono
thf(fact_648_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ X @ B3 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% le_infI
thf(fact_649_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B3 ) )
=> ~ ( ( ord_less_eq @ A @ X @ A3 )
=> ~ ( ord_less_eq @ A @ X @ B3 ) ) ) ) ).
% le_infE
thf(fact_650_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_le2
thf(fact_651_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_le1
thf(fact_652_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_sup_ord(1)
thf(fact_653_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_sup_ord(2)
thf(fact_654_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% inf.strict_coboundedI2
thf(fact_655_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ A3 @ C3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% inf.strict_coboundedI1
thf(fact_656_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] :
( ( A5
= ( inf_inf @ A @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ) ).
% inf.strict_order_iff
thf(fact_657_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ ( inf_inf @ A @ B3 @ C3 ) )
=> ~ ( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% inf.strict_boundedE
thf(fact_658_inf_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb4
thf(fact_659_inf_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( inf_inf @ A @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb3
thf(fact_660_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,X: A,A3: A] :
( ( ord_less @ A @ B3 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X ) ) ) ).
% less_infI2
thf(fact_661_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B3: A] :
( ( ord_less @ A @ A3 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X ) ) ) ).
% less_infI1
thf(fact_662_sup__inf__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y: A,Z: A,X: A] :
( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z ) @ X )
= ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z @ X ) ) ) ) ).
% sup_inf_distrib2
thf(fact_663_sup__inf__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_inf_distrib1
thf(fact_664_inf__sup__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y: A,Z: A,X: A] :
( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z ) @ X )
= ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z @ X ) ) ) ) ).
% inf_sup_distrib2
thf(fact_665_inf__sup__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% inf_sup_distrib1
thf(fact_666_distrib__imp2,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ! [X3: A,Y2: A,Z3: A] :
( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z3 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ ( sup_sup @ A @ X3 @ Z3 ) ) )
=> ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ) ).
% distrib_imp2
thf(fact_667_distrib__imp1,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ! [X3: A,Y2: A,Z3: A] :
( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z3 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ ( inf_inf @ A @ X3 @ Z3 ) ) )
=> ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ) ).
% distrib_imp1
thf(fact_668_boolean__algebra_Odisj__conj__distrib2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,Z: A,X: A] :
( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z ) @ X )
= ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z @ X ) ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_669_boolean__algebra_Oconj__disj__distrib2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,Z: A,X: A] :
( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z ) @ X )
= ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z @ X ) ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_670_boolean__algebra_Odisj__conj__distrib,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_671_boolean__algebra_Oconj__disj__distrib,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_672_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_673_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_674_disjoint__iff__not__equal,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ B2 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_675_Int__empty__right,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_676_Int__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_677_disjoint__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ~ ( member @ A @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_678_Int__emptyI,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ~ ( member @ A @ X3 @ B2 ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_679_Int__Collect__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B2 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_680_Int__greatest,axiom,
! [A: $tType,C2: set @ A,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ C2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_681_Int__absorb2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= A4 ) ) ).
% Int_absorb2
thf(fact_682_Int__absorb1,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_683_Int__lower2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_684_Int__lower1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ A4 ) ).
% Int_lower1
thf(fact_685_Int__mono,axiom,
! [A: $tType,A4: set @ A,C2: set @ A,B2: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ ( inf_inf @ ( set @ A ) @ C2 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_686_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_687_Int__insert__right,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) )
& ( ~ ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_688_Int__insert__left,axiom,
! [A: $tType,A3: A,C2: set @ A,B2: set @ A] :
( ( ( member @ A @ A3 @ C2 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B2 ) @ C2 )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) )
& ( ~ ( member @ A @ A3 @ C2 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B2 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_689_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_690_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N @ M )
= ( zero_zero @ nat ) )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_691_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ord_less @ nat @ M @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_692_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_693_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ( minus_minus @ nat @ M @ K )
= ( minus_minus @ nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_694_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_695_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_696_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_697_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_698_le__diff__iff_H,axiom,
! [A3: nat,C3: nat,B3: nat] :
( ( ord_less_eq @ nat @ A3 @ C3 )
=> ( ( ord_less_eq @ nat @ B3 @ C3 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C3 @ A3 ) @ ( minus_minus @ nat @ C3 @ B3 ) )
= ( ord_less_eq @ nat @ B3 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_699_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_700_Un__Int__distrib2,axiom,
! [A: $tType,B2: set @ A,C2: set @ A,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) @ A4 )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B2 @ A4 ) @ ( sup_sup @ ( set @ A ) @ C2 @ A4 ) ) ) ).
% Un_Int_distrib2
thf(fact_701_Int__Un__distrib2,axiom,
! [A: $tType,B2: set @ A,C2: set @ A,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) @ A4 )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B2 @ A4 ) @ ( inf_inf @ ( set @ A ) @ C2 @ A4 ) ) ) ).
% Int_Un_distrib2
thf(fact_702_Un__Int__distrib,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ ( sup_sup @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_703_Int__Un__distrib,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_704_Un__Int__crazy,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) @ ( inf_inf @ ( set @ A ) @ C2 @ A4 ) )
= ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) @ ( sup_sup @ ( set @ A ) @ C2 @ A4 ) ) ) ).
% Un_Int_crazy
thf(fact_705_Diff__Int__distrib2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ C2 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_706_Diff__Int__distrib,axiom,
! [A: $tType,C2: set @ A,A4: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C2 @ A4 ) @ ( inf_inf @ ( set @ A ) @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_707_Diff__Diff__Int,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_708_Diff__Int2,axiom,
! [A: $tType,A4: set @ A,C2: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_709_Int__Diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_710_card__Diff__subset,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_711_diff__card__le__card__Diff,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_712_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A4 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_713_card__Diff__singleton,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_714_of__nat__0__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_nat_0_le_iff
thf(fact_715_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_716_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% distrib_inf_le
thf(fact_717_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% distrib_sup_le
thf(fact_718_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_719_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% of_nat_less_imp_less
thf(fact_720_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_721_infinite__arbitrarily__large,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ A4 )
=> ? [B4: set @ A] :
( ( finite_finite2 @ A @ B4 )
& ( ( finite_card @ A @ B4 )
= N )
& ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_722_card__subset__eq,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ( finite_card @ A @ A4 )
= ( finite_card @ A @ B2 ) )
=> ( A4 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_723_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B2: set @ A,A4: set @ B,R2: B > A > $o] :
( ( finite_finite2 @ A @ B2 )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ A4 )
=> ? [B11: A] :
( ( member @ A @ B11 @ B2 )
& ( R2 @ A7 @ B11 ) ) )
=> ( ! [A1: B,A22: B,B7: A] :
( ( member @ B @ A1 @ A4 )
=> ( ( member @ B @ A22 @ A4 )
=> ( ( member @ A @ B7 @ B2 )
=> ( ( R2 @ A1 @ B7 )
=> ( ( R2 @ A22 @ B7 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A4 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_724_of__nat__diff,axiom,
! [A: $tType] :
( ( semiring_1_cancel @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N ) )
= ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% of_nat_diff
thf(fact_725_card__insert__le,axiom,
! [A: $tType,A4: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ ( insert2 @ A @ X @ A4 ) ) ) ).
% card_insert_le
thf(fact_726_Diff__triv,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A4 @ B2 )
= A4 ) ) ).
% Diff_triv
thf(fact_727_Int__Diff__disjoint,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_Diff_disjoint
thf(fact_728_Un__Int__assoc__eq,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) )
= ( ord_less_eq @ ( set @ A ) @ C2 @ A4 ) ) ).
% Un_Int_assoc_eq
thf(fact_729_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_730_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less @ nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_731_diff__less__mono,axiom,
! [A3: nat,B3: nat,C3: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( ( ord_less_eq @ nat @ C3 @ A3 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A3 @ C3 ) @ ( minus_minus @ nat @ B3 @ C3 ) ) ) ) ).
% diff_less_mono
thf(fact_732_Diff__Un,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) )
= ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% Diff_Un
thf(fact_733_Diff__Int,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% Diff_Int
thf(fact_734_Int__Diff__Un,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= A4 ) ).
% Int_Diff_Un
thf(fact_735_Un__Diff__Int,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= A4 ) ).
% Un_Diff_Int
thf(fact_736_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_737_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y: set @ A,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert2 @ A @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_738_is__singleton__altdef,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
( ( finite_card @ A @ A6 )
= ( one_one @ nat ) ) ) ) ).
% is_singleton_altdef
thf(fact_739_card__eq__0__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ( finite_card @ A @ A4 )
= ( zero_zero @ nat ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A4 ) ) ) ).
% card_eq_0_iff
thf(fact_740_card__ge__0__finite,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A4 ) )
=> ( finite_finite2 @ A @ A4 ) ) ).
% card_ge_0_finite
thf(fact_741_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F4: set @ A,C2: nat] :
( ! [G4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G4 @ F4 )
=> ( ( finite_finite2 @ A @ G4 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G4 ) @ C2 ) ) )
=> ( ( finite_finite2 @ A @ F4 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F4 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_742_card__seteq,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B2 ) @ ( finite_card @ A @ A4 ) )
=> ( A4 = B2 ) ) ) ) ).
% card_seteq
thf(fact_743_card__mono,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) ) ) ) ).
% card_mono
thf(fact_744_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S: set @ A] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S ) )
=> ~ ! [T5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T5 @ S )
=> ( ( ( finite_card @ A @ T5 )
= N )
=> ~ ( finite_finite2 @ A @ T5 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_745_card__less__sym__Diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_746_card__le__sym__Diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_747_card__1__singletonE,axiom,
! [A: $tType,A4: set @ A] :
( ( ( finite_card @ A @ A4 )
= ( one_one @ nat ) )
=> ~ ! [X3: A] :
( A4
!= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_748_psubset__card__mono,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_749_card__gt__0__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A4 ) )
= ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A4 ) ) ) ).
% card_gt_0_iff
thf(fact_750_card__Diff1__le,axiom,
! [A: $tType,A4: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) ) ).
% card_Diff1_le
thf(fact_751_card__psubset,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) )
=> ( ord_less @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_752_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B2 ) @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_753_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_754_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] : ( member @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_755_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ ( lattic7752659483105999362nf_fin @ A @ B2 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_756_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_757_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_758_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N3: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% reals_Archimedean2
thf(fact_759_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N3: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% real_arch_simple
thf(fact_760_finite__enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,M: nat,N: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ M @ ( finite_card @ A @ S ) )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S ) )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ M ) @ ( infini527867602293511546merate @ A @ S @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_761_finite__enum__subset,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X4: set @ A,Y6: set @ A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( finite_card @ A @ X4 ) )
=> ( ( infini527867602293511546merate @ A @ X4 @ I2 )
= ( infini527867602293511546merate @ A @ Y6 @ I2 ) ) )
=> ( ( finite_finite2 @ A @ X4 )
=> ( ( finite_finite2 @ A @ Y6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ X4 ) @ ( finite_card @ A @ Y6 ) )
=> ( ord_less_eq @ ( set @ A ) @ X4 @ Y6 ) ) ) ) ) ) ).
% finite_enum_subset
thf(fact_762_card_Oremove,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ A4 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_763_card_Oinsert__remove,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X @ A4 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_764_card__Suc__Diff1,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A4 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_765_inverse__of__nat__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_766_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_767_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_768_Int__iff,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( ( member @ A @ C3 @ A4 )
& ( member @ A @ C3 @ B2 ) ) ) ).
% Int_iff
thf(fact_769_IntI,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ A4 )
=> ( ( member @ A @ C3 @ B2 )
=> ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% IntI
thf(fact_770_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_771_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_772_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_773_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_774_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_775_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq @ nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_776_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_777_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_778_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_779_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_780_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_781_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_782_zle__diff1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq @ int @ W2 @ ( minus_minus @ int @ Z @ ( one_one @ int ) ) )
= ( ord_less @ int @ W2 @ Z ) ) ).
% zle_diff1_eq
thf(fact_783_Suc__pred,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
= N ) ) ).
% Suc_pred
thf(fact_784_card__insert__disjoint,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X @ A4 ) )
= ( suc @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_785_enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,M: nat,N: nat] :
( ~ ( finite_finite2 @ A @ S )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ M ) @ ( infini527867602293511546merate @ A @ S @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ) ).
% enumerate_mono_iff
thf(fact_786_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= N ) ) ).
% Suc_diff_1
thf(fact_787_int__ops_I6_J,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B3 ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B3 ) )
= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% int_ops(6)
thf(fact_788_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less @ int @ I @ K )
=> ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I2: int] :
( ( ord_less @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_789_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq @ int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_790_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_791_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_792_Int__left__commute,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
= ( inf_inf @ ( set @ A ) @ B2 @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_793_Int__left__absorb,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ).
% Int_left_absorb
thf(fact_794_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_795_Int__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Int_absorb
thf(fact_796_Int__assoc,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_797_IntD2,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
=> ( member @ A @ C3 @ B2 ) ) ).
% IntD2
thf(fact_798_IntD1,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
=> ( member @ A @ C3 @ A4 ) ) ).
% IntD1
thf(fact_799_IntE,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
=> ~ ( ( member @ A @ C3 @ A4 )
=> ~ ( member @ A @ C3 @ B2 ) ) ) ).
% IntE
thf(fact_800_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_801_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_802_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ Z )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_803_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_804_verit__la__generic,axiom,
! [A3: int,X: int] :
( ( ord_less_eq @ int @ A3 @ X )
| ( A3 = X )
| ( ord_less_eq @ int @ X @ A3 ) ) ).
% verit_la_generic
thf(fact_805_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_806_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_807_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_808_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_809_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_810_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_811_enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ N ) @ ( infini527867602293511546merate @ A @ S @ ( suc @ N ) ) ) ) ) ).
% enumerate_step
thf(fact_812_vebt__buildup_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ( ( X
!= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ! [Va: nat] :
( X
!= ( suc @ ( suc @ Va ) ) ) ) ) ).
% vebt_buildup.cases
thf(fact_813_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_814_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_815_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_816_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_817_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_818_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y2: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_819_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_820_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_821_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_822_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_823_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_824_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( ( zero_zero @ nat )
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_825_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_826_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_827_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_828_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_829_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_830_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_831_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less @ nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_832_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_833_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_834_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( ord_less @ nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_835_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_836_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_837_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_838_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_839_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_840_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_841_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_842_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_843_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N ) ) )
= ( minus_minus @ nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_844_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_845_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_846_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_847_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq @ nat @ M @ N ) )
= ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_848_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_849_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq @ nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_850_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M7 )
=> ? [M4: nat] :
( M7
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_851_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_852_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_853_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_854_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_855_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_856_enumerate__in__set,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S )
=> ( member @ A @ ( infini527867602293511546merate @ A @ S @ N ) @ S ) ) ) ).
% enumerate_in_set
thf(fact_857_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ~ ! [N3: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ E2 ) ) ) ).
% nat_approx_posE
thf(fact_858_enumerate__Ex,axiom,
! [S: set @ nat,S3: nat] :
( ~ ( finite_finite2 @ nat @ S )
=> ( ( member @ nat @ S3 @ S )
=> ? [N3: nat] :
( ( infini527867602293511546merate @ nat @ S @ N3 )
= S3 ) ) ) ).
% enumerate_Ex
thf(fact_859_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_860_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less @ nat @ N @ N7 )
=> ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ N7 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_861_of__nat__neq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ N ) )
!= ( zero_zero @ A ) ) ) ).
% of_nat_neq_0
thf(fact_862_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ ( suc @ N3 ) ) @ ( F3 @ N3 ) )
=> ( ( ord_less_eq @ nat @ N @ N7 )
=> ( ord_less_eq @ A @ ( F3 @ N7 ) @ ( F3 @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_863_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N7 )
=> ( ord_less_eq @ A @ ( F3 @ N ) @ ( F3 @ N7 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_864_finite__enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ N ) @ ( infini527867602293511546merate @ A @ S @ ( suc @ N ) ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_865_enumerate__Suc_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S @ ( suc @ N ) )
= ( infini527867602293511546merate @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ ( infini527867602293511546merate @ A @ S @ ( zero_zero @ nat ) ) @ ( bot_bot @ ( set @ A ) ) ) ) @ N ) ) ) ).
% enumerate_Suc'
thf(fact_866_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( M
= ( zero_zero @ nat ) )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_867_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_868_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_869_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_870_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_871_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_872_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_873_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N2: nat] : ( ord_less_eq @ nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_874_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_875_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_876_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_877_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_878_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_879_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_880_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_881_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_882_le__enumerate,axiom,
! [S: set @ nat,N: nat] :
( ~ ( finite_finite2 @ nat @ S )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S @ N ) ) ) ).
% le_enumerate
thf(fact_883_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_884_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_885_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N )
= ( minus_minus @ nat @ M @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_886_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( N
= ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_pred'
thf(fact_887_card__insert__if,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X @ A4 ) )
= ( finite_card @ A @ A4 ) ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X @ A4 ) )
= ( suc @ ( finite_card @ A @ A4 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_888_card__Suc__eq__finite,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( ( finite_card @ A @ A4 )
= ( suc @ K ) )
= ( ? [B5: A,B6: set @ A] :
( ( A4
= ( insert2 @ A @ B5 @ B6 ) )
& ~ ( member @ A @ B5 @ B6 )
& ( ( finite_card @ A @ B6 )
= K )
& ( finite_finite2 @ A @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_889_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% zle_int
thf(fact_890_invar__vebt_Ointros_I1_J,axiom,
! [A3: $o,B3: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) ) ).
% invar_vebt.intros(1)
thf(fact_891_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_892_enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M: nat,N: nat,S: set @ A] :
( ( ord_less @ nat @ M @ N )
=> ( ~ ( finite_finite2 @ A @ S )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ M ) @ ( infini527867602293511546merate @ A @ S @ N ) ) ) ) ) ).
% enumerate_mono
thf(fact_893_finite__le__enumerate,axiom,
! [S: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ S )
=> ( ( ord_less @ nat @ N @ ( finite_card @ nat @ S ) )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_894_finite__enumerate__in__set,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S ) )
=> ( member @ A @ ( infini527867602293511546merate @ A @ S @ N ) @ S ) ) ) ) ).
% finite_enumerate_in_set
thf(fact_895_finite__enumerate__Ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,S3: A] :
( ( finite_finite2 @ A @ S )
=> ( ( member @ A @ S3 @ S )
=> ? [N3: nat] :
( ( ord_less @ nat @ N3 @ ( finite_card @ A @ S ) )
& ( ( infini527867602293511546merate @ A @ S @ N3 )
= S3 ) ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_896_finite__enum__ext,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X4: set @ A,Y6: set @ A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( finite_card @ A @ X4 ) )
=> ( ( infini527867602293511546merate @ A @ X4 @ I2 )
= ( infini527867602293511546merate @ A @ Y6 @ I2 ) ) )
=> ( ( finite_finite2 @ A @ X4 )
=> ( ( finite_finite2 @ A @ Y6 )
=> ( ( ( finite_card @ A @ X4 )
= ( finite_card @ A @ Y6 ) )
=> ( X4 = Y6 ) ) ) ) ) ) ).
% finite_enum_ext
thf(fact_897_card__Suc__eq,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( ( finite_card @ A @ A4 )
= ( suc @ K ) )
= ( ? [B5: A,B6: set @ A] :
( ( A4
= ( insert2 @ A @ B5 @ B6 ) )
& ~ ( member @ A @ B5 @ B6 )
& ( ( finite_card @ A @ B6 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B6
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_898_card__eq__SucD,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( ( finite_card @ A @ A4 )
= ( suc @ K ) )
=> ? [B7: A,B4: set @ A] :
( ( A4
= ( insert2 @ A @ B7 @ B4 ) )
& ~ ( member @ A @ B7 @ B4 )
& ( ( finite_card @ A @ B4 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_899_card__1__singleton__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ( finite_card @ A @ A4 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X2: A] :
( A4
= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_900_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ A4 )
=> ( X2 = Y3 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_901_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A4: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A4 ) )
= ( ? [A5: A,B6: set @ A] :
( ( A4
= ( insert2 @ A @ A5 @ B6 ) )
& ~ ( member @ A @ A5 @ B6 )
& ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B6 ) )
& ( finite_finite2 @ A @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_902_finite__enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M: nat,N: nat,S: set @ A] :
( ( ord_less @ nat @ M @ N )
=> ( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ M ) @ ( infini527867602293511546merate @ A @ S @ N ) ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_903_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_904_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_905_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_906_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_907_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_908_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_909_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_910_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_911_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_912_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_913_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_914_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ( divide_divide @ A @ C3 @ A3 )
= ( divide_divide @ A @ C3 @ B3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B3 ) ) ) ) ).
% divide_cancel_left
thf(fact_915_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ( divide_divide @ A @ A3 @ C3 )
= ( divide_divide @ A @ B3 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B3 ) ) ) ) ).
% divide_cancel_right
thf(fact_916_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_917_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( ( divide_divide @ A @ A3 @ B3 )
= ( one_one @ A ) )
= ( ( B3
!= ( zero_zero @ A ) )
& ( A3 = B3 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_918_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A3 @ B3 ) )
= ( ( B3
!= ( zero_zero @ A ) )
& ( A3 = B3 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_919_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_920_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_921_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ( divide_divide @ A @ B3 @ A3 )
= ( one_one @ A ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B3 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_922_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B3 @ A3 ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B3 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_923_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_924_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_925_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_926_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_927_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X5: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X5 ) ) ).
% linordered_field_no_lb
thf(fact_928_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X5: A] :
? [X_1: A] : ( ord_less @ A @ X5 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_929_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ ( divide_divide @ A @ A3 @ C3 ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_930_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_931_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_932_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_933_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_934_zero__le__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_divide_iff
thf(fact_935_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) ) ) ) ) ).
% divide_right_mono
thf(fact_936_divide__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% divide_le_0_iff
thf(fact_937_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_neg_neg
thf(fact_938_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_939_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_940_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_pos_pos
thf(fact_941_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_942_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A3 ) )
& ( C3
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_943_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_944_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_945_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_946_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B3 )
= ( one_one @ A ) )
= ( A3 = B3 ) ) ) ) ).
% right_inverse_eq
thf(fact_947_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_948_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_949_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_950_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_951_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_952_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W2: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less @ A @ W2 @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_953_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W2: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less
thf(fact_954_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,W2: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_le
thf(fact_955_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B3 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A3 @ B3 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_956_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ B3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_957_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ A3 @ B3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_958_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B3 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A3 @ B3 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_959_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_960_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_961_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ nat @ M @ N )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_962_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_963_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_964_bits__div__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% bits_div_0
thf(fact_965_bits__div__by__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_div_by_0
thf(fact_966_Suc__if__eq,axiom,
! [A: $tType,F3: nat > A,H: nat > A,G2: A,N: nat] :
( ! [N3: nat] :
( ( F3 @ ( suc @ N3 ) )
= ( H @ N3 ) )
=> ( ( ( F3 @ ( zero_zero @ nat ) )
= G2 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( ( F3 @ N )
= G2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( F3 @ N )
= ( H @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% Suc_if_eq
thf(fact_967_real__of__nat__div3,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) ) @ ( one_one @ real ) ) ).
% real_of_nat_div3
thf(fact_968_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_969_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_970_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_971_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_972_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( ord_less @ nat @ M @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_973_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ ( divide_divide @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_974_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_975_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N ) )
= ( ( ord_less_eq @ nat @ N @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_976_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ( divide_divide @ nat @ M @ N )
= M )
= ( N
= ( one_one @ nat ) ) ) ) ).
% div_eq_dividend_iff
thf(fact_977_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_978_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M2 @ N2 )
| ( N2
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% div_if
thf(fact_979_div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ) ) ).
% div_geq
thf(fact_980_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K )
=> ( ord_less @ int @ ( divide_divide @ int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_981_nonneg1__imp__zdiv__pos__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B3 ) )
= ( ( ord_less_eq @ int @ B3 @ A3 )
& ( ord_less @ int @ ( zero_zero @ int ) @ B3 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_982_pos__imp__zdiv__nonneg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B3 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_983_neg__imp__zdiv__nonneg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B3 ) )
= ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_984_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_985_div__nonpos__pos__le0,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonpos_pos_le0
thf(fact_986_div__nonneg__neg__le0,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonneg_neg_le0
thf(fact_987_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_988_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( ln_ln @ real @ X )
= ( ln_ln @ real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_989_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_990_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_991_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_less_zero_iff
thf(fact_992_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
= ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_993_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ln_ln @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% ln_eq_zero_iff
thf(fact_994_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_995_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_le_zero_iff
thf(fact_996_ln__div,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ln_ln @ real @ ( divide_divide @ real @ X @ Y ) )
= ( minus_minus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_997_ln__diff__le,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ X @ Y ) @ Y ) ) ) ) ).
% ln_diff_le
thf(fact_998_ln__bound,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ X ) ) ).
% ln_bound
thf(fact_999_ln__less__self,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( ln_ln @ real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_1000_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X2: real,Y3: real] :
( ( ord_less @ real @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% less_eq_real_def
thf(fact_1001_complete__real,axiom,
! [S: set @ real] :
( ? [X5: real] : ( member @ real @ X5 @ S )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member @ real @ X3 @ S )
=> ( ord_less_eq @ real @ X3 @ Z4 ) )
=> ? [Y2: real] :
( ! [X5: real] :
( ( member @ real @ X5 @ S )
=> ( ord_less_eq @ real @ X5 @ Y2 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ S )
=> ( ord_less_eq @ real @ X3 @ Z4 ) )
=> ( ord_less_eq @ real @ Y2 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1002_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1003_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ln_ln @ real @ X )
= ( minus_minus @ real @ X @ ( one_one @ real ) ) )
=> ( X
= ( one_one @ real ) ) ) ) ).
% ln_eq_minus_one
thf(fact_1004_ln__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% ln_less_zero
thf(fact_1005_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) ) ) ).
% ln_gt_zero
thf(fact_1006_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) ) ) ).
% ln_ge_zero
thf(fact_1007_ln__le__minus__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( minus_minus @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_le_minus_one
thf(fact_1008_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_1009_pos__imp__zdiv__neg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1010_neg__imp__zdiv__neg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1011_div__neg__pos__less0,axiom,
! [A3: int,B3: int] :
( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( zero_zero @ int ) ) ) ) ).
% div_neg_pos_less0
thf(fact_1012_div__positive,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_positive
thf(fact_1013_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_1014_zdiv__mono1,axiom,
! [A3: int,A11: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ A11 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( divide_divide @ int @ A11 @ B3 ) ) ) ) ).
% zdiv_mono1
thf(fact_1015_zdiv__mono2,axiom,
! [A3: int,B10: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B10 )
=> ( ( ord_less_eq @ int @ B10 @ B3 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( divide_divide @ int @ A3 @ B10 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1016_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_1017_zdiv__mono1__neg,axiom,
! [A3: int,A11: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ A11 )
=> ( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A11 @ B3 ) @ ( divide_divide @ int @ A3 @ B3 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1018_zdiv__mono2__neg,axiom,
! [A3: int,B10: int,B3: int] :
( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B10 )
=> ( ( ord_less_eq @ int @ B10 @ B3 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B10 ) @ ( divide_divide @ int @ A3 @ B3 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1019_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_1020_Bolzano,axiom,
! [A3: real,B3: real,P: real > real > $o] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [A7: real,B7: real,C5: real] :
( ( P @ A7 @ B7 )
=> ( ( P @ B7 @ C5 )
=> ( ( ord_less_eq @ real @ A7 @ B7 )
=> ( ( ord_less_eq @ real @ B7 @ C5 )
=> ( P @ A7 @ C5 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B3 )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [A7: real,B7: real] :
( ( ( ord_less_eq @ real @ A7 @ X3 )
& ( ord_less_eq @ real @ X3 @ B7 )
& ( ord_less @ real @ ( minus_minus @ real @ B7 @ A7 ) @ D4 ) )
=> ( P @ A7 @ B7 ) ) ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Bolzano
thf(fact_1021_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_1022_nat__ivt__aux,axiom,
! [N: nat,F3: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I2 ) ) @ ( F3 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F3 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
& ( ( F3 @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1023_list__decode_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_1024_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_12: A] : ( P @ ( zero_zero @ nat ) @ X_12 )
=> ( ! [X3: A,N3: nat] :
( ( P @ N3 @ X3 )
=> ? [Y5: A] :
( ( P @ ( suc @ N3 ) @ Y5 )
& ( Q @ N3 @ X3 @ Y5 ) ) )
=> ? [F6: nat > A] :
! [N4: nat] :
( ( P @ N4 @ ( F6 @ N4 ) )
& ( Q @ N4 @ ( F6 @ N4 ) @ ( F6 @ ( suc @ N4 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_1025_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_1026_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F3: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq @ nat @ M @ I2 )
& ( ord_less @ nat @ I2 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I2 ) ) @ ( F3 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ int @ ( F3 @ M ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F3 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ M @ I2 )
& ( ord_less_eq @ nat @ I2 @ N )
& ( ( F3 @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1027_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z ) ) ).
% one_less_nat_eq
thf(fact_1028_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C3 ) )
= ( B3 = C3 ) ) ) ).
% add_left_cancel
thf(fact_1029_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
= ( B3 = C3 ) ) ) ).
% add_right_cancel
thf(fact_1030_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_idempotent
thf(fact_1031_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_cancel_right
thf(fact_1032_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_cancel_left
thf(fact_1033_double__eq__0__iff,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( plus_plus @ A @ A3 @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_eq_0_iff
thf(fact_1034_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add_0
thf(fact_1035_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1036_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1037_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( plus_plus @ A @ A3 @ B3 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_1038_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( plus_plus @ A @ B3 @ A3 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_1039_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= A3 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_1040_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B3: A,A3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= A3 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_1041_double__zero__sym,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A3 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_1042_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.right_neutral
thf(fact_1043_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% add_less_cancel_left
thf(fact_1044_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% add_less_cancel_right
thf(fact_1045_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% add_diff_cancel
thf(fact_1046_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% diff_add_cancel
thf(fact_1047_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) )
= ( minus_minus @ A @ A3 @ B3 ) ) ) ).
% add_diff_cancel_left
thf(fact_1048_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ A3 )
= B3 ) ) ).
% add_diff_cancel_left'
thf(fact_1049_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ A3 @ B3 ) ) ) ).
% add_diff_cancel_right
thf(fact_1050_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% add_diff_cancel_right'
thf(fact_1051_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_1052_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ( abs_abs @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_1053_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_1054_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_1055_of__nat__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_add
thf(fact_1056_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_add_abs
thf(fact_1057_abs__of__nat,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat] :
( ( abs_abs @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% abs_of_nat
thf(fact_1058_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B3 @ A3 ) @ B3 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_1059_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_1060_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% le_add_same_cancel1
thf(fact_1061_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ B3 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% le_add_same_cancel2
thf(fact_1062_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1063_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1064_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B3 @ A3 ) @ B3 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_1065_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_1066_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel1
thf(fact_1067_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ B3 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel2
thf(fact_1068_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1069_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1070_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( plus_plus @ A @ B3 @ ( minus_minus @ A @ A3 @ B3 ) )
= A3 ) ) ) ).
% le_add_diff_inverse
thf(fact_1071_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ) ).
% le_add_diff_inverse2
thf(fact_1072_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_1073_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_nonneg
thf(fact_1074_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% abs_le_self_iff
thf(fact_1075_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_1076_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_1077_divide__le__0__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B3 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_0_abs_iff
thf(fact_1078_zero__le__divide__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B3 ) ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_1079_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_1080_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_1081_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_1082_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_1083_zless__nat__conj,axiom,
! [W2: int,Z: int] :
( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
& ( ord_less @ int @ W2 @ Z ) ) ) ).
% zless_nat_conj
thf(fact_1084_zle__add1__eq__le,axiom,
! [W2: int,Z: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z @ ( one_one @ int ) ) )
= ( ord_less_eq @ int @ W2 @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1085_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
= ( zero_zero @ int ) ) ) ) ).
% int_nat_eq
thf(fact_1086_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less @ int @ ( abs_abs @ int @ Z ) @ ( one_one @ int ) )
= ( Z
= ( zero_zero @ int ) ) ) ).
% zabs_less_one_iff
thf(fact_1087_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ).
% zero_less_nat_eq
thf(fact_1088_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B3 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_1089_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_1090_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1091_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_1092_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A,K: A,A3: A,B3: A] :
( ( A4
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( plus_plus @ A @ A4 @ B3 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_1093_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B2: A,K: A,B3: A,A3: A] :
( ( B2
= ( plus_plus @ A @ K @ B3 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_1094_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add.assoc
thf(fact_1095_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C3 ) )
= ( B3 = C3 ) ) ) ).
% add.left_cancel
thf(fact_1096_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
= ( B3 = C3 ) ) ) ).
% add.right_cancel
thf(fact_1097_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B5: A] : ( plus_plus @ A @ B5 @ A5 ) ) ) ) ).
% add.commute
thf(fact_1098_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add.left_commute
thf(fact_1099_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C3 ) )
=> ( B3 = C3 ) ) ) ).
% add_left_imp_eq
thf(fact_1100_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
=> ( B3 = C3 ) ) ) ).
% add_right_imp_eq
thf(fact_1101_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R2 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ R2 ) @ X )
& ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A3 @ R2 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_1102_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( plus_plus @ A @ C3 @ D2 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ C3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_1103_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B3 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_1104_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ R2 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ A3 @ R2 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1105_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_self
thf(fact_1106_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% abs_le_D1
thf(fact_1107_abs__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( ( abs_abs @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0_iff
thf(fact_1108_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B3 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B3 @ A3 ) ) ) ) ).
% abs_minus_commute
thf(fact_1109_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_imp_le_right
thf(fact_1110_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_imp_le_left
thf(fact_1111_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
? [C6: A] :
( B5
= ( plus_plus @ A @ A5 @ C6 ) ) ) ) ) ).
% le_iff_add
thf(fact_1112_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add_right_mono
thf(fact_1113_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ~ ! [C5: A] :
( B3
!= ( plus_plus @ A @ A3 @ C5 ) ) ) ) ).
% less_eqE
thf(fact_1114_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% add_left_mono
thf(fact_1115_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_mono
thf(fact_1116_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_1117_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_1118_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_1119_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add.group_left_neutral
thf(fact_1120_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.comm_neutral
thf(fact_1121_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_1122_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% verit_sum_simplify
thf(fact_1123_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_1124_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_1125_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_1126_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_strict_mono
thf(fact_1127_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% add_strict_left_mono
thf(fact_1128_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add_strict_right_mono
thf(fact_1129_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ).
% add_less_imp_less_left
thf(fact_1130_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ).
% add_less_imp_less_right
thf(fact_1131_infinite__int__iff__unbounded__le,axiom,
! [S: set @ int] :
( ( ~ ( finite_finite2 @ int @ S ) )
= ( ! [M2: int] :
? [N2: int] :
( ( ord_less_eq @ int @ M2 @ ( abs_abs @ int @ N2 ) )
& ( member @ int @ N2 @ S ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_1132_infinite__int__iff__unbounded,axiom,
! [S: set @ int] :
( ( ~ ( finite_finite2 @ int @ S ) )
= ( ! [M2: int] :
? [N2: int] :
( ( ord_less @ int @ M2 @ ( abs_abs @ int @ N2 ) )
& ( member @ int @ N2 @ S ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_1133_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,K: A,A3: A,B3: A] :
( ( A4
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( minus_minus @ A @ A4 @ B3 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_1134_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= C3 )
= ( A3
= ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% diff_eq_eq
thf(fact_1135_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( A3
= ( minus_minus @ A @ C3 @ B3 ) )
= ( ( plus_plus @ A @ A3 @ B3 )
= C3 ) ) ) ).
% eq_diff_eq
thf(fact_1136_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% add_diff_eq
thf(fact_1137_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( minus_minus @ A @ A3 @ ( minus_minus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B3 ) ) ) ).
% diff_diff_eq2
thf(fact_1138_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B3 ) ) ) ).
% diff_add_eq
thf(fact_1139_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C3 ) @ B3 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1140_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ( plus_plus @ A @ C3 @ B3 )
= A3 )
=> ( C3
= ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% add_implies_diff
thf(fact_1141_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 )
= ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% diff_diff_eq
thf(fact_1142_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_1143_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_zero
thf(fact_1144_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_pos
thf(fact_1145_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_1146_nat__abs__int__diff,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) )
= ( minus_minus @ nat @ B3 @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) )
= ( minus_minus @ nat @ A3 @ B3 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_1147_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_1148_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_1149_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B3 @ A3 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_1150_nonzero__abs__divide,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ) ).
% nonzero_abs_divide
thf(fact_1151_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( suc @ ( nat2 @ Z ) )
= ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1152_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_1153_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ X @ Y )
=> ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1154_eq__nat__nat__iff,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z5 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z5 ) )
= ( Z = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1155_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
! [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_1156_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
& ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_1157_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B3 ) ) ) ) ).
% add_decreasing
thf(fact_1158_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_increasing
thf(fact_1159_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B3 ) ) ) ) ).
% add_decreasing2
thf(fact_1160_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_increasing2
thf(fact_1161_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1162_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_1163_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1164_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1165_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_1166_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_1167_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_1168_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_1169_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_1170_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% add_pos_pos
thf(fact_1171_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ! [C5: A] :
( ( B3
= ( plus_plus @ A @ A3 @ C5 ) )
=> ( C5
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1172_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_add_strict
thf(fact_1173_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X @ Y ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_1174_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( minus_minus @ A @ B3 @ A3 )
= C3 )
= ( B3
= ( plus_plus @ A @ C3 @ A3 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1175_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B3 @ A3 ) )
= B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1176_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( minus_minus @ A @ C3 @ ( minus_minus @ A @ B3 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A3 ) @ B3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1177_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C3 ) @ A3 )
= ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A3 ) @ C3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1178_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A3 ) @ C3 )
= ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C3 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1179_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C3 @ B3 ) @ A3 )
= ( plus_plus @ A @ C3 @ ( minus_minus @ A @ B3 @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1180_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( plus_plus @ A @ C3 @ ( minus_minus @ A @ B3 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C3 @ B3 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1181_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ ( minus_minus @ A @ B3 @ A3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ B3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1182_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C3 ) @ A3 ) ) ) ) ).
% le_add_diff
thf(fact_1183_diff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A3 ) @ A3 )
= B3 ) ) ) ).
% diff_add
thf(fact_1184_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( minus_minus @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% le_diff_eq
thf(fact_1185_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 )
= ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% diff_le_eq
thf(fact_1186_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_1187_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N: A,J: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K ) @ J ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1188_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A] : ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_1189_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B3 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_1190_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 )
= ( ord_less @ A @ A3 @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% diff_less_eq
thf(fact_1191_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( minus_minus @ A @ C3 @ B3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% less_diff_eq
thf(fact_1192_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B3: A] :
( ~ ( ord_less @ A @ A3 @ B3 )
=> ( ( plus_plus @ A @ B3 @ ( minus_minus @ A @ A3 @ B3 ) )
= A3 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1193_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1194_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less @ int @ K @ I )
=> ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I2: int] :
( ( ord_less @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1195_zless__add1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z @ ( one_one @ int ) ) )
= ( ( ord_less @ int @ W2 @ Z )
| ( W2 = Z ) ) ) ).
% zless_add1_eq
thf(fact_1196_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W3: int,Z6: int] :
? [N2: nat] :
( Z6
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1197_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X2: A] : ( plus_plus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_1198_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_1199_abs__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( divide_divide @ A @ ( abs_abs @ A @ X ) @ Y )
= ( abs_abs @ A @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% abs_div_pos
thf(fact_1200_nat__mono__iff,axiom,
! [Z: int,W2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ W2 @ Z ) ) ) ).
% nat_mono_iff
thf(fact_1201_zless__nat__eq__int__zless,axiom,
! [M: nat,Z: int] :
( ( ord_less @ nat @ M @ ( nat2 @ Z ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_1202_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_1203_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_1204_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_1205_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_1206_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_1207_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_1208_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y @ E ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_epsilon
thf(fact_1209_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_le_iff
thf(fact_1210_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_1211_int__eq__iff,axiom,
! [M: nat,Z: int] :
( ( ( semiring_1_of_nat @ int @ M )
= Z )
= ( ( M
= ( nat2 @ Z ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% int_eq_iff
thf(fact_1212_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_1213_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_1214_div__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_1215_div__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B3 @ A3 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_1216_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B3 ) ) ) ).
% gt_half_sum
thf(fact_1217_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ A3 @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_1218_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W3: int,Z6: int] :
? [N2: nat] :
( Z6
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1219_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) @ Z ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ Z @ ( zero_zero @ int ) ) ) ).
% odd_less_0_iff
thf(fact_1220_add1__zle__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z )
= ( ord_less @ int @ W2 @ Z ) ) ).
% add1_zle_eq
thf(fact_1221_zless__imp__add1__zle,axiom,
! [W2: int,Z: int] :
( ( ord_less @ int @ W2 @ Z )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1222_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1223_nat__less__eq__zless,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ W2 @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_1224_nat__le__eq__zle,axiom,
! [W2: int,Z: int] :
( ( ( ord_less @ int @ ( zero_zero @ int ) @ W2 )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) )
=> ( ( ord_less_eq @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_eq @ int @ W2 @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_1225_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_1226_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_1227_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N2: nat] :
( ( I
= ( semiring_1_of_nat @ int @ N2 ) )
=> ( P @ N2 ) )
& ( ( ord_less @ int @ I @ ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ) ).
% split_nat
thf(fact_1228_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1229_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( minus_minus @ int @ X @ Y ) )
= ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_1230_nat__diff__distrib,axiom,
! [Z5: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z5 )
=> ( ( ord_less_eq @ int @ Z5 @ Z )
=> ( ( nat2 @ ( minus_minus @ int @ Z @ Z5 ) )
= ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1231_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_1232_nat__div__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_1233_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1234_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_1235_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
= ( power_power @ A @ A3 @ ( minus_minus @ nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1236_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_1237_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ B3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B3 @ M ) @ ( power_power @ A @ B3 @ N ) )
= ( ord_less_eq @ nat @ N @ M ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_1238_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) )
= ( ( A3
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_1239_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_1240_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B3 @ X ) @ ( power_power @ A @ B3 @ Y ) )
= ( ord_less_eq @ nat @ X @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_1241_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ B3 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B3 @ M ) @ ( power_power @ A @ B3 @ N ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1242_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: nat,W2: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ W2 ) @ X ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_1243_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B3: nat,W2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W2 ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B3 @ W2 ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_1244_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: nat,W2: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ B3 @ W2 ) @ X ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_1245_even__odd__cases,axiom,
! [X: nat] :
( ! [N3: nat] :
( X
!= ( plus_plus @ nat @ N3 @ N3 ) )
=> ~ ! [N3: nat] :
( X
!= ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% even_odd_cases
thf(fact_1246_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1247_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_1248_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_1249_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1250_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1251_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_1252_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ( power_power @ A @ A3 @ M )
= ( power_power @ A @ A3 @ N ) )
= ( M = N ) ) ) ) ).
% power_inject_exp
thf(fact_1253_power__0__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N ) )
= ( zero_zero @ A ) ) ) ).
% power_0_Suc
thf(fact_1254_power__Suc0__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% power_Suc0_right
thf(fact_1255_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% add_gr_0
thf(fact_1256_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_1257_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power @ nat @ X @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( X
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1258_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_1259_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_1260_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_1261_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1262_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B3 )
=> ( ( ord_less @ A @ ( power_power @ A @ B3 @ X ) @ ( power_power @ A @ B3 @ Y ) )
= ( ord_less @ nat @ X @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_1263_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A,N: nat] :
( ( ( power_power @ A @ A3 @ N )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% power_eq_0_iff
thf(fact_1264_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_1265_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_1266_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B3: nat,W2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W2 ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ B3 @ W2 ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_1267_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( plus_plus @ nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1268_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1269_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A3: nat] :
( ( A4
= ( plus_plus @ nat @ K @ A3 ) )
=> ( ( suc @ A4 )
= ( plus_plus @ nat @ K @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1270_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1271_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= M )
=> ( N
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_1272_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_1273_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_1274_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1275_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1276_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_1277_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_1278_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_1279_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1280_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M @ N )
=> ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1281_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_1282_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_1283_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% add_leD1
thf(fact_1284_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ).
% add_leD2
thf(fact_1285_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_1286_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_1287_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_1288_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_1289_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_1290_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1291_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1292_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1293_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1294_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1295_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1296_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ K @ ( power_power @ nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1297_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_1298_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% add_is_1
thf(fact_1299_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ M @ N ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% one_is_add
thf(fact_1300_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ? [N3: nat] : ( ord_less @ real @ Y @ ( power_power @ real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1301_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1302_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1303_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1304_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1305_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1306_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_1307_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F3: A > nat,N: nat] :
( ( P @ K )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ? [Y5: A] :
( ( P @ Y5 )
& ~ ( ord_less_eq @ nat @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) ) )
=> ? [Y2: A] :
( ( P @ Y2 )
& ~ ( ord_less @ nat @ ( F3 @ Y2 ) @ ( plus_plus @ nat @ ( F3 @ K ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_1308_mono__nat__linear__lb,axiom,
! [F3: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less @ nat @ M4 @ N3 )
=> ( ord_less @ nat @ ( F3 @ M4 ) @ ( F3 @ N3 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F3 @ M ) @ K ) @ ( F3 @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1309_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M ) )
= ( zero_zero @ nat ) ) ).
% diff_add_0
thf(fact_1310_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_1311_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less @ nat @ M @ N )
=> ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1312_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_1313_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1314_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_1315_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_1316_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_1317_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_1318_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_1319_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_1320_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ? [N3: nat] : ( ord_less @ real @ ( power_power @ real @ X @ N3 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1321_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B3: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B3 ) )
= ( ( ( ord_less @ nat @ A3 @ B3 )
=> ( P @ ( zero_zero @ nat ) ) )
& ! [D5: nat] :
( ( A3
= ( plus_plus @ nat @ B3 @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1322_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B3: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B3 ) )
= ( ~ ( ( ( ord_less @ nat @ A3 @ B3 )
& ~ ( P @ ( zero_zero @ nat ) ) )
| ? [D5: nat] :
( ( A3
= ( plus_plus @ nat @ B3 @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1323_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_1324_card__Un__le,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) ) ) ).
% card_Un_le
thf(fact_1325_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N2
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1326_nat__less__real__le,axiom,
( ( ord_less @ nat )
= ( ^ [N2: nat,M2: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ).
% nat_less_real_le
thf(fact_1327_nat__le__real__less,axiom,
( ( ord_less_eq @ nat )
= ( ^ [N2: nat,M2: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( one_one @ real ) ) ) ) ) ).
% nat_le_real_less
thf(fact_1328_card__Un__Int,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_1329_ln__add__one__self__le__self,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self
thf(fact_1330_nat__power__eq,axiom,
! [Z: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( nat2 @ ( power_power @ int @ Z @ N ) )
= ( power_power @ nat @ ( nat2 @ Z ) @ N ) ) ) ).
% nat_power_eq
thf(fact_1331_nat__add__distrib,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z5 )
=> ( ( nat2 @ ( plus_plus @ int @ Z @ Z5 ) )
= ( plus_plus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1332_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_1333_card__Un__disjoint,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1334_power__not__zero,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A,N: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A3 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_not_zero
thf(fact_1335_nat0__intermed__int__val,axiom,
! [N: nat,F3: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( F3 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F3 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
& ( ( F3 @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1336_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% zero_le_power
thf(fact_1337_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ).
% power_mono
thf(fact_1338_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% zero_less_power
thf(fact_1339_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% one_le_power
thf(fact_1340_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_1341_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B3: A] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_1342_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_1343_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( power_power @ A @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% power_le_imp_le_base
thf(fact_1344_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B3: A] :
( ( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( power_power @ A @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( A3 = B3 ) ) ) ) ) ).
% power_inject_base
thf(fact_1345_power__0__left,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ) ).
% power_0_left
thf(fact_1346_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_1347_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_1348_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less @ nat @ N @ N6 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_1349_zero__le__power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) ) ) ).
% zero_le_power_abs
thf(fact_1350_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% power_increasing
thf(fact_1351_zero__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ).
% zero_power
thf(fact_1352_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ) ).
% power_Suc_le_self
thf(fact_1353_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_1354_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less @ nat @ N @ N6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1355_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_1356_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_1357_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B3: A] :
( ( ( power_power @ A @ A3 @ N )
= ( power_power @ A @ B3 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A3 = B3 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1358_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,A3: A,B3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ( power_power @ A @ A3 @ N )
= ( power_power @ A @ B3 @ N ) )
= ( A3 = B3 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1359_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% self_le_power
thf(fact_1360_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_1361_power__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( power_power @ A @ A3 @ ( minus_minus @ nat @ M @ N ) )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_diff
thf(fact_1362_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_1363_lemma__interval,axiom,
! [A3: real,X: real,B3: real] :
( ( ord_less @ real @ A3 @ X )
=> ( ( ord_less @ real @ X @ B3 )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [Y5: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y5 ) ) @ D6 )
=> ( ( ord_less_eq @ real @ A3 @ Y5 )
& ( ord_less_eq @ real @ Y5 @ B3 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_1364_realpow__pos__nth,axiom,
! [N: nat,A3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ N )
= A3 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1365_realpow__pos__nth__unique,axiom,
! [N: nat,A3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
& ( ( power_power @ real @ X3 @ N )
= A3 )
& ! [Y5: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y5 )
& ( ( power_power @ real @ Y5 @ N )
= A3 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1366_lemma__interval__lt,axiom,
! [A3: real,X: real,B3: real] :
( ( ord_less @ real @ A3 @ X )
=> ( ( ord_less @ real @ X @ B3 )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [Y5: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y5 ) ) @ D6 )
=> ( ( ord_less @ real @ A3 @ Y5 )
& ( ord_less @ real @ Y5 @ B3 ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_1367_realpow__pos__nth2,axiom,
! [A3: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ ( suc @ N ) )
= A3 ) ) ) ).
% realpow_pos_nth2
thf(fact_1368_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_1369_finite__maxlen,axiom,
! [A: $tType,M5: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M5 )
=> ? [N3: nat] :
! [X5: list @ A] :
( ( member @ ( list @ A ) @ X5 @ M5 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_1370_sin__bound__lemma,axiom,
! [X: real,Y: real,U: real,V2: real] :
( ( X = Y )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ U ) @ V2 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( plus_plus @ real @ X @ U ) @ Y ) ) @ V2 ) ) ) ).
% sin_bound_lemma
thf(fact_1371_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A7: nat,B7: nat] :
( ( P @ A7 @ B7 )
= ( P @ B7 @ A7 ) )
=> ( ! [A7: nat] : ( P @ A7 @ ( zero_zero @ nat ) )
=> ( ! [A7: nat,B7: nat] :
( ( P @ A7 @ B7 )
=> ( P @ A7 @ ( plus_plus @ nat @ A7 @ B7 ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Euclid_induct
thf(fact_1372_add__0__iff,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [B3: A,A3: A] :
( ( B3
= ( plus_plus @ A @ B3 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_1373_ln__root,axiom,
! [N: nat,B3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( ln_ln @ real @ ( root @ N @ B3 ) )
= ( divide_divide @ real @ ( ln_ln @ real @ B3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% ln_root
thf(fact_1374_log__of__power__le,axiom,
! [M: nat,B3: real,N: nat] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B3 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less_eq @ real @ ( log @ B3 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_1375_decr__lemma,axiom,
! [D2: int,X: int,Z: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z ) ) @ ( one_one @ int ) ) @ D2 ) ) @ Z ) ) ).
% decr_lemma
thf(fact_1376_incr__lemma,axiom,
! [D2: int,Z: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ Z @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z ) ) @ ( one_one @ int ) ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_1377_linear__plus__1__le__power,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) @ ( one_one @ real ) ) @ ( power_power @ real @ ( plus_plus @ real @ X @ ( one_one @ real ) ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_1378_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_1379_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_1380_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B3: A] :
( ( ( times_times @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_1381_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ( times_times @ A @ C3 @ A3 )
= ( times_times @ A @ C3 @ B3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B3 ) ) ) ) ).
% mult_cancel_left
thf(fact_1382_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ( times_times @ A @ A3 @ C3 )
= ( times_times @ A @ B3 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B3 ) ) ) ) ).
% mult_cancel_right
thf(fact_1383_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% mult_1
thf(fact_1384_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.right_neutral
thf(fact_1385_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mult
thf(fact_1386_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,B3: A] :
( ( C3
= ( times_times @ A @ C3 @ B3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( B3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_1387_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,A3: A] :
( ( ( times_times @ A @ C3 @ A3 )
= C3 )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_1388_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,B3: A] :
( ( C3
= ( times_times @ A @ B3 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( B3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_1389_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A,C3: A] :
( ( ( times_times @ A @ A3 @ C3 )
= C3 )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_1390_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1391_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1392_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1393_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1394_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1395_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1396_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B3 ) @ A3 )
= B3 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1397_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1398_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% div_mult_mult1
thf(fact_1399_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% div_mult_mult2
thf(fact_1400_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1401_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less @ real @ ( zero_zero @ real ) @ ( times_times @ real @ X @ X ) ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% not_real_square_gt_zero
thf(fact_1402_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_1403_root__0,axiom,
! [X: real] :
( ( root @ ( zero_zero @ nat ) @ X )
= ( zero_zero @ real ) ) ).
% root_0
thf(fact_1404_real__root__eq__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( root @ N @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_1405_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B3 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1406_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B3 @ ( times_times @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1407_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C3 @ B3 ) ) @ B3 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_mult_self1
thf(fact_1408_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) ) @ B3 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_mult_self2
thf(fact_1409_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C3 @ B3 ) @ A3 ) @ B3 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_mult_self3
thf(fact_1410_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ C3 ) @ A3 ) @ B3 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_mult_self4
thf(fact_1411_real__root__eq__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% real_root_eq_0_iff
thf(fact_1412_real__root__less__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_1413_real__root__le__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_1414_real__root__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( one_one @ real ) )
= ( one_one @ real ) ) ) ).
% real_root_one
thf(fact_1415_real__root__eq__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( one_one @ real ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% real_root_eq_1_iff
thf(fact_1416_log__eq__one,axiom,
! [A3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log @ A3 @ A3 )
= ( one_one @ real ) ) ) ) ).
% log_eq_one
thf(fact_1417_log__less__cancel__iff,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( log @ A3 @ X ) @ ( log @ A3 @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_1418_log__less__one__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ A3 @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ A3 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_1419_one__less__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( log @ A3 @ X ) )
= ( ord_less @ real @ A3 @ X ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_1420_log__less__zero__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ A3 @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_1421_zero__less__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( log @ A3 @ X ) )
= ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_1422_real__root__gt__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_1423_real__root__lt__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_lt_0_iff
thf(fact_1424_real__root__le__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_le_0_iff
thf(fact_1425_real__root__ge__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_1426_real__root__gt__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_1427_real__root__lt__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_lt_1_iff
thf(fact_1428_real__root__le__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_le_1_iff
thf(fact_1429_real__root__ge__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_1430_zero__le__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ A3 @ X ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_1431_log__le__zero__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ A3 @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_1432_one__le__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( log @ A3 @ X ) )
= ( ord_less_eq @ real @ A3 @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_1433_log__le__one__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ A3 @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ A3 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_1434_log__le__cancel__iff,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( log @ A3 @ X ) @ ( log @ A3 @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_1435_real__root__pow__pos2,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_1436_log__pow__cancel,axiom,
! [A3: real,B3: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log @ A3 @ ( power_power @ real @ A3 @ B3 ) )
= ( semiring_1_of_nat @ real @ B3 ) ) ) ) ).
% log_pow_cancel
thf(fact_1437_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( times_times @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) )
= ( times_times @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% mult.left_commute
thf(fact_1438_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A5: A,B5: A] : ( times_times @ A @ B5 @ A5 ) ) ) ) ).
% mult.commute
thf(fact_1439_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B3 ) @ C3 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% mult.assoc
thf(fact_1440_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B3 ) @ C3 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1441_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A,B3: A] :
( ( ( times_times @ A @ A3 @ B3 )
!= ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
& ( B3
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_1442_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B3: A] :
( ( ( times_times @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
=> ( ( A3
= ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_1443_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ B3 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_1444_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C3 @ A3 )
= ( times_times @ A @ C3 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% mult_left_cancel
thf(fact_1445_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C3 )
= ( times_times @ A @ B3 @ C3 ) )
= ( A3 = B3 ) ) ) ) ).
% mult_right_cancel
thf(fact_1446_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.comm_neutral
thf(fact_1447_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1448_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: nat,Y: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X ) @ Y )
= ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X ) ) ) ) ).
% mult_of_nat_commute
thf(fact_1449_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R2: A,A3: A,B3: A,C3: A,D2: A] :
( ( R2
!= ( zero_zero @ A ) )
=> ( ( ( A3 = B3 )
& ( C3 != D2 ) )
=> ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R2 @ C3 ) )
!= ( plus_plus @ A @ B3 @ ( times_times @ A @ R2 @ D2 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1450_log__base__root,axiom,
! [N: nat,B3: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( log @ ( root @ N @ B3 ) @ X )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B3 @ X ) ) ) ) ) ).
% log_base_root
thf(fact_1451_log__mult,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( log @ A3 @ ( times_times @ real @ X @ Y ) )
= ( plus_plus @ real @ ( log @ A3 @ X ) @ ( log @ A3 @ Y ) ) ) ) ) ) ) ).
% log_mult
thf(fact_1452_log__nat__power,axiom,
! [X: real,B3: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ B3 @ ( power_power @ real @ X @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B3 @ X ) ) ) ) ).
% log_nat_power
thf(fact_1453_real__root__pos__pos__le,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ).
% real_root_pos_pos_le
thf(fact_1454_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1455_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1456_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B3 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1457_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1458_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1459_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1460_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B3: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_1461_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_1462_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% mult_right_mono
thf(fact_1463_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1464_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% mult_left_mono
thf(fact_1465_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1466_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1467_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B3: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) ) ) ) ).
% split_mult_pos_le
thf(fact_1468_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ A3 ) ) ) ).
% zero_le_square
thf(fact_1469_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1470_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_1471_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_1472_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( times_times @ A @ A3 @ A3 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_1473_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_1474_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_1475_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_1476_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_1477_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B3 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_1478_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1479_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_1480_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B3 @ A3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_1481_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1482_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1483_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1484_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1485_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A3 @ B3 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1486_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1487_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1488_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A3 @ B3 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1489_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1490_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N ) ) ) ) ) ).
% less_1_mult
thf(fact_1491_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X @ Y )
= ( divide_divide @ A @ W2 @ Z ) )
= ( ( times_times @ A @ X @ Z )
= ( times_times @ A @ W2 @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1492_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ( divide_divide @ A @ B3 @ C3 )
= A3 )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A3 @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_1493_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3
= ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C3 )
= B3 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_1494_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( B3
= ( times_times @ A @ A3 @ C3 ) )
=> ( ( divide_divide @ A @ B3 @ C3 )
= A3 ) ) ) ) ).
% divide_eq_imp
thf(fact_1495_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C3 )
= B3 )
=> ( A3
= ( divide_divide @ A @ B3 @ C3 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_1496_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B3 @ C3 )
= A3 )
= ( B3
= ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1497_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( A3
= ( divide_divide @ A @ B3 @ C3 ) )
= ( ( times_times @ A @ A3 @ C3 )
= B3 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1498_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C3: A,B3: A,D2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ C3 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B3 ) @ D2 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) @ ( times_times @ A @ C3 @ D2 ) ) ) ) ) ).
% abs_mult_less
thf(fact_1499_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_1500_log__eq__div__ln__mult__log,axiom,
! [A3: real,B3: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( B3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ A3 @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B3 ) @ ( ln_ln @ real @ A3 ) ) @ ( log @ B3 @ X ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_1501_log__root,axiom,
! [N: nat,A3: real,B3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( log @ B3 @ ( root @ N @ A3 ) )
= ( divide_divide @ real @ ( log @ B3 @ A3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_root
thf(fact_1502_real__root__less__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_1503_real__root__le__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_1504_real__root__power,axiom,
! [N: nat,X: real,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X @ K ) )
= ( power_power @ real @ ( root @ N @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_1505_real__root__abs,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( abs_abs @ real @ X ) )
= ( abs_abs @ real @ ( root @ N @ X ) ) ) ) ).
% real_root_abs
thf(fact_1506_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1507_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1508_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_1509_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_1510_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1511_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1512_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B3 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1513_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1514_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_1515_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B3 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1516_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1517_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_1518_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1519_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1520_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C3: A,A3: A] :
( ( ord_less_eq @ A @ C3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ A3 ) ) ) ) ).
% mult_left_le
thf(fact_1521_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_1522_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X @ Y ) @ X ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1523_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X ) @ X ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1524_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1525_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_1526_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1527_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_1528_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_1529_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1530_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_1531_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_1532_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_1533_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A3 )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_1534_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_1535_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ X @ ( times_times @ A @ Z @ Y ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ Z ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1536_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ ( times_times @ A @ Z @ Y ) @ X )
=> ( ord_less @ A @ Z @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1537_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B3 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_1538_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B3 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_1539_ordered__ring__class_Ole__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C3: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E2 ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B3 ) @ E2 ) @ C3 ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1540_ordered__ring__class_Ole__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C3: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E2 ) @ D2 ) )
= ( ord_less_eq @ A @ C3 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A3 ) @ E2 ) @ D2 ) ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1541_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B3 )
= B3 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B3 )
= ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B3 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1542_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z ) )
= A3 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1543_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_1544_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ Z )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z @ Y ) ) @ Y ) ) ) ) ).
% add_frac_num
thf(fact_1545_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z @ ( divide_divide @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z @ Y ) ) @ Y ) ) ) ) ).
% add_num_frac
thf(fact_1546_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z ) @ Y ) @ Z ) ) ) ) ).
% add_divide_eq_iff
thf(fact_1547_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% divide_add_eq_iff
thf(fact_1548_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C3: A,B3: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E2 ) @ D2 ) )
= ( ord_less @ A @ C3 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A3 ) @ E2 ) @ D2 ) ) ) ) ).
% less_add_iff2
thf(fact_1549_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C3: A,B3: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E2 ) @ D2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B3 ) @ E2 ) @ C3 ) @ D2 ) ) ) ).
% less_add_iff1
thf(fact_1550_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z ) )
= A3 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1551_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1552_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X @ ( divide_divide @ A @ Y @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ Y ) @ Z ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_1553_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Z ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_1554_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_1555_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_1556_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_1557_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A3: A,B3: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
& ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
| ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) )
=> ( ( abs_abs @ A @ ( times_times @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_1558_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y ) @ X )
= ( abs_abs @ A @ ( times_times @ A @ Y @ X ) ) ) ) ) ).
% abs_mult_pos
thf(fact_1559_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ! [Y5: real] :
? [N3: nat] : ( ord_less @ real @ Y5 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1560_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M )
=> ( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
= ( ( M
= ( one_one @ int ) )
& ( N
= ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1561_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1562_plusinfinity,axiom,
! [D2: int,P4: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1563_zdiv__zmult2__eq,axiom,
! [C3: int,A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C3 )
=> ( ( divide_divide @ int @ A3 @ ( times_times @ int @ B3 @ C3 ) )
= ( divide_divide @ int @ ( divide_divide @ int @ A3 @ B3 ) @ C3 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1564_real__root__gt__zero,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_1565_real__root__strict__decreasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N6 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( root @ N6 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_1566_log__base__change,axiom,
! [A3: real,B3: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log @ B3 @ X )
= ( divide_divide @ real @ ( log @ A3 @ X ) @ ( log @ A3 @ B3 ) ) ) ) ) ).
% log_base_change
thf(fact_1567_root__abs__power,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( abs_abs @ real @ ( root @ N @ ( power_power @ real @ Y @ N ) ) )
= ( abs_abs @ real @ Y ) ) ) ).
% root_abs_power
thf(fact_1568_less__log__of__power,axiom,
! [B3: real,N: nat,M: real] :
( ( ord_less @ real @ ( power_power @ real @ B3 @ N ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B3 @ M ) ) ) ) ).
% less_log_of_power
thf(fact_1569_log__of__power__eq,axiom,
! [M: nat,B3: real,N: nat] :
( ( ( semiring_1_of_nat @ real @ M )
= ( power_power @ real @ B3 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( semiring_1_of_nat @ real @ N )
= ( log @ B3 @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ).
% log_of_power_eq
thf(fact_1570_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less @ A @ Z3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ Y ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_1571_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C3: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ C3 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1572_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( times_times @ A @ B3 @ C3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1573_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A3: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ C3 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1574_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( times_times @ A @ C3 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1575_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ C3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1576_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ ( times_times @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1577_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ C3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1578_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ ( times_times @ A @ C3 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1579_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B3 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_1580_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ Y ) @ X )
=> ( ord_less_eq @ A @ Z @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1581_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ ( times_times @ A @ Z @ Y ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ Z ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1582_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_1583_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A3 )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_1584_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_1585_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_1586_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B3 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_1587_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_1588_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_1589_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A3: A,Y: A,U: A,V2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V2 @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1590_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1591_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_1592_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_Suc_less
thf(fact_1593_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_1594_q__pos__lemma,axiom,
! [B10: int,Q4: int,R4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B10 @ Q4 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B10 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B10 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q4 ) ) ) ) ).
% q_pos_lemma
thf(fact_1595_zdiv__mono2__lemma,axiom,
! [B3: int,Q5: int,R2: int,B10: int,Q4: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B3 @ Q5 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B10 @ Q4 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B10 @ Q4 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B10 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B10 )
=> ( ( ord_less_eq @ int @ B10 @ B3 )
=> ( ord_less_eq @ int @ Q5 @ Q4 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_1596_zdiv__mono2__neg__lemma,axiom,
! [B3: int,Q5: int,R2: int,B10: int,Q4: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B3 @ Q5 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B10 @ Q4 ) @ R4 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B10 @ Q4 ) @ R4 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R2 @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B10 )
=> ( ( ord_less_eq @ int @ B10 @ B3 )
=> ( ord_less_eq @ int @ Q4 @ Q5 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_1597_unique__quotient__lemma,axiom,
! [B3: int,Q4: int,R4: int,Q5: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q5 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ R4 @ B3 )
=> ( ( ord_less @ int @ R2 @ B3 )
=> ( ord_less_eq @ int @ Q4 @ Q5 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_1598_unique__quotient__lemma__neg,axiom,
! [B3: int,Q4: int,R4: int,Q5: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q5 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ R2 )
=> ( ( ord_less @ int @ B3 @ R4 )
=> ( ord_less_eq @ int @ Q5 @ Q4 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_1599_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( plus_plus @ int @ X5 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1600_decr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1601_ln__mult,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ln_ln @ real @ ( times_times @ real @ X @ Y ) )
= ( plus_plus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_1602_real__root__pos__pos,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_1603_real__root__strict__increasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N6 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_1604_real__root__decreasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( root @ N6 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_1605_real__root__pow__pos,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_1606_real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N @ ( power_power @ real @ X @ N ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_1607_real__root__pos__unique,axiom,
! [N: nat,Y: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( power_power @ real @ Y @ N )
= X )
=> ( ( root @ N @ X )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_1608_log__divide,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( log @ A3 @ ( divide_divide @ real @ X @ Y ) )
= ( minus_minus @ real @ ( log @ A3 @ X ) @ ( log @ A3 @ Y ) ) ) ) ) ) ) ).
% log_divide
thf(fact_1609_le__log__of__power,axiom,
! [B3: real,N: nat,M: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ B3 @ N ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B3 @ M ) ) ) ) ).
% le_log_of_power
thf(fact_1610_log__base__pow,axiom,
! [A3: real,N: nat,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( log @ ( power_power @ real @ A3 @ N ) @ X )
= ( divide_divide @ real @ ( log @ A3 @ X ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log_base_pow
thf(fact_1611_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A3: A,Y: A,U: A,V2: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ( ord_less @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V2 @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1612_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V2: A,R2: A,S3: A] :
( ( ord_less_eq @ A @ U @ V2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 )
=> ( ( ord_less_eq @ A @ R2 @ S3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R2 @ ( minus_minus @ A @ V2 @ U ) ) @ S3 ) ) @ V2 ) ) ) ) ) ).
% scaling_mono
thf(fact_1613_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P5: A,M2: nat] :
( if @ A
@ ( M2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P5 @ ( power_power @ A @ P5 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1614_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A3 )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus_mult
thf(fact_1615_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C3 )
=> ( ! [M4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M4 ) @ X ) @ C3 ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1616_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_zdiv
thf(fact_1617_int__div__neg__eq,axiom,
! [A3: int,B3: int,Q5: int,R2: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q5 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ R2 )
=> ( ( divide_divide @ int @ A3 @ B3 )
= Q5 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1618_int__div__pos__eq,axiom,
! [A3: int,B3: int,Q5: int,R2: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q5 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B3 )
=> ( ( divide_divide @ int @ A3 @ B3 )
= Q5 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1619_ln__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( power_power @ real @ X @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_realpow
thf(fact_1620_real__root__increasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_1621_log__of__power__less,axiom,
! [M: nat,B3: real,N: nat] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B3 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ real @ ( log @ B3 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_1622_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ X ) @ ( times_times @ A @ Z @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1623_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ Y @ Z ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1624_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ Y @ Z ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% mult_less_iff1
thf(fact_1625_arctan__add,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( plus_plus @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( arctan @ ( divide_divide @ real @ ( plus_plus @ real @ X @ Y ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( times_times @ real @ X @ Y ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_1626_root__powr__inverse,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N @ X )
= ( powr @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_1627_split__root,axiom,
! [P: real > $o,N: nat,X: real] :
( ( P @ ( root @ N @ X ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ real ) ) )
& ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ! [Y3: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ Y3 ) @ ( power_power @ real @ ( abs_abs @ real @ Y3 ) @ N ) )
= X )
=> ( P @ Y3 ) ) ) ) ) ).
% split_root
thf(fact_1628_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_1629_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_1630_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A6: A > B,X2: A] : ( uminus_uminus @ B @ ( A6 @ X2 ) ) ) ) ) ).
% uminus_apply
thf(fact_1631_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
= X ) ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_1632_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( uminus_uminus @ A @ X )
= ( uminus_uminus @ A @ Y ) )
= ( X = Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_1633_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
= A3 ) ) ).
% add.inverse_inverse
thf(fact_1634_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B3 ) )
= ( A3 = B3 ) ) ) ).
% neg_equal_iff_equal
thf(fact_1635_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% neg_le_iff_le
thf(fact_1636_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_1637_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= A3 )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_1638_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( A3
= ( uminus_uminus @ A @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_1639_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_1640_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A3 ) )
= ( ( zero_zero @ A )
= A3 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_1641_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_1642_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% compl_less_compl_iff
thf(fact_1643_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% neg_less_iff_less
thf(fact_1644_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) )
= B3 ) ) ).
% add_minus_cancel
thf(fact_1645_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B3 ) )
= B3 ) ) ).
% minus_add_cancel
thf(fact_1646_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ).
% minus_add_distrib
thf(fact_1647_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B3 ) )
= ( minus_minus @ A @ B3 @ A3 ) ) ) ).
% minus_diff_eq
thf(fact_1648_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_minus_cancel
thf(fact_1649_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times @ nat @ M @ K )
= ( times_times @ nat @ N @ K ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_1650_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_1651_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_1652_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_1653_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_1654_powr__eq__0__iff,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [W2: A,Z: A] :
( ( ( powr @ A @ W2 @ Z )
= ( zero_zero @ A ) )
= ( W2
= ( zero_zero @ A ) ) ) ) ).
% powr_eq_0_iff
thf(fact_1655_powr__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [Z: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_1656_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M @ N ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1657_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( one_one @ nat ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1658_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_1659_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_le_0_iff_le
thf(fact_1660_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_1661_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_1662_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_1663_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_pos
thf(fact_1664_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_1665_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_0_iff_less
thf(fact_1666_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_1667_ab__left__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_left_minus
thf(fact_1668_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ).
% diff_0
thf(fact_1669_verit__minus__simplify_I3_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B3: B] :
( ( minus_minus @ B @ ( zero_zero @ B ) @ B3 )
= ( uminus_uminus @ B @ B3 ) ) ) ).
% verit_minus_simplify(3)
thf(fact_1670_mult__minus1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z )
= ( uminus_uminus @ A @ Z ) ) ) ).
% mult_minus1
thf(fact_1671_mult__minus1__right,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: A] :
( ( times_times @ A @ Z @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ Z ) ) ) ).
% mult_minus1_right
thf(fact_1672_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( plus_plus @ A @ A3 @ B3 ) ) ) ).
% diff_minus_eq_add
thf(fact_1673_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( minus_minus @ A @ B3 @ A3 ) ) ) ).
% uminus_add_conv_diff
thf(fact_1674_inf__compl__bot__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( inf_inf @ A @ X @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left1
thf(fact_1675_inf__compl__bot__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left2
thf(fact_1676_inf__compl__bot__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ ( uminus_uminus @ A @ X ) ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_right
thf(fact_1677_boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ X )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1678_boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( uminus_uminus @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1679_abs__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( one_one @ A ) ) ) ).
% abs_neg_one
thf(fact_1680_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_greater
thf(fact_1681_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_1682_boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( inf_inf @ A @ X @ Y ) )
= ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_1683_boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( sup_sup @ A @ X @ Y ) )
= ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_1684_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( times_times @ nat @ M @ N ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1685_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1686_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1687_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1688_powr__zero__eq__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [X: A] :
( ( ( X
= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) )
& ( ( X
!= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% powr_zero_eq_one
thf(fact_1689_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times @ nat @ M @ ( suc @ N ) )
= ( plus_plus @ nat @ M @ ( times_times @ nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1690_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_1691_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ int @ M ) )
= ( ( N
= ( zero_zero @ nat ) )
& ( M
= ( zero_zero @ nat ) ) ) ) ).
% negative_eq_positive
thf(fact_1692_powr__gt__zero,axiom,
! [X: real,A3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ A3 ) )
= ( X
!= ( zero_zero @ real ) ) ) ).
% powr_gt_zero
thf(fact_1693_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_1694_powr__nonneg__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_eq @ real @ ( powr @ real @ A3 @ X ) @ ( zero_zero @ real ) )
= ( A3
= ( zero_zero @ real ) ) ) ).
% powr_nonneg_iff
thf(fact_1695_gbinomial__Suc0,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% gbinomial_Suc0
thf(fact_1696_powr__less__cancel__iff,axiom,
! [X: real,A3: real,B3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ X @ B3 ) )
= ( ord_less @ real @ A3 @ B3 ) ) ) ).
% powr_less_cancel_iff
thf(fact_1697_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zle
thf(fact_1698_arctan__less__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( arctan @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% arctan_less_zero_iff
thf(fact_1699_zero__less__arctan__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( arctan @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_less_arctan_iff
thf(fact_1700_zero__le__arctan__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arctan @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_le_arctan_iff
thf(fact_1701_arctan__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( arctan @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% arctan_le_zero_iff
thf(fact_1702_dbl__inc__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% dbl_inc_simps(4)
thf(fact_1703_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_1704_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_1705_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_1706_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_nonpos
thf(fact_1707_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_1708_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) )
= ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
& ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1709_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_1710_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1711_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1712_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1713_powr__eq__one__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ( powr @ real @ A3 @ X )
= ( one_one @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% powr_eq_one_iff
thf(fact_1714_powr__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( one_one @ real ) )
= X ) ) ).
% powr_one
thf(fact_1715_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( ( powr @ real @ X @ ( one_one @ real ) )
= X )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% powr_one_gt_zero_iff
thf(fact_1716_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zless
thf(fact_1717_powr__le__cancel__iff,axiom,
! [X: real,A3: real,B3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ X @ B3 ) )
= ( ord_less_eq @ real @ A3 @ B3 ) ) ) ).
% powr_le_cancel_iff
thf(fact_1718_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_1719_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_1720_powr__log__cancel,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ A3 @ ( log @ A3 @ X ) )
= X ) ) ) ) ).
% powr_log_cancel
thf(fact_1721_log__powr__cancel,axiom,
! [A3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log @ A3 @ ( powr @ real @ A3 @ Y ) )
= Y ) ) ) ).
% log_powr_cancel
thf(fact_1722_sgn__not__eq__imp,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: A,A3: A] :
( ( ( sgn_sgn @ A @ B3 )
!= ( sgn_sgn @ A @ A3 ) )
=> ( ( ( sgn_sgn @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( sgn_sgn @ A @ B3 )
!= ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ B3 ) ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_1723_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A6: A > B,X2: A] : ( uminus_uminus @ B @ ( A6 @ X2 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_1724_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( uminus_uminus @ A @ B3 ) )
= ( B3
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% equation_minus_iff
thf(fact_1725_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( ( uminus_uminus @ A @ A3 )
= B3 )
= ( ( uminus_uminus @ A @ B3 )
= A3 ) ) ) ).
% minus_equation_iff
thf(fact_1726_sgn__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% sgn_eq_0_iff
thf(fact_1727_sgn__0__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% sgn_0_0
thf(fact_1728_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_imp_neg_le
thf(fact_1729_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ A3 ) ) ) ).
% minus_le_iff
thf(fact_1730_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_minus_iff
thf(fact_1731_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_1732_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_1733_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_1734_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_less_swap1
thf(fact_1735_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_less_swap2
thf(fact_1736_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( ord_less @ A @ B3 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% less_minus_iff
thf(fact_1737_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ A3 ) ) ) ).
% minus_less_iff
thf(fact_1738_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1739_one__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ( ( one_one @ A )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% one_neq_neg_one
thf(fact_1740_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% is_num_normalize(8)
thf(fact_1741_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,K: A,A3: A] :
( ( A4
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( uminus_uminus @ A @ A4 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_1742_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1743_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B3: A,A3: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B3 ) @ A3 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ).
% minus_diff_commute
thf(fact_1744_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ ( suc @ K ) @ M )
= ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1745_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_1746_arctan__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% arctan_less_iff
thf(fact_1747_arctan__monotone,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone
thf(fact_1748_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_1749_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_1750_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_1751_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_1752_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1753_arctan__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% arctan_le_iff
thf(fact_1754_arctan__monotone_H,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone'
thf(fact_1755_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1756_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M @ N ) @ K )
= ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1757_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ M @ N ) @ K )
= ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1758_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
= ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1759_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_1760_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_1761_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X2: A] :
( if @ A
@ ( X2
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_1762_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_1763_powr__non__neg,axiom,
! [A3: real,X: real] :
~ ( ord_less @ real @ ( powr @ real @ A3 @ X ) @ ( zero_zero @ real ) ) ).
% powr_non_neg
thf(fact_1764_powr__less__mono2__neg,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( powr @ real @ Y @ A3 ) @ ( powr @ real @ X @ A3 ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_1765_powr__ge__pzero,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ Y ) ) ).
% powr_ge_pzero
thf(fact_1766_powr__mono2,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y @ A3 ) ) ) ) ) ).
% powr_mono2
thf(fact_1767_powr__less__cancel,axiom,
! [X: real,A3: real,B3: real] :
( ( ord_less @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ X @ B3 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ A3 @ B3 ) ) ) ).
% powr_less_cancel
thf(fact_1768_powr__less__mono,axiom,
! [A3: real,B3: real,X: real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ X @ B3 ) ) ) ) ).
% powr_less_mono
thf(fact_1769_powr__mono,axiom,
! [A3: real,B3: real,X: real] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ X @ B3 ) ) ) ) ).
% powr_mono
thf(fact_1770_nat__mult__distrib__neg,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z5 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z5 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_1771_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_1772_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_1773_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_1774_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( ( uminus_uminus @ A @ A3 )
= B3 )
= ( ( plus_plus @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1775_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( uminus_uminus @ A @ B3 ) )
= ( ( plus_plus @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1776_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A3 )
= B3 ) ) ) ).
% add.inverse_unique
thf(fact_1777_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1778_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( B3
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add_eq_0_iff
thf(fact_1779_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_1780_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_1781_nonzero__minus__divide__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_1782_nonzero__minus__divide__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_1783_gbinomial__of__nat__symmetric,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_1784_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B5: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1785_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B5: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1786_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,K: A,B3: A,A3: A] :
( ( B2
= ( plus_plus @ A @ K @ B3 ) )
=> ( ( minus_minus @ A @ A3 @ B2 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_1787_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B3 ) ) ) ) ).
% abs_leI
thf(fact_1788_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ).
% abs_le_D2
thf(fact_1789_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B3 )
= ( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ) ).
% abs_le_iff
thf(fact_1790_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_minus_self
thf(fact_1791_abs__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ B3 )
= ( ( ord_less @ A @ A3 @ B3 )
& ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ) ).
% abs_less_iff
thf(fact_1792_diff__eq,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( minus_minus @ A )
= ( ^ [X2: A,Y3: A] : ( inf_inf @ A @ X2 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ) ).
% diff_eq
thf(fact_1793_inf__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ A3 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ B3 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left1
thf(fact_1794_inf__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( inf_inf @ A @ X @ B3 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left2
thf(fact_1795_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1796_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_1797_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_1798_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1799_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times @ nat @ ( suc @ M ) @ N )
= ( plus_plus @ nat @ N @ ( times_times @ nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1800_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times @ nat @ M @ N ) )
=> ( ( N
= ( one_one @ nat ) )
| ( M
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1801_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less @ nat @ M @ ( times_times @ nat @ I @ N ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1802_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1803_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1804_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1805_gbinomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,A3: A] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_1806_powr__less__mono2,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y @ A3 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_1807_powr__mono2_H,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ Y @ A3 ) @ ( powr @ real @ X @ A3 ) ) ) ) ) ).
% powr_mono2'
thf(fact_1808_gr__one__powr,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less @ real @ ( one_one @ real ) @ ( powr @ real @ X @ Y ) ) ) ) ).
% gr_one_powr
thf(fact_1809_powr__inj,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ( powr @ real @ A3 @ X )
= ( powr @ real @ A3 @ Y ) )
= ( X = Y ) ) ) ) ).
% powr_inj
thf(fact_1810_powr__le1,axiom,
! [A3: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A3 ) @ ( one_one @ real ) ) ) ) ) ).
% powr_le1
thf(fact_1811_powr__mono__both,axiom,
! [A3: real,B3: real,X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y @ B3 ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_1812_ge__one__powr__ge__zero,axiom,
! [X: real,A3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( powr @ real @ X @ A3 ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_1813_powr__divide,axiom,
! [X: real,Y: real,A3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( divide_divide @ real @ X @ Y ) @ A3 )
= ( divide_divide @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y @ A3 ) ) ) ) ) ).
% powr_divide
thf(fact_1814_powr__mult,axiom,
! [X: real,Y: real,A3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( times_times @ real @ X @ Y ) @ A3 )
= ( times_times @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y @ A3 ) ) ) ) ) ).
% powr_mult
thf(fact_1815_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_1_pos
thf(fact_1816_sgn__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( sgn_sgn @ real @ ( root @ N @ X ) )
= ( sgn_sgn @ real @ X ) ) ) ).
% sgn_root
thf(fact_1817_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_1818_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_1819_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_1820_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_1821_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_1822_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( C3
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) ) )
= ( ( times_times @ A @ C3 @ B3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_1823_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= C3 )
= ( ( uminus_uminus @ A @ A3 )
= ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_1824_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) )
= A3 )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B3 )
= ( times_times @ A @ A3 @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_1825_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3
= ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C3 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_1826_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( ( divide_divide @ A @ A3 @ B3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( B3
!= ( zero_zero @ A ) )
& ( A3
= ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_1827_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A3 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_1828_abs__eq__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B3: A] :
( ( ( abs_abs @ A @ A3 )
= B3 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
& ( ( A3 = B3 )
| ( A3
= ( uminus_uminus @ A @ B3 ) ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_1829_eq__abs__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( abs_abs @ A @ B3 ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ( B3 = A3 )
| ( B3
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_1830_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_neg
thf(fact_1831_abs__if__raw,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% abs_if_raw
thf(fact_1832_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% abs_if
thf(fact_1833_inf__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% inf_shunt
thf(fact_1834_shunt1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y ) @ Z ) ) ) ) ).
% shunt1
thf(fact_1835_shunt2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ ( uminus_uminus @ A @ Y ) ) @ Z )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% shunt2
thf(fact_1836_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P6: A,Q5: A,R2: A] :
( ( ord_less_eq @ A @ P6 @ ( sup_sup @ A @ Q5 @ R2 ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P6 @ ( uminus_uminus @ A @ Q5 ) ) @ R2 ) ) ) ).
% sup_neg_inf
thf(fact_1837_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N @ ( times_times @ nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1838_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N @ ( times_times @ nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1839_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1840_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_1841_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) )
= ( ( N
= ( zero_zero @ nat ) )
& ( M
= ( zero_zero @ nat ) ) ) ) ).
% int_zle_neg
thf(fact_1842_div__less__iff__less__mult,axiom,
! [Q5: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q5 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q5 ) @ N )
= ( ord_less @ nat @ M @ ( times_times @ nat @ N @ Q5 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1843_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( zero_zero @ int ) ) ).
% negative_zle_0
thf(fact_1844_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_1845_zabs__def,axiom,
( ( abs_abs @ int )
= ( ^ [I4: int] : ( if @ int @ ( ord_less @ int @ I4 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_1846_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,M: nat,A3: A] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( times_times @ A @ ( gbinomial @ A @ A3 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_1847_powr__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( semiring_1_of_nat @ real @ N ) )
= ( power_power @ real @ X @ N ) ) ) ).
% powr_realpow
thf(fact_1848_powr__less__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ B3 @ Y ) @ X )
= ( ord_less @ real @ Y @ ( log @ B3 @ X ) ) ) ) ) ).
% powr_less_iff
thf(fact_1849_less__powr__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( powr @ real @ B3 @ Y ) )
= ( ord_less @ real @ ( log @ B3 @ X ) @ Y ) ) ) ) ).
% less_powr_iff
thf(fact_1850_log__less__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ B3 @ X ) @ Y )
= ( ord_less @ real @ X @ ( powr @ real @ B3 @ Y ) ) ) ) ) ).
% log_less_iff
thf(fact_1851_less__log__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ Y @ ( log @ B3 @ X ) )
= ( ord_less @ real @ ( powr @ real @ B3 @ Y ) @ X ) ) ) ) ).
% less_log_iff
thf(fact_1852_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_1853_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_1854_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_1855_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_1856_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_1857_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_1858_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z ) ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_1859_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B3 )
= B3 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B3 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B3 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1860_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z ) ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_1861_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B3 )
= ( divide_divide @ A @ ( minus_minus @ A @ A3 @ ( times_times @ A @ B3 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_1862_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B3 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B3 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_1863_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_1864_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1865_negative__zless__0,axiom,
! [N: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( zero_zero @ int ) ) ).
% negative_zless_0
thf(fact_1866_negD,axiom,
! [X: int] :
( ( ord_less @ int @ X @ ( zero_zero @ int ) )
=> ? [N3: nat] :
( X
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1867_div__nat__eqI,axiom,
! [N: nat,Q5: nat,M: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q5 ) @ M )
=> ( ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q5 ) ) )
=> ( ( divide_divide @ nat @ M @ N )
= Q5 ) ) ) ).
% div_nat_eqI
thf(fact_1868_less__eq__div__iff__mult__less__eq,axiom,
! [Q5: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q5 )
=> ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N @ Q5 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q5 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1869_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I4 ) @ J3 ) )
=> ( P @ I4 ) ) ) ) ) ) ).
% split_div
thf(fact_1870_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1871_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1872_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1873_verit__less__mono__div__int2,axiom,
! [A4: int,B2: int,N: int] :
( ( ord_less_eq @ int @ A4 @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B2 @ N ) @ ( divide_divide @ int @ A4 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1874_div__eq__minus1,axiom,
! [B3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B3 )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ).
% div_eq_minus1
thf(fact_1875_nat__mult__distrib,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z5 ) )
= ( times_times @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1876_powr__mult__base,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( times_times @ real @ X @ ( powr @ real @ X @ Y ) )
= ( powr @ real @ X @ ( plus_plus @ real @ ( one_one @ real ) @ Y ) ) ) ) ).
% powr_mult_base
thf(fact_1877_sgn__power__injE,axiom,
! [A3: real,N: nat,X: real,B3: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A3 ) @ ( power_power @ real @ ( abs_abs @ real @ A3 ) @ N ) )
= X )
=> ( ( X
= ( times_times @ real @ ( sgn_sgn @ real @ B3 ) @ ( power_power @ real @ ( abs_abs @ real @ B3 ) @ N ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A3 = B3 ) ) ) ) ).
% sgn_power_injE
thf(fact_1878_le__log__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ Y @ ( log @ B3 @ X ) )
= ( ord_less_eq @ real @ ( powr @ real @ B3 @ Y ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_1879_log__le__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ B3 @ X ) @ Y )
= ( ord_less_eq @ real @ X @ ( powr @ real @ B3 @ Y ) ) ) ) ) ).
% log_le_iff
thf(fact_1880_le__powr__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( powr @ real @ B3 @ Y ) )
= ( ord_less_eq @ real @ ( log @ B3 @ X ) @ Y ) ) ) ) ).
% le_powr_iff
thf(fact_1881_powr__le__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ B3 @ Y ) @ X )
= ( ord_less_eq @ real @ Y @ ( log @ B3 @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_1882_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_1883_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_1884_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_1885_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_1886_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_1887_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_1888_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_1889_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_1890_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
& ( P @ ( zero_zero @ nat ) ) )
| ? [Q6: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q6 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q6 ) ) )
& ( P @ Q6 ) ) ) ) ).
% split_div'
thf(fact_1891_ln__powr__bound,axiom,
! [X: real,A3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( divide_divide @ real @ ( powr @ real @ X @ A3 ) @ A3 ) ) ) ) ).
% ln_powr_bound
thf(fact_1892_ln__powr__bound2,axiom,
! [X: real,A3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X ) @ A3 ) @ ( times_times @ real @ ( powr @ real @ A3 @ A3 ) @ X ) ) ) ) ).
% ln_powr_bound2
thf(fact_1893_add__log__eq__powr,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( B3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ Y @ ( log @ B3 @ X ) )
= ( log @ B3 @ ( times_times @ real @ ( powr @ real @ B3 @ Y ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_1894_log__add__eq__powr,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( B3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ ( log @ B3 @ X ) @ Y )
= ( log @ B3 @ ( times_times @ real @ X @ ( powr @ real @ B3 @ Y ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_1895_minus__log__eq__powr,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( B3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( minus_minus @ real @ Y @ ( log @ B3 @ X ) )
= ( log @ B3 @ ( divide_divide @ real @ ( powr @ real @ B3 @ Y ) @ X ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_1896_root__sgn__power,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N ) ) )
= Y ) ) ).
% root_sgn_power
thf(fact_1897_sgn__power__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N @ X ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N @ X ) ) @ N ) )
= X ) ) ).
% sgn_power_root
thf(fact_1898_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1899_sgn__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sgn_sgn @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% sgn_le_0_iff
thf(fact_1900_zero__le__sgn__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sgn_sgn @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_le_sgn_iff
thf(fact_1901_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K
= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( zero_zero @ nat ) ) )
& ( ( K
!= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( divide_divide @ nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1902_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1903_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N @ K ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1904_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1905_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1906_Compl__subset__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1907_Compl__anti__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% Compl_anti_mono
thf(fact_1908_Compl__disjoint,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_1909_Compl__disjoint2,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_1910_sgn__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_zero
thf(fact_1911_Diff__Compl,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ).
% Diff_Compl
thf(fact_1912_Compl__Diff__eq,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ B2 ) ) ).
% Compl_Diff_eq
thf(fact_1913_subset__Compl__singleton,axiom,
! [A: $tType,A4: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B3 @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_1914_artanh__minus__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( artanh @ real @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( artanh @ real @ X ) ) ) ) ).
% artanh_minus_real
thf(fact_1915_subset__Compl__self__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_1916_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( times_times @ real @ U @ U ) ) @ ( times_times @ real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1917_Compl__Un,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) ) ) ).
% Compl_Un
thf(fact_1918_Compl__Int,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) ) ) ).
% Compl_Int
thf(fact_1919_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_1920_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1921_Compl__insert,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_1922_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( plus_plus @ real @ X @ Y ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ Y @ ( uminus_uminus @ real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_1923_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ Y ) )
= ( ord_less @ real @ ( uminus_uminus @ real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1924_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ Y ) )
= ( ord_less_eq @ real @ ( uminus_uminus @ real @ X ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1925_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ X @ Y ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ Y @ ( uminus_uminus @ real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_1926_abs__real__def,axiom,
( ( abs_abs @ real )
= ( ^ [A5: real] : ( if @ real @ ( ord_less @ real @ A5 @ ( zero_zero @ real ) ) @ ( uminus_uminus @ real @ A5 ) @ A5 ) ) ) ).
% abs_real_def
thf(fact_1927_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I4: int] :
( if @ int
@ ( I4
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I4 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_1928_sgn__real__def,axiom,
( ( sgn_sgn @ real )
= ( ^ [A5: real] :
( if @ real
@ ( A5
= ( zero_zero @ real ) )
@ ( zero_zero @ real )
@ ( if @ real @ ( ord_less @ real @ ( zero_zero @ real ) @ A5 ) @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ) ).
% sgn_real_def
thf(fact_1929_powr__neg__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ X ) ) ) ).
% powr_neg_one
thf(fact_1930_ln__add__one__self__le__self2,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self2
thf(fact_1931_ln__one__minus__pos__upper__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X ) ) @ ( uminus_uminus @ real @ X ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_1932_Bernoulli__inequality,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_1933_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1934_sgn__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( sgn_sgn @ A @ X )
= ( zero_zero @ A ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% sgn_zero_iff
thf(fact_1935_log__minus__eq__powr,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( B3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( minus_minus @ real @ ( log @ B3 @ X ) @ Y )
= ( log @ B3 @ ( times_times @ real @ X @ ( powr @ real @ B3 @ ( uminus_uminus @ real @ Y ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_1936_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1937_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1938_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1939_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( divide_divide @ nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1940_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1941_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1942_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1943_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1944_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1945_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1946_ceiling__log__eq__powr__iff,axiom,
! [X: real,B3: real,K: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ( archimedean_ceiling @ real @ ( log @ B3 @ X ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) ) )
= ( ( ord_less @ real @ ( powr @ real @ B3 @ ( semiring_1_of_nat @ real @ K ) ) @ X )
& ( ord_less_eq @ real @ X @ ( powr @ real @ B3 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_1947_powr__int,axiom,
! [X: real,I: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I ) )
= ( power_power @ real @ X @ ( nat2 @ I ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( nat2 @ ( uminus_uminus @ int @ I ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_1948_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_1949_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ ( uminus_uminus @ real @ X ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_1950_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ X ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_1951_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( gcd_Gcd @ A @ A4 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_1952_ComplI,axiom,
! [A: $tType,C3: A,A4: set @ A] :
( ~ ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% ComplI
thf(fact_1953_Compl__iff,axiom,
! [A: $tType,C3: A,A4: set @ A] :
( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( ~ ( member @ A @ C3 @ A4 ) ) ) ).
% Compl_iff
thf(fact_1954_Compl__eq__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A4 )
= ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( A4 = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_1955_exp__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) ) ) ).
% exp_less_mono
thf(fact_1956_exp__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% exp_less_cancel_iff
thf(fact_1957_exp__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% exp_le_cancel_iff
thf(fact_1958_dbl__dec__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_dec_simps(3)
thf(fact_1959_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( ring_1_of_int @ A @ Z )
= ( zero_zero @ A ) )
= ( Z
= ( zero_zero @ int ) ) ) ) ).
% of_int_eq_0_iff
thf(fact_1960_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( zero_zero @ A )
= ( ring_1_of_int @ A @ Z ) )
= ( Z
= ( zero_zero @ int ) ) ) ) ).
% of_int_0_eq_iff
thf(fact_1961_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_1962_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_1963_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ W2 @ Z ) ) ) ).
% of_int_le_iff
thf(fact_1964_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ W2 @ Z ) ) ) ).
% of_int_less_iff
thf(fact_1965_ceiling__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_1966_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_1967_one__less__exp__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% one_less_exp_iff
thf(fact_1968_exp__less__one__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% exp_less_one_iff
thf(fact_1969_exp__le__one__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% exp_le_one_iff
thf(fact_1970_one__le__exp__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( exp @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% one_le_exp_iff
thf(fact_1971_exp__ln__iff,axiom,
! [X: real] :
( ( ( exp @ real @ ( ln_ln @ real @ X ) )
= X )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% exp_ln_iff
thf(fact_1972_exp__ln,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( exp @ real @ ( ln_ln @ real @ X ) )
= X ) ) ).
% exp_ln
thf(fact_1973_of__int__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% of_int_0_le_iff
thf(fact_1974_of__int__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% of_int_le_0_iff
thf(fact_1975_of__int__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
= ( ord_less @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% of_int_less_0_iff
thf(fact_1976_of__int__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% of_int_0_less_iff
thf(fact_1977_of__int__1__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( one_one @ int ) @ Z ) ) ) ).
% of_int_1_le_iff
thf(fact_1978_of__int__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
= ( ord_less_eq @ int @ Z @ ( one_one @ int ) ) ) ) ).
% of_int_le_1_iff
thf(fact_1979_of__int__1__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z ) ) ) ).
% of_int_1_less_iff
thf(fact_1980_of__int__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
= ( ord_less @ int @ Z @ ( one_one @ int ) ) ) ) ).
% of_int_less_1_iff
thf(fact_1981_ceiling__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_le_zero
thf(fact_1982_zero__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_less_ceiling
thf(fact_1983_ceiling__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_less_one
thf(fact_1984_one__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% one_le_ceiling
thf(fact_1985_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_1986_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_less_ceiling
thf(fact_1987_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: int,W2: nat,X: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B3 @ W2 ) @ X ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_1988_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B3: int,W2: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W2 ) )
= ( ord_less_eq @ int @ X @ ( power_power @ int @ B3 @ W2 ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_1989_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: int,W2: nat,X: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( power_power @ int @ B3 @ W2 ) @ X ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_1990_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B3: int,W2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W2 ) )
= ( ord_less @ int @ X @ ( power_power @ int @ B3 @ W2 ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_1991_of__nat__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ A @ ( nat2 @ Z ) )
= ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_nat_nat
thf(fact_1992_nat__ceiling__le__eq,axiom,
! [X: real,A3: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A3 )
= ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A3 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_1993_ceiling__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_zero
thf(fact_1994_zero__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X ) ) ) ).
% zero_le_ceiling
thf(fact_1995_ComplD,axiom,
! [A: $tType,C3: A,A4: set @ A] :
( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
=> ~ ( member @ A @ C3 @ A4 ) ) ).
% ComplD
thf(fact_1996_double__complement,axiom,
! [A: $tType,A4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= A4 ) ).
% double_complement
thf(fact_1997_le__of__int__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% le_of_int_ceiling
thf(fact_1998_ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ A3 ) ) ) ).
% ceiling_le
thf(fact_1999_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
= ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% ceiling_le_iff
thf(fact_2000_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ int @ Z @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ).
% less_ceiling_iff
thf(fact_2001_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_le_of_int
thf(fact_2002_exp__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
=> ( ord_less @ real @ X @ Y ) ) ).
% exp_less_cancel
thf(fact_2003_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ).
% ex_of_int_less
thf(fact_2004_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_less_of_int
thf(fact_2005_exp__not__eq__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( exp @ A @ X )
!= ( zero_zero @ A ) ) ) ).
% exp_not_eq_zero
thf(fact_2006_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_2007_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_2008_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I4: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I4 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% ceiling_split
thf(fact_2009_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archimedean_ceiling @ A @ X )
= A3 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_2010_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) )
=> ( ( archimedean_ceiling @ A @ X )
= Z ) ) ) ) ).
% ceiling_unique
thf(fact_2011_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ) ).
% ceiling_correct
thf(fact_2012_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_2013_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ int @ Z @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X ) ) ) ).
% le_ceiling_iff
thf(fact_2014_exp__total,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ? [X3: real] :
( ( exp @ real @ X3 )
= Y ) ) ).
% exp_total
thf(fact_2015_exp__gt__zero,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( exp @ real @ X ) ) ).
% exp_gt_zero
thf(fact_2016_not__exp__less__zero,axiom,
! [X: real] :
~ ( ord_less @ real @ ( exp @ real @ X ) @ ( zero_zero @ real ) ) ).
% not_exp_less_zero
thf(fact_2017_not__exp__le__zero,axiom,
! [X: real] :
~ ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( zero_zero @ real ) ) ).
% not_exp_le_zero
thf(fact_2018_exp__ge__zero,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( exp @ real @ X ) ) ).
% exp_ge_zero
thf(fact_2019_ceiling__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% ceiling_mono
thf(fact_2020_ceiling__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% ceiling_less_cancel
thf(fact_2021_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q5: A,P6: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q5 )
=> ( ord_less_eq @ A @ P6 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P6 @ Q5 ) ) ) @ Q5 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_2022_Gcd__int__greater__eq__0,axiom,
! [K4: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Gcd @ int @ K4 ) ) ).
% Gcd_int_greater_eq_0
thf(fact_2023_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q5: A,P6: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q5 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P6 @ Q5 ) ) ) @ ( one_one @ A ) ) @ Q5 ) @ P6 ) ) ) ).
% ceiling_divide_lower
thf(fact_2024_ceiling__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int,X: A] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ N ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N ) @ ( one_one @ A ) ) )
=> ( ( archimedean_ceiling @ A @ X )
= ( plus_plus @ int @ N @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_eq
thf(fact_2025_exp__gt__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X ) ) ) ).
% exp_gt_one
thf(fact_2026_exp__ge__add__one__self,axiom,
! [X: real] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( exp @ real @ X ) ) ).
% exp_ge_add_one_self
thf(fact_2027_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_2028_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) ) ) ) ).
% ceiling_add_le
thf(fact_2029_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_2030_real__of__int__div4,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) ) ).
% real_of_int_div4
thf(fact_2031_exp__ge__add__one__self__aux,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( exp @ real @ X ) ) ) ).
% exp_ge_add_one_self_aux
thf(fact_2032_of__int__nonneg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_nonneg
thf(fact_2033_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_leD
thf(fact_2034_of__int__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_pos
thf(fact_2035_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_lessD
thf(fact_2036_lemma__exp__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ Y )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( minus_minus @ real @ Y @ ( one_one @ real ) ) )
& ( ( exp @ real @ X3 )
= Y ) ) ) ).
% lemma_exp_total
thf(fact_2037_ln__ge__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ Y @ ( ln_ln @ real @ X ) )
= ( ord_less_eq @ real @ ( exp @ real @ Y ) @ X ) ) ) ).
% ln_ge_iff
thf(fact_2038_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [X3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) ) )
& ! [Y5: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y5 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y5 @ ( one_one @ int ) ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% floor_exists1
thf(fact_2039_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_2040_ln__x__over__x__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( ln_ln @ real @ Y ) @ Y ) @ ( divide_divide @ real @ ( ln_ln @ real @ X ) @ X ) ) ) ) ).
% ln_x_over_x_mono
thf(fact_2041_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,X: int] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ X ) ) ) ).
% of_nat_less_of_int_iff
thf(fact_2042_int__le__real__less,axiom,
( ( ord_less_eq @ int )
= ( ^ [N2: int,M2: int] : ( ord_less @ real @ ( ring_1_of_int @ real @ N2 ) @ ( plus_plus @ real @ ( ring_1_of_int @ real @ M2 ) @ ( one_one @ real ) ) ) ) ) ).
% int_le_real_less
thf(fact_2043_int__less__real__le,axiom,
( ( ord_less @ int )
= ( ^ [N2: int,M2: int] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N2 ) @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ M2 ) ) ) ) ).
% int_less_real_le
thf(fact_2044_powr__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( powr @ A )
= ( ^ [X2: A,A5: A] :
( if @ A
@ ( X2
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( exp @ A @ ( times_times @ A @ A5 @ ( ln_ln @ A @ X2 ) ) ) ) ) ) ) ).
% powr_def
thf(fact_2045_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A3 @ B3 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A3 ) @ ( archimedean_ceiling @ A @ B3 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_2046_exp__divide__power__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X @ ( semiring_1_of_nat @ A @ N ) ) ) @ N )
= ( exp @ A @ X ) ) ) ) ).
% exp_divide_power_eq
thf(fact_2047_real__of__int__div2,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) ) ) ).
% real_of_int_div2
thf(fact_2048_real__of__int__div3,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) ) @ ( one_one @ real ) ) ).
% real_of_int_div3
thf(fact_2049_Gcd__remove0__nat,axiom,
! [M5: set @ nat] :
( ( finite_finite2 @ nat @ M5 )
=> ( ( gcd_Gcd @ nat @ M5 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M5 @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_2050_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_2051_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X2: A] : ( minus_minus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_2052_powr__real__of__int,axiom,
! [X: real,N: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N ) )
= ( power_power @ real @ X @ ( nat2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N ) )
= ( inverse_inverse @ real @ ( power_power @ real @ X @ ( nat2 @ ( uminus_uminus @ int @ N ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_2053_floor__log__eq__powr__iff,axiom,
! [X: real,B3: real,K: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ B3 @ X ) )
= K )
= ( ( ord_less_eq @ real @ ( powr @ real @ B3 @ ( ring_1_of_int @ real @ K ) ) @ X )
& ( ord_less @ real @ X @ ( powr @ real @ B3 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_2054_sinh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sinh @ A @ X )
= ( zero_zero @ A ) )
= ( member @ A @ ( exp @ A @ X ) @ ( insert2 @ A @ ( one_one @ A ) @ ( insert2 @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% sinh_zero_iff
thf(fact_2055_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B3: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A3 @ B3 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A3 ) @ ( archimedean_ceiling @ B @ B3 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_2056_power_Opower__eq__if,axiom,
! [A: $tType] :
( ( power2 @ A )
= ( ^ [One: A,Times: A > A > A,P5: A,M2: nat] :
( if @ A
@ ( M2
= ( zero_zero @ nat ) )
@ One
@ ( Times @ P5 @ ( power2 @ A @ One @ Times @ P5 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% power.power_eq_if
thf(fact_2057_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q5: A,P6: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q5 )
=> ( ord_less @ A @ P6 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P6 @ Q5 ) ) ) @ ( one_one @ A ) ) @ Q5 ) ) ) ) ).
% floor_divide_upper
thf(fact_2058_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_2059_sinh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( sinh @ real @ X ) @ ( sinh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% sinh_real_less_iff
thf(fact_2060_sinh__real__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( sinh @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% sinh_real_le_iff
thf(fact_2061_sinh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% sinh_real_pos_iff
thf(fact_2062_sinh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sinh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% sinh_real_neg_iff
thf(fact_2063_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_2064_inverse__nonzero__iff__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_2065_sinh__real__nonneg__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% sinh_real_nonneg_iff
thf(fact_2066_sinh__real__nonpos__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% sinh_real_nonpos_iff
thf(fact_2067_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_2068_inverse__nonpositive__iff__nonpositive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_2069_inverse__nonnegative__iff__nonnegative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_2070_inverse__less__iff__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% inverse_less_iff_less
thf(fact_2071_inverse__less__iff__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_2072_inverse__negative__iff__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% inverse_negative_iff_negative
thf(fact_2073_inverse__positive__iff__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% inverse_positive_iff_positive
thf(fact_2074_floor__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_2075_inverse__le__iff__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_2076_inverse__le__iff__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% inverse_le_iff_le
thf(fact_2077_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_2078_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( inverse_inverse @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_2079_zero__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_le_floor
thf(fact_2080_floor__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% floor_less_zero
thf(fact_2081_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% zero_less_floor
thf(fact_2082_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_2083_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_le_floor
thf(fact_2084_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_2085_nonzero__imp__inverse__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
!= ( zero_zero @ A ) ) ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_2086_nonzero__inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A3 ) )
= A3 ) ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_2087_nonzero__inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( inverse_inverse @ A @ B3 ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( A3 = B3 ) ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_2088_inverse__zero__imp__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ).
% inverse_zero_imp_zero
thf(fact_2089_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_2090_Ints__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_0
thf(fact_2091_floor__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ).
% floor_mono
thf(fact_2092_of__int__floor__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X ) ) ).
% of_int_floor_le
thf(fact_2093_inverse__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ B3 @ A3 ) ) ) ) ).
% inverse_less_imp_less
thf(fact_2094_less__imp__inverse__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% less_imp_inverse_less
thf(fact_2095_inverse__less__imp__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A3 ) ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_2096_less__imp__inverse__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_2097_inverse__negative__imp__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% inverse_negative_imp_negative
thf(fact_2098_inverse__positive__imp__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% inverse_positive_imp_positive
thf(fact_2099_negative__imp__inverse__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% negative_imp_inverse_negative
thf(fact_2100_positive__imp__inverse__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ).
% positive_imp_inverse_positive
thf(fact_2101_floor__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% floor_less_cancel
thf(fact_2102_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_2103_nonzero__inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_2104_nonzero__abs__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( inverse_inverse @ A @ A3 ) )
= ( inverse_inverse @ A @ ( abs_abs @ A @ A3 ) ) ) ) ) ).
% nonzero_abs_inverse
thf(fact_2105_floor__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% floor_le_ceiling
thf(fact_2106_Ints__double__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( ( plus_plus @ A @ A3 @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_2107_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B3: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A3 ) @ ( archim6421214686448440834_floor @ B @ B3 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A3 @ B3 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_2108_power_Opower_Opower__0,axiom,
! [A: $tType,One2: A,Times2: A > A > A,A3: A] :
( ( power2 @ A @ One2 @ Times2 @ A3 @ ( zero_zero @ nat ) )
= One2 ) ).
% power.power.power_0
thf(fact_2109_le__imp__inverse__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_2110_inverse__le__imp__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_2111_le__imp__inverse__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% le_imp_inverse_le
thf(fact_2112_inverse__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% inverse_le_imp_le
thf(fact_2113_inverse__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_le_1_iff
thf(fact_2114_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_less_inverse
thf(fact_2115_one__less__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_less_inverse_iff
thf(fact_2116_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_2117_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B3 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_2118_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ).
% inverse_add
thf(fact_2119_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ B3 @ A3 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_2120_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_2121_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ int @ Z @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ).
% le_floor_iff
thf(fact_2122_floor__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z )
= ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% floor_less_iff
thf(fact_2123_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% le_floor_add
thf(fact_2124_inverse__powr,axiom,
! [Y: real,A3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( inverse_inverse @ real @ Y ) @ A3 )
= ( inverse_inverse @ real @ ( powr @ real @ Y @ A3 ) ) ) ) ).
% inverse_powr
thf(fact_2125_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 )
!= ( zero_zero @ A ) ) ) ) ).
% Ints_odd_nonzero
thf(fact_2126_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_2127_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less @ A @ B3 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_2128_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_2129_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_2130_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_less_1_iff
thf(fact_2131_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_le_inverse
thf(fact_2132_inverse__diff__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ A3 @ B3 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_2133_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ X ) ) ) ).
% reals_Archimedean
thf(fact_2134_le__mult__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( nat2 @ ( archim6421214686448440834_floor @ A @ A3 ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ B3 ) ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% le_mult_nat_floor
thf(fact_2135_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= ( zero_zero @ nat ) ) ) ).
% nat_floor_neg
thf(fact_2136_floor__eq3,axiom,
! [N: nat,X: real] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N ) ) ) ).
% floor_eq3
thf(fact_2137_le__nat__floor,axiom,
! [X: nat,A3: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X ) @ A3 )
=> ( ord_less_eq @ nat @ X @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A3 ) ) ) ) ).
% le_nat_floor
thf(fact_2138_ceiling__diff__floor__le__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) @ ( one_one @ int ) ) ) ).
% ceiling_diff_floor_le_1
thf(fact_2139_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_2140_floor__eq,axiom,
! [N: int,X: real] :
( ( ord_less @ real @ ( ring_1_of_int @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X )
= N ) ) ) ).
% floor_eq
thf(fact_2141_real__of__int__floor__add__one__ge,axiom,
! [R2: real] : ( ord_less_eq @ real @ R2 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R2 ) ) @ ( one_one @ real ) ) ) ).
% real_of_int_floor_add_one_ge
thf(fact_2142_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_2143_real__of__int__floor__ge__diff__one,axiom,
! [R2: real] : ( ord_less_eq @ real @ ( minus_minus @ real @ R2 @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R2 ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_2144_forall__pos__mono__1,axiom,
! [P: real > $o,E2: real] :
( ! [D6: real,E: real] :
( ( ord_less @ real @ D6 @ E )
=> ( ( P @ D6 )
=> ( P @ E ) ) )
=> ( ! [N3: nat] : ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_2145_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
= ( ? [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) )
& ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_2146_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D6: real,E: real] :
( ( ord_less @ real @ D6 @ E )
=> ( ( P @ D6 )
=> ( P @ E ) ) )
=> ( ! [N3: nat] :
( ( N3
!= ( zero_zero @ nat ) )
=> ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_2147_ln__inverse,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( inverse_inverse @ real @ X ) )
= ( uminus_uminus @ real @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_inverse
thf(fact_2148_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_2149_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_2150_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_2151_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y @ ( ring_1_Ints @ A ) )
=> ( ( X = Y )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_2152_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X )
=> ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X )
= Z ) ) ) ) ).
% floor_unique
thf(fact_2153_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archim6421214686448440834_floor @ A @ X )
= A3 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_2154_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I4: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I4 ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% floor_split
thf(fact_2155_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A3 ) @ ( archim6421214686448440834_floor @ A @ B3 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_2156_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ int @ Z @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X ) ) ) ).
% less_floor_iff
thf(fact_2157_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_2158_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_2159_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N3 ) ) @ X ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_2160_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( power_power @ A @ X @ ( minus_minus @ nat @ N @ M ) )
= ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_2161_floor__eq4,axiom,
! [N: nat,X: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N ) ) ) ).
% floor_eq4
thf(fact_2162_floor__eq2,axiom,
! [N: int,X: real] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X )
= N ) ) ) ).
% floor_eq2
thf(fact_2163_floor__divide__real__eq__div,axiom,
! [B3: int,A3: real] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ A3 @ ( ring_1_of_int @ real @ B3 ) ) )
= ( divide_divide @ int @ ( archim6421214686448440834_floor @ real @ A3 ) @ B3 ) ) ) ).
% floor_divide_real_eq_div
thf(fact_2164_log__inverse,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ A3 @ ( inverse_inverse @ real @ X ) )
= ( uminus_uminus @ real @ ( log @ A3 @ X ) ) ) ) ) ) ).
% log_inverse
thf(fact_2165_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q5: A,P6: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q5 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P6 @ Q5 ) ) ) @ Q5 ) @ P6 ) ) ) ).
% floor_divide_lower
thf(fact_2166_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X8: nat > real] :
! [J3: nat] :
? [M8: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M8 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M8 @ N2 )
=> ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_2167_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_2168_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: A] :
( ( ( archimedean_frac @ A @ X )
= A3 )
= ( ( member @ A @ ( minus_minus @ A @ X @ A3 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_2169_frac__neg,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( zero_zero @ A ) ) )
& ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X ) ) ) ) ) ) ).
% frac_neg
thf(fact_2170_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_2171_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_2172_verit__le__mono__div__int,axiom,
! [A4: int,B2: int,N: int] :
( ( ord_less @ int @ A4 @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A4 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B2 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B2 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_2173_bits__mod__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_0
thf(fact_2174_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_2175_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% mod_by_0
thf(fact_2176_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_2177_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B3 @ A3 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_2178_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B3 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_2179_bits__mod__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_mod_by_1
thf(fact_2180_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_2181_mod__div__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% mod_div_trivial
thf(fact_2182_bits__mod__div__trivial,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,B3: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_div_trivial
thf(fact_2183_frac__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int] :
( ( archimedean_frac @ A @ ( ring_1_of_int @ A @ Z ) )
= ( zero_zero @ A ) ) ) ).
% frac_of_int
thf(fact_2184_frac__eq__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= ( zero_zero @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_eq_0_iff
thf(fact_2185_mod__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% mod_minus1_right
thf(fact_2186_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_2187_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_2188_frac__gt__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) )
= ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ) ).
% frac_gt_0_iff
thf(fact_2189_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ A3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2190_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ B3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2191_mod__eq__self__iff__div__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B3: A] :
( ( ( modulo_modulo @ A @ A3 @ B3 )
= A3 )
= ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_2192_zmod__le__nonneg__dividend,axiom,
! [M: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ M @ K ) @ M ) ) ).
% zmod_le_nonneg_dividend
thf(fact_2193_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_2194_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_2195_frac__ge__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) ) ) ).
% frac_ge_0
thf(fact_2196_frac__lt__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X ) @ ( one_one @ A ) ) ) ).
% frac_lt_1
thf(fact_2197_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( ( modulo_modulo @ A @ A3 @ B3 )
= A3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2198_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2199_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_2200_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_2201_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_2202_pos__mod__conj,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A3 @ B3 ) )
& ( ord_less @ int @ ( modulo_modulo @ int @ A3 @ B3 ) @ B3 ) ) ) ).
% pos_mod_conj
thf(fact_2203_neg__mod__conj,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A3 @ B3 ) @ ( zero_zero @ int ) )
& ( ord_less @ int @ B3 @ ( modulo_modulo @ int @ A3 @ B3 ) ) ) ) ).
% neg_mod_conj
thf(fact_2204_zdiv__mono__strict,axiom,
! [A4: int,B2: int,N: int] :
( ( ord_less @ int @ A4 @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( modulo_modulo @ int @ A4 @ N )
= ( zero_zero @ int ) )
=> ( ( ( modulo_modulo @ int @ B2 @ N )
= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( divide_divide @ int @ A4 @ N ) @ ( divide_divide @ int @ B2 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_2205_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_2206_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_2207_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_2208_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C3 ) ) @ ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2209_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= X )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_2210_int__mod__pos__eq,axiom,
! [A3: int,B3: int,Q5: int,R2: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q5 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B3 )
=> ( ( modulo_modulo @ int @ A3 @ B3 )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_2211_int__mod__neg__eq,axiom,
! [A3: int,B3: int,Q5: int,R2: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q5 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ R2 )
=> ( ( modulo_modulo @ int @ A3 @ B3 )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_2212_split__zmod,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( modulo_modulo @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ N ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_2213_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
= ( minus_minus @ int @ ( minus_minus @ int @ L @ ( one_one @ int ) ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) @ L ) ) ) ) ).
% minus_mod_int_eq
thf(fact_2214_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_2215_zmod__minus1,axiom,
! [B3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B3 )
= ( minus_minus @ int @ B3 @ ( one_one @ int ) ) ) ) ).
% zmod_minus1
thf(fact_2216_zmod__zmult2__eq,axiom,
! [C3: int,A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C3 )
=> ( ( modulo_modulo @ int @ A3 @ ( times_times @ int @ B3 @ C3 ) )
= ( plus_plus @ int @ ( times_times @ int @ B3 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A3 @ B3 ) @ C3 ) ) @ ( modulo_modulo @ int @ A3 @ B3 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_2217_fact__reduce,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% fact_reduce
thf(fact_2218_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_2219_verit__le__mono__div,axiom,
! [A4: nat,B2: nat,N: nat] :
( ( ord_less @ nat @ A4 @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A4 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B2 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B2 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_2220_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_2221_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_2222_norm__power__diff,axiom,
! [A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z: A,W2: A,M: nat] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ W2 ) @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( power_power @ A @ Z @ M ) @ ( power_power @ A @ W2 @ M ) ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Z @ W2 ) ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_2223_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( zero_zero @ A ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
& ( finite_finite2 @ A @ A4 ) ) ) ) ).
% Gcd_fin_0_iff
thf(fact_2224_numeral__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: num,N: num] :
( ( ( numeral_numeral @ A @ M )
= ( numeral_numeral @ A @ N ) )
= ( M = N ) ) ) ).
% numeral_eq_iff
thf(fact_2225_numeral__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ) ).
% numeral_le_iff
thf(fact_2226_numeral__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N: num] :
( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ M @ N ) ) ) ).
% numeral_less_iff
thf(fact_2227_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [V2: num,W2: num,Z: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Z ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W2 ) ) @ Z ) ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_2228_numeral__times__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% numeral_times_numeral
thf(fact_2229_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V2: num,W2: num,Z: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Z ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W2 ) ) @ Z ) ) ) ).
% add_numeral_left
thf(fact_2230_numeral__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% numeral_plus_numeral
thf(fact_2231_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_2232_neg__numeral__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( M = N ) ) ) ).
% neg_numeral_eq_iff
thf(fact_2233_of__nat__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: num] :
( ( semiring_1_of_nat @ A @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% of_nat_numeral
thf(fact_2234_abs__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
( ( abs_abs @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% abs_numeral
thf(fact_2235_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( modulo_modulo @ nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_2236_neg__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less_eq @ num @ N @ M ) ) ) ).
% neg_numeral_le_iff
thf(fact_2237_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V2: num,B3: A,C3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ B3 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C3 ) ) ) ) ).
% distrib_left_numeral
thf(fact_2238_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A3: A,B3: A,V2: num] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( numeral_numeral @ A @ V2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_2239_neg__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less @ num @ N @ M ) ) ) ).
% neg_numeral_less_iff
thf(fact_2240_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [V2: num,B3: A,C3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( minus_minus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C3 ) ) ) ) ).
% right_diff_distrib_numeral
thf(fact_2241_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A3: A,B3: A,V2: num] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B3 ) @ ( numeral_numeral @ A @ V2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_2242_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_2243_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_2244_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_2245_add__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_2246_diff__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_2247_diff__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% diff_numeral_simps(2)
thf(fact_2248_abs__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% abs_neg_numeral
thf(fact_2249_norm__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( zero_zero @ A ) )
= ( zero_zero @ real ) ) ) ).
% norm_zero
thf(fact_2250_norm__eq__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( real_V7770717601297561774m_norm @ A @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% norm_eq_zero
thf(fact_2251_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_2252_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_2253_numeral__less__real__of__nat__iff,axiom,
! [W2: num,N: nat] :
( ( ord_less @ real @ ( numeral_numeral @ real @ W2 ) @ ( semiring_1_of_nat @ real @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ W2 ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_2254_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W2: num] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( numeral_numeral @ real @ W2 ) )
= ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ W2 ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_2255_numeral__le__real__of__nat__iff,axiom,
! [N: num,M: nat] :
( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N ) @ ( semiring_1_of_nat @ real @ M ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_2256_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_2257_Gcd__fin_Oempty,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_fin.empty
thf(fact_2258_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% Gcd_fin.infinite
thf(fact_2259_Gcd__fin__eq__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( gcd_Gcd @ A @ A4 ) ) ) ) ).
% Gcd_fin_eq_Gcd
thf(fact_2260_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W2: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B3 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_2261_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_2262_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,W2: num] :
( ( A3
= ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) )
= B3 ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_2263_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,W2: num,A3: A] :
( ( ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) )
= A3 )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_2264_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W2: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B3 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_2265_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_2266_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num] :
( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_2267_zero__less__norm__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
= ( X
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_norm_iff
thf(fact_2268_norm__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( zero_zero @ real ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% norm_le_zero_iff
thf(fact_2269_of__int__le__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int,N: num] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ int @ Z @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_le_numeral_iff
thf(fact_2270_of__int__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z: int] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( numeral_numeral @ int @ N ) @ Z ) ) ) ).
% of_int_numeral_le_iff
thf(fact_2271_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int,N: num] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ int @ Z @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_less_numeral_iff
thf(fact_2272_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z: int] :
( ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( numeral_numeral @ int @ N ) @ Z ) ) ) ).
% of_int_numeral_less_iff
thf(fact_2273_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_2274_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V2 ) @ X ) ) ) ).
% numeral_le_floor
thf(fact_2275_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% floor_less_numeral
thf(fact_2276_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% ceiling_le_numeral
thf(fact_2277_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V2 ) @ X ) ) ) ).
% numeral_less_ceiling
thf(fact_2278_powr__numeral,axiom,
! [X: real,N: num] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% powr_numeral
thf(fact_2279_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B3 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_2280_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W2: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_2281_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,W2: num,A3: A] :
( ( ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= A3 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_2282_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,W2: num] :
( ( A3
= ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= B3 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_2283_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B3 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_2284_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W2: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_2285_dbl__dec__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_2286_dbl__inc__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_2287_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_2288_nat__less__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_2289_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_2290_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_2291_nat__le__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_2292_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_2293_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_2294_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_2295_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_2296_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_less_floor
thf(fact_2297_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_2298_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_2299_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_le_ceiling
thf(fact_2300_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X ) ) ) ).
% neg_numeral_le_floor
thf(fact_2301_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_2302_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_2303_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_2304_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_2305_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_2306_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_2307_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_2308_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_less_floor
thf(fact_2309_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_2310_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_2311_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_2312_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_2313_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A3 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_2314_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A3 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_2315_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_2316_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( semiring_char_0_fact @ nat @ N ) ) ).
% fact_ge_self
thf(fact_2317_fact__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ).
% fact_mono_nat
thf(fact_2318_fact__nonzero,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ N )
!= ( zero_zero @ A ) ) ) ).
% fact_nonzero
thf(fact_2319_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N ) @ M ) ).
% mod_less_eq_dividend
thf(fact_2320_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_2321_numeral__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( numeral_numeral @ A @ M )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_2322_neg__numeral__neq__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
!= ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_neq_numeral
thf(fact_2323_fact__less__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_2324_fact__ge__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% fact_ge_zero
thf(fact_2325_fact__gt__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% fact_gt_zero
thf(fact_2326_fact__not__neg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] :
~ ( ord_less @ A @ ( semiring_char_0_fact @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% fact_not_neg
thf(fact_2327_norm__not__less__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
~ ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( zero_zero @ real ) ) ) ).
% norm_not_less_zero
thf(fact_2328_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo @ nat @ M @ N ) )
= N )
=> ( ( modulo_modulo @ nat @ ( suc @ M ) @ N )
= ( zero_zero @ nat ) ) )
& ( ( ( suc @ ( modulo_modulo @ nat @ M @ N ) )
!= N )
=> ( ( modulo_modulo @ nat @ ( suc @ M ) @ N )
= ( suc @ ( modulo_modulo @ nat @ M @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_2329_norm__ge__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) ) ) ).
% norm_ge_zero
thf(fact_2330_fact__ge__1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% fact_ge_1
thf(fact_2331_mod__induct,axiom,
! [P: nat > $o,N: nat,P6: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less @ nat @ N @ P6 )
=> ( ( ord_less @ nat @ M @ P6 )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ P6 )
=> ( ( P @ N3 )
=> ( P @ ( modulo_modulo @ nat @ ( suc @ N3 ) @ P6 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_2332_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M4: nat] : ( P @ M4 @ ( zero_zero @ nat ) )
=> ( ! [M4: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo @ nat @ M4 @ N3 ) )
=> ( P @ M4 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_2333_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( modulo_modulo @ nat @ M @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_2334_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_2335_not__numeral__le__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_le_zero
thf(fact_2336_zero__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% zero_le_numeral
thf(fact_2337_zero__less__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% zero_less_numeral
thf(fact_2338_not__numeral__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_less_zero
thf(fact_2339_fact__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_mono
thf(fact_2340_one__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% one_le_numeral
thf(fact_2341_mod__eq__0D,axiom,
! [M: nat,D2: nat] :
( ( ( modulo_modulo @ nat @ M @ D2 )
= ( zero_zero @ nat ) )
=> ? [Q3: nat] :
( M
= ( times_times @ nat @ D2 @ Q3 ) ) ) ).
% mod_eq_0D
thf(fact_2342_not__numeral__less__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) ) ) ).
% not_numeral_less_one
thf(fact_2343_mod__geq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less @ nat @ M @ N )
=> ( ( modulo_modulo @ nat @ M @ N )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ).
% mod_geq
thf(fact_2344_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M2: nat,N2: nat] : ( if @ nat @ ( ord_less @ nat @ M2 @ N2 ) @ M2 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% mod_if
thf(fact_2345_not__numeral__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_2346_neg__numeral__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_le_numeral
thf(fact_2347_zero__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% zero_neq_neg_numeral
thf(fact_2348_le__mod__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( modulo_modulo @ nat @ M @ N )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_2349_not__numeral__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_2350_neg__numeral__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_less_numeral
thf(fact_2351_one__plus__numeral__commute,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% one_plus_numeral_commute
thf(fact_2352_numeral__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ N )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% numeral_neq_neg_one
thf(fact_2353_one__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( one_one @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% one_neq_neg_numeral
thf(fact_2354_fact__ge__Suc__0__nat,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( semiring_char_0_fact @ nat @ N ) ) ).
% fact_ge_Suc_0_nat
thf(fact_2355_nonzero__norm__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B3 ) ) ) ) ) ).
% nonzero_norm_divide
thf(fact_2356_power__eq__imp__eq__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W2: A,N: nat,Z: A] :
( ( ( power_power @ A @ W2 @ N )
= ( power_power @ A @ Z @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( real_V7770717601297561774m_norm @ A @ W2 )
= ( real_V7770717601297561774m_norm @ A @ Z ) ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_2357_norm__mult__less,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A,R2: real,Y: A,S3: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y ) ) @ ( times_times @ real @ R2 @ S3 ) ) ) ) ) ).
% norm_mult_less
thf(fact_2358_norm__mult__ineq,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ).
% norm_mult_ineq
thf(fact_2359_mod__le__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_2360_norm__add__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,R2: real,Y: A,S3: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ real @ R2 @ S3 ) ) ) ) ) ).
% norm_add_less
thf(fact_2361_norm__triangle__lt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E2: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ E2 ) ) ) ).
% norm_triangle_lt
thf(fact_2362_norm__triangle__mono,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,R2: real,B3: A,S3: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ R2 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B3 ) @ S3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A3 @ B3 ) ) @ ( plus_plus @ real @ R2 @ S3 ) ) ) ) ) ).
% norm_triangle_mono
thf(fact_2363_norm__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ).
% norm_triangle_ineq
thf(fact_2364_norm__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E2: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ E2 ) ) ) ).
% norm_triangle_le
thf(fact_2365_norm__add__leD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A,C3: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A3 @ B3 ) ) @ C3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B3 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ C3 ) ) ) ) ).
% norm_add_leD
thf(fact_2366_norm__power__ineq,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A,N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X @ N ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ N ) ) ) ).
% norm_power_ineq
thf(fact_2367_norm__diff__triangle__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E1: real,Z: A,E22: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) @ E1 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ Z ) ) @ E22 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_2368_norm__triangle__sub,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) ) ) ) ).
% norm_triangle_sub
thf(fact_2369_norm__triangle__ineq4,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ B3 ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B3 ) ) ) ) ).
% norm_triangle_ineq4
thf(fact_2370_norm__diff__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E1: real,Z: A,E22: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) @ E1 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ Z ) ) @ E22 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_2371_norm__triangle__le__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E2: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) @ E2 ) ) ) ).
% norm_triangle_le_diff
thf(fact_2372_not__zero__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_2373_neg__numeral__le__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_le_zero
thf(fact_2374_fact__less__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N )
=> ( ord_less @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ) ).
% fact_less_mono
thf(fact_2375_neg__numeral__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_less_zero
thf(fact_2376_not__zero__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_2377_norm__diff__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B3 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ).
% norm_diff_ineq
thf(fact_2378_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B3: A,C3: A] :
( ( ( numeral_numeral @ A @ W2 )
= ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 )
= B3 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_2379_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C3: A,W2: num] :
( ( ( divide_divide @ A @ B3 @ C3 )
= ( numeral_numeral @ A @ W2 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_2380_div__less__mono,axiom,
! [A4: nat,B2: nat,N: nat] :
( ( ord_less @ nat @ A4 @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A4 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B2 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A4 @ N ) @ ( divide_divide @ nat @ B2 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_2381_norm__triangle__ineq2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B3 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% norm_triangle_ineq2
thf(fact_2382_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_2383_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_2384_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_2385_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_2386_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_2387_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_2388_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_2389_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_2390_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_2391_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_2392_mod__eq__nat1E,axiom,
! [M: nat,Q5: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q5 )
= ( modulo_modulo @ nat @ N @ Q5 ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ~ ! [S4: nat] :
( M
!= ( plus_plus @ nat @ N @ ( times_times @ nat @ Q5 @ S4 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_2393_mod__eq__nat2E,axiom,
! [M: nat,Q5: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q5 )
= ( modulo_modulo @ nat @ N @ Q5 ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ~ ! [S4: nat] :
( N
!= ( plus_plus @ nat @ M @ ( times_times @ nat @ Q5 @ S4 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_2394_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N )
= ( modulo_modulo @ nat @ Y @ N ) )
=> ( ( ord_less_eq @ nat @ Y @ X )
=> ? [Q3: nat] :
( X
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q3 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_2395_fact__mod,axiom,
! [A: $tType] :
( ( ( linordered_semidom @ A )
& ( semidom_modulo @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M ) )
= ( zero_zero @ A ) ) ) ) ).
% fact_mod
thf(fact_2396_nonzero__norm__inverse,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ A3 ) )
= ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_2397_fact__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ N @ N ) ) ) ) ).
% fact_le_power
thf(fact_2398_norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ X ) ) @ ( exp @ real @ ( real_V7770717601297561774m_norm @ A @ X ) ) ) ) ).
% norm_exp
thf(fact_2399_powr__neg__numeral,axiom,
! [X: real,N: num] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ N ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_2400_fact__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) )
= ( times_times @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_2401_fact__div__fact__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq @ nat @ R2 @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ R2 ) ) ) @ ( power_power @ nat @ N @ R2 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_2402_power__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W2: A,N: nat] :
( ( ( power_power @ A @ W2 @ N )
= ( one_one @ A ) )
=> ( ( ( real_V7770717601297561774m_norm @ A @ W2 )
= ( one_one @ real ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% power_eq_1_iff
thf(fact_2403_norm__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( plus_plus @ A @ C3 @ D2 ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ C3 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_2404_norm__sgn,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( X
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X ) )
= ( zero_zero @ real ) ) )
& ( ( X
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X ) )
= ( one_one @ real ) ) ) ) ) ).
% norm_sgn
thf(fact_2405_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C3 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_2406_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C3: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_2407_split__mod,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( modulo_modulo @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ M ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I4 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_2408_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C3: A,W2: num] :
( ( ( divide_divide @ A @ B3 @ C3 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_2409_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B3: A,C3: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 )
= B3 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_2410_norm__triangle__ineq3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B3 ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% norm_triangle_ineq3
thf(fact_2411_nat__mod__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( modulo_modulo @ int @ X @ Y ) )
= ( modulo_modulo @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_2412_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X4: A > B] :
( ( ? [K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
& ! [N2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X4 @ N2 ) ) @ K5 ) ) )
= ( ? [N5: nat] :
! [N2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X4 @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_2413_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X4: A > B] :
( ( ? [K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
& ! [N2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X4 @ N2 ) ) @ K5 ) ) )
= ( ? [N5: nat] :
! [N2: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X4 @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_2414_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C3: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_2415_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_2416_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M @ N ) ) @ M )
= ( one_one @ nat ) ) ) ).
% Suc_times_mod_eq
thf(fact_2417_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_2418_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_2419_norm__inverse__le__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [R2: real,X: A] :
( ( ord_less_eq @ real @ R2 @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ X ) ) @ ( inverse_inverse @ real @ R2 ) ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_2420_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X4: nat > A,E2: real] :
( ( topolo3814608138187158403Cauchy @ A @ X4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M9: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M9 @ M3 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ M9 @ N4 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X4 @ M3 ) @ ( X4 @ N4 ) ) ) @ E2 ) ) ) ) ) ) ).
% CauchyD
thf(fact_2421_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X4: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M10 @ M4 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M10 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X4 @ M4 ) @ ( X4 @ N3 ) ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X4 ) ) ) ).
% CauchyI
thf(fact_2422_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M8: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M8 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M8 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_2423_assms_I2_J,axiom,
ord_less @ nat @ x @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ n ) ).
% assms(2)
thf(fact_2424_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_2425_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_2426_lemma__termdiff3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [H: A,Z: A,K4: real,N: nat] :
( ( H
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K4 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z @ H ) ) @ K4 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H ) @ N ) @ ( power_power @ A @ Z @ N ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) @ ( times_times @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( power_power @ real @ K4 @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_2427_bounded__linear__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( real_V4916620083959148203axioms @ A @ B )
= ( ^ [F2: A > B] :
? [K5: real] :
! [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ K5 ) ) ) ) ) ).
% bounded_linear_axioms_def
thf(fact_2428_bounded__linear__axioms_Ointro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ? [K6: real] :
! [X3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X3 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K6 ) )
=> ( real_V4916620083959148203axioms @ A @ B @ F3 ) ) ) ).
% bounded_linear_axioms.intro
thf(fact_2429_i0__less,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( N
!= ( zero_zero @ extended_enat ) ) ) ).
% i0_less
thf(fact_2430_pow__sum,axiom,
! [A3: nat,B3: nat] :
( ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ A3 @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 ) ) ).
% pow_sum
thf(fact_2431_member__bound,axiom,
! [Tree: vEBT_VEBT,X: nat,N: nat] :
( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_invar_vebt @ Tree @ N )
=> ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% member_bound
thf(fact_2432_valid__pres__insert,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ T2 @ X ) @ N ) ) ) ).
% valid_pres_insert
thf(fact_2433_valid__insert__both__member__options__pres,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ Y ) @ X ) ) ) ) ) ).
% valid_insert_both_member_options_pres
thf(fact_2434_valid__insert__both__member__options__add,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ X ) @ X ) ) ) ).
% valid_insert_both_member_options_add
thf(fact_2435_post__member__pre__member,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T2 @ X ) @ Y )
=> ( ( vEBT_vebt_member @ T2 @ Y )
| ( X = Y ) ) ) ) ) ) ).
% post_member_pre_member
thf(fact_2436_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_2437_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_2438_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less @ num @ M @ one2 ) ).
% semiring_norm(75)
thf(fact_2439_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% semiring_norm(68)
thf(fact_2440_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H2: nat,L2: nat,D5: nat] : ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ D5 ) ) @ L2 ) ) ) ).
% bit_concat_def
thf(fact_2441_numeral__eq__one__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( ( numeral_numeral @ A @ N )
= ( one_one @ A ) )
= ( N = one2 ) ) ) ).
% numeral_eq_one_iff
thf(fact_2442_one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ N ) )
= ( one2 = N ) ) ) ).
% one_eq_numeral_iff
thf(fact_2443_num__double,axiom,
! [N: num] :
( ( times_times @ num @ ( bit0 @ one2 ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_2444_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_2445_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% semiring_norm(69)
thf(fact_2446_neg__one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( ( uminus_uminus @ A @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( N = one2 ) ) ) ).
% neg_one_eq_numeral_iff
thf(fact_2447_numeral__eq__neg__one__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( N = one2 ) ) ) ).
% numeral_eq_neg_one_iff
thf(fact_2448_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% Suc_numeral
thf(fact_2449_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_2450_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_2451_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_2452_zero__eq__power2,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A] :
( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_power2
thf(fact_2453_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_2454_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_2455_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_2456_numeral__plus__one,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ N @ one2 ) ) ) ) ).
% numeral_plus_one
thf(fact_2457_one__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% one_plus_numeral
thf(fact_2458_numeral__le__one__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% numeral_le_one_iff
thf(fact_2459_one__less__numeral__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ one2 @ N ) ) ) ).
% one_less_numeral_iff
thf(fact_2460_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_2461_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_2462_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( X = Y ) ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_2463_power2__less__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% power2_less_eq_zero_iff
thf(fact_2464_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_2465_zero__less__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_power2
thf(fact_2466_diff__numeral__special_I10_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_2467_diff__numeral__special_I11_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% diff_numeral_special(11)
thf(fact_2468_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_2469_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_2470_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_2471_not__mod2__eq__Suc__0__eq__0,axiom,
! [N: nat] :
( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
!= ( suc @ ( zero_zero @ nat ) ) )
= ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ nat ) ) ) ).
% not_mod2_eq_Suc_0_eq_0
thf(fact_2472_diff__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% diff_numeral_special(3)
thf(fact_2473_diff__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ one2 ) ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_2474_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_2475_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_2476_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_2477_one__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% one_less_floor
thf(fact_2478_floor__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% floor_le_one
thf(fact_2479_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_2480_add__diff__assoc__enat,axiom,
! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
( ( ord_less_eq @ extended_enat @ Z @ Y )
=> ( ( plus_plus @ extended_enat @ X @ ( minus_minus @ extended_enat @ Y @ Z ) )
= ( minus_minus @ extended_enat @ ( plus_plus @ extended_enat @ X @ Y ) @ Z ) ) ) ).
% add_diff_assoc_enat
thf(fact_2481_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_less_eq @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= ( N
= ( zero_zero @ extended_enat ) ) ) ).
% ile0_eq
thf(fact_2482_i0__lb,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% i0_lb
thf(fact_2483_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq @ num @ X @ one2 )
= ( X = one2 ) ) ).
% le_num_One_iff
thf(fact_2484_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N3: extended_enat] :
( ! [M3: extended_enat] :
( ( ord_less @ extended_enat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_2485_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_less @ extended_enat @ N @ ( zero_zero @ extended_enat ) ) ).
% not_iless0
thf(fact_2486_add__One__commute,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ N )
= ( plus_plus @ num @ N @ one2 ) ) ).
% add_One_commute
thf(fact_2487_enat__0__less__mult__iff,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ ( times_times @ extended_enat @ M @ N ) )
= ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ M )
& ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ) ) ).
% enat_0_less_mult_iff
thf(fact_2488_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_2489_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_2490_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_2491_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_2492_less__exp,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% less_exp
thf(fact_2493_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_2494_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_2495_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_2496_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_2497_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_2498_power2__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% power2_le_imp_le
thf(fact_2499_power2__eq__imp__eq,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( X = Y ) ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_2500_zero__le__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% zero_le_power2
thf(fact_2501_power2__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ).
% power2_less_0
thf(fact_2502_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ B3 ) ) ) ).
% left_add_twice
thf(fact_2503_mult__2__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z: A] :
( ( times_times @ A @ Z @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ Z @ Z ) ) ) ).
% mult_2_right
thf(fact_2504_mult__2,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z )
= ( plus_plus @ A @ Z @ Z ) ) ) ).
% mult_2
thf(fact_2505_abs__le__square__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( abs_abs @ A @ Y ) )
= ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_2506_less__2__cases,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( N
= ( zero_zero @ nat ) )
| ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases
thf(fact_2507_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases_iff
thf(fact_2508_card__2__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X2: A,Y3: A] :
( ( S
= ( insert2 @ A @ X2 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X2 != Y3 ) ) ) ) ).
% card_2_iff
thf(fact_2509_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_2510_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_2511_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ).
% two_realpow_ge_one
thf(fact_2512_square__fact__le__2__fact,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_char_0_fact @ real @ N ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% square_fact_le_2_fact
thf(fact_2513_realpow__square__minus__le,axiom,
! [U: real,X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( power_power @ real @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% realpow_square_minus_le
thf(fact_2514_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ ( minus_minus @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_2515_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_2516_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ int ) ) ).
% not_exp_less_eq_0_int
thf(fact_2517_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ X @ Y ) ) ) ) ).
% power2_less_imp_less
thf(fact_2518_half__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% half_gt_zero
thf(fact_2519_half__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% half_gt_zero_iff
thf(fact_2520_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_2521_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_2522_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_2523_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_2524_field__less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ X @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% field_less_half_sum
thf(fact_2525_square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% square_le_1
thf(fact_2526_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ Y ) ) ) ) ).
% power2_le_iff_abs_le
thf(fact_2527_of__nat__less__two__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat] : ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% of_nat_less_two_power
thf(fact_2528_exp__add__not__zero__imp__right,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_right
thf(fact_2529_exp__add__not__zero__imp__left,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_left
thf(fact_2530_zero__le__even__power_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% zero_le_even_power'
thf(fact_2531_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,M: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) )
!= ( zero_zero @ A ) ) ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_2532_abs__square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_le_1
thf(fact_2533_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_2534_nat__bit__induct,axiom,
! [P: nat > $o,N: 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 @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_2535_div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_2536_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_2537_L2__set__mult__ineq__lemma,axiom,
! [A3: real,C3: real,B3: real,D2: real] : ( ord_less_eq @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( times_times @ real @ A3 @ C3 ) ) @ ( times_times @ real @ B3 @ D2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( power_power @ real @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ real @ ( power_power @ real @ B3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ C3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_2538_numeral__Bit0,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit0 @ N ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_Bit0
thf(fact_2539_exp__half__le2,axiom,
ord_less_eq @ real @ ( exp @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% exp_half_le2
thf(fact_2540_exp__plus__inverse__exp,axiom,
! [X: real] : ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) ) ).
% exp_plus_inverse_exp
thf(fact_2541_mult__numeral__1,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A3 )
= A3 ) ) ).
% mult_numeral_1
thf(fact_2542_mult__numeral__1__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
= A3 ) ) ).
% mult_numeral_1_right
thf(fact_2543_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_2544_divide__numeral__1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
= A3 ) ) ).
% divide_numeral_1
thf(fact_2545_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_2546_Suc__nat__number__of__add,axiom,
! [V2: num,N: nat] :
( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V2 @ one2 ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_2547_inverse__numeral__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( numeral_numeral @ A @ one2 ) )
= ( numeral_numeral @ A @ one2 ) ) ) ).
% inverse_numeral_1
thf(fact_2548_sum__squares__bound,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_squares_bound
thf(fact_2549_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ B3 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_2550_bits__stable__imp__add__self,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) ) ) ) ).
% bits_stable_imp_add_self
thf(fact_2551_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_2552_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_2553_ex__power__ivl1,axiom,
! [B3: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ? [N3: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ N3 ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_2554_ex__power__ivl2,axiom,
! [B3: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B3 @ N3 ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_2555_plus__inverse__ge__2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ X @ ( inverse_inverse @ real @ X ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_2556_exp__bound__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ Z ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% exp_bound_half
thf(fact_2557_less__log2__of__power,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% less_log2_of_power
thf(fact_2558_le__log2__of__power,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% le_log2_of_power
thf(fact_2559_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ B3 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_2560_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_2561_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q5: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q5 ) ) )
= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q5 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(2)
thf(fact_2562_num_Osize_I5_J,axiom,
! [X23: num] :
( ( size_size @ num @ ( bit0 @ X23 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X23 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(5)
thf(fact_2563_log2__of__power__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_less
thf(fact_2564_exp__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% exp_bound
thf(fact_2565_pos__zdiv__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ B3 @ A3 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_2566_neg__zdiv__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ ( plus_plus @ int @ B3 @ ( one_one @ int ) ) @ A3 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_2567_pos__zmod__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B3 @ A3 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_2568_real__le__x__sinh,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_x_sinh
thf(fact_2569_mult__1s__ring__1_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) ) ).
% mult_1s_ring_1(1)
thf(fact_2570_mult__1s__ring__1_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B3: A] :
( ( times_times @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) )
= ( uminus_uminus @ A @ B3 ) ) ) ).
% mult_1s_ring_1(2)
thf(fact_2571_uminus__numeral__One,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% uminus_numeral_One
thf(fact_2572_real__le__abs__sinh,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( abs_abs @ real @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_abs_sinh
thf(fact_2573_cong__exp__iff__simps_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ one2 ) )
= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(1)
thf(fact_2574_arith__geo__mean,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,X: A,Y: A] :
( ( ( power_power @ A @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ X @ Y ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ U @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_2575_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ A @ X @ M ) )
| ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M ) @ M ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2576_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_2577_log2__of__power__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less_eq @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_le
thf(fact_2578_exp__bound__lemma,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ Z ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( real_V7770717601297561774m_norm @ A @ Z ) ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_2579_real__exp__bound__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) ) ) ) ) ).
% real_exp_bound_lemma
thf(fact_2580_exp__lower__Taylor__quadratic,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( divide_divide @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ X ) ) ) ).
% exp_lower_Taylor_quadratic
thf(fact_2581_ln__one__plus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ X @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) ) ) ) ).
% ln_one_plus_pos_lower_bound
thf(fact_2582_neg__zmod__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B3 @ ( one_one @ int ) ) @ A3 ) ) @ ( one_one @ int ) ) ) ) ).
% neg_zmod_mult_2
thf(fact_2583_floor__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% floor_log2_div2
thf(fact_2584_sinh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( sinh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( minus_minus @ real @ X @ ( inverse_inverse @ real @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% sinh_ln_real
thf(fact_2585_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_2586_arctan__double,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ X ) )
= ( arctan @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arctan_double
thf(fact_2587_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_2588_ln__one__minus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( uminus_uminus @ real @ X ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X ) ) ) ) ) ).
% ln_one_minus_pos_lower_bound
thf(fact_2589_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound
thf(fact_2590_floor__log__nat__eq__if,axiom,
! [B3: nat,N: nat,K: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ N ) @ K )
=> ( ( ord_less @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B3 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_2591_floor__log__nat__eq__powr__iff,axiom,
! [B3: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B3 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( semiring_1_of_nat @ int @ N ) )
= ( ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ N ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_2592_ceiling__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% ceiling_log2_div2
thf(fact_2593_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_2594_ceiling__log__nat__eq__if,axiom,
! [B3: nat,N: nat,K: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B3 @ N ) @ K )
=> ( ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B3 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_2595_ceiling__log__nat__eq__powr__iff,axiom,
! [B3: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B3 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) )
= ( ( ord_less @ nat @ ( power_power @ nat @ B3 @ N ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_2596_inrange,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ).
% inrange
thf(fact_2597_abs__sqrt__wlog,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A > A > $o,X: A] :
( ! [X3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( P @ X3 @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ ( abs_abs @ A @ X ) @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_2598_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_2599_low__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X ) @ N )
= X ) ) ).
% low_inv
thf(fact_2600_set__n__deg__not__0,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,M: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ N ) ) ) ).
% set_n_deg_not_0
thf(fact_2601_high__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X ) @ N )
= Y ) ) ).
% high_inv
thf(fact_2602_bit__split__inv,axiom,
! [X: nat,D2: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D2 ) @ ( vEBT_VEBT_low @ X @ D2 ) @ D2 )
= X ) ).
% bit_split_inv
thf(fact_2603_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X2: nat,N2: nat] : ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% high_def
thf(fact_2604_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X2: nat,N2: nat] : ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% low_def
thf(fact_2605_high__bound__aux,axiom,
! [Ma: nat,N: nat,M: nat] :
( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% high_bound_aux
thf(fact_2606_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,H: A,L3: A,H3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ L @ H )
= ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H = H3 ) )
| ( ~ ( ord_less_eq @ A @ L @ H )
& ~ ( ord_less_eq @ A @ L3 @ H3 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_2607_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_2608_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_2609_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_2610_set__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% set_bit_negative_int_iff
thf(fact_2611_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_2612_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_2613_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_2614_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B3 )
| ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_2615_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_2616_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% infinite_Icc_iff
thf(fact_2617_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( insert2 @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B3 )
& ( B3 = C3 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_2618_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
( ( set_or1337092689740270186AtMost @ A @ A3 @ A3 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastAtMost_singleton
thf(fact_2619_finite__list,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [Xs2: list @ A] :
( ( set2 @ A @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_2620_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B2 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_2621_set__bit__greater__eq,axiom,
! [K: int,N: nat] : ( ord_less_eq @ int @ K @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) ) ).
% set_bit_greater_eq
thf(fact_2622_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) ) ) ) ).
% infinite_Icc
thf(fact_2623_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_2624_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( A3 = B3 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_2625_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less
thf(fact_2626_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less
thf(fact_2627_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D2 ) )
= ( ( ~ ( ord_less_eq @ A @ A3 @ B3 )
| ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 )
& ( ( ord_less @ A @ C3 @ A3 )
| ( ord_less @ A @ B3 @ D2 ) ) ) )
& ( ord_less_eq @ A @ C3 @ D2 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_2628_length__pos__if__in__set,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_2629_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_2630_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert2 @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_2631_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( insert2 @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_2632_atLeastAtMostSuc__conv,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) )
= ( insert2 @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_2633_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ N )
= ( insert2 @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_2634_subset__eq__atLeast0__atMost__finite,axiom,
! [N6: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N6 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_2635_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_2636_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_2637_complex__mod__minus__le__complex__mod,axiom,
! [X: complex] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ X ) ) @ ( real_V7770717601297561774m_norm @ complex @ X ) ) ).
% complex_mod_minus_le_complex_mod
thf(fact_2638_complex__mod__triangle__ineq2,axiom,
! [B3: complex,A3: complex] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ ( plus_plus @ complex @ B3 @ A3 ) ) @ ( real_V7770717601297561774m_norm @ complex @ B3 ) ) @ ( real_V7770717601297561774m_norm @ complex @ A3 ) ) ).
% complex_mod_triangle_ineq2
thf(fact_2639_set__encode__insert,axiom,
! [A4: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ A4 )
=> ( ~ ( member @ nat @ N @ A4 )
=> ( ( nat_set_encode @ ( insert2 @ nat @ N @ A4 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).
% set_encode_insert
thf(fact_2640_unset__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% unset_bit_0
thf(fact_2641_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N2: nat,A5: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_2642_set__union,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( set2 @ A @ ( union @ A @ Xs @ Ys2 ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) ) ) ).
% set_union
thf(fact_2643_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y ) )
=> ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X )
= Y ) ) ) ) ).
% round_unique
thf(fact_2644_dbl__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% dbl_simps(4)
thf(fact_2645_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_2646_unset__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% unset_bit_negative_int_iff
thf(fact_2647_signed__take__bit__of__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% signed_take_bit_of_0
thf(fact_2648_dbl__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% dbl_simps(2)
thf(fact_2649_round__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_2650_dbl__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% dbl_simps(5)
thf(fact_2651_dbl__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_simps(1)
thf(fact_2652_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_2653_dbl__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% dbl_simps(3)
thf(fact_2654_signed__take__bit__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_2655_unset__bit__less__eq,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) @ K ) ).
% unset_bit_less_eq
thf(fact_2656_set__encode__eq,axiom,
! [A4: set @ nat,B2: set @ nat] :
( ( finite_finite2 @ nat @ A4 )
=> ( ( finite_finite2 @ nat @ B2 )
=> ( ( ( nat_set_encode @ A4 )
= ( nat_set_encode @ B2 ) )
= ( A4 = B2 ) ) ) ) ).
% set_encode_eq
thf(fact_2657_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X2: A] : ( plus_plus @ A @ X2 @ X2 ) ) ) ) ).
% dbl_def
thf(fact_2658_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
=> ( ( set_or1337092689740270186AtMost @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
= ( insert2 @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N ) @ ( set_or1337092689740270186AtMost @ int @ M @ N ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_2659_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I4: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I4 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert2 @ int @ I4 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_2660_round__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y ) ) ) ) ).
% round_mono
thf(fact_2661_floor__le__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_round @ A @ X ) ) ) ).
% floor_le_round
thf(fact_2662_ceiling__ge__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% ceiling_ge_round
thf(fact_2663_set__encode__inf,axiom,
! [A4: set @ nat] :
( ~ ( finite_finite2 @ nat @ A4 )
=> ( ( nat_set_encode @ A4 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_2664_periodic__finite__ex,axiom,
! [D2: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
& ( P @ X2 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_2665_aset_I7_J,axiom,
! [D3: int,A4: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X5 )
=> ( ord_less @ int @ T2 @ ( plus_plus @ int @ X5 @ D3 ) ) ) ) ) ).
% aset(7)
thf(fact_2666_aset_I5_J,axiom,
! [D3: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ T2 @ A4 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less @ int @ X5 @ T2 )
=> ( ord_less @ int @ ( plus_plus @ int @ X5 @ D3 ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_2667_aset_I4_J,axiom,
! [D3: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ T2 @ A4 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( X5 != T2 )
=> ( ( plus_plus @ int @ X5 @ D3 )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_2668_aset_I3_J,axiom,
! [D3: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A4 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( X5 = T2 )
=> ( ( plus_plus @ int @ X5 @ D3 )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_2669_bset_I7_J,axiom,
! [D3: int,T2: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ T2 @ B2 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B2 )
=> ( X5
!= ( plus_plus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X5 )
=> ( ord_less @ int @ T2 @ ( minus_minus @ int @ X5 @ D3 ) ) ) ) ) ) ).
% bset(7)
thf(fact_2670_bset_I5_J,axiom,
! [D3: int,B2: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B2 )
=> ( X5
!= ( plus_plus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less @ int @ X5 @ T2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X5 @ D3 ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_2671_bset_I4_J,axiom,
! [D3: int,T2: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ T2 @ B2 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B2 )
=> ( X5
!= ( plus_plus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( X5 != T2 )
=> ( ( minus_minus @ int @ X5 @ D3 )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_2672_bset_I3_J,axiom,
! [D3: int,T2: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B2 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B2 )
=> ( X5
!= ( plus_plus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( X5 = T2 )
=> ( ( minus_minus @ int @ X5 @ D3 )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_2673_signed__take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% signed_take_bit_int_less_exp
thf(fact_2674_round__diff__minimal,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: A,M: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z @ ( ring_1_of_int @ A @ M ) ) ) ) ) ).
% round_diff_minimal
thf(fact_2675_bset_I6_J,axiom,
! [D3: int,B2: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B2 )
=> ( X5
!= ( plus_plus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_eq @ int @ X5 @ T2 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X5 @ D3 ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_2676_bset_I8_J,axiom,
! [D3: int,T2: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B2 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B2 )
=> ( X5
!= ( plus_plus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X5 )
=> ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X5 @ D3 ) ) ) ) ) ) ).
% bset(8)
thf(fact_2677_aset_I6_J,axiom,
! [D3: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A4 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_eq @ int @ X5 @ T2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X5 @ D3 ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_2678_aset_I8_J,axiom,
! [D3: int,A4: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X5 )
=> ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X5 @ D3 ) ) ) ) ) ).
% aset(8)
thf(fact_2679_cpmi,axiom,
! [D3: int,P: int > $o,P4: int > $o,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B2 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D3 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ( P4 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ? [Y3: int] :
( ( member @ int @ Y3 @ B2 )
& ( P @ ( plus_plus @ int @ Y3 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_2680_cppi,axiom,
! [D3: int,P: int > $o,P4: int > $o,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D3 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ( P4 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ? [Y3: int] :
( ( member @ int @ Y3 @ A4 )
& ( P @ ( minus_minus @ int @ Y3 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_2681_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_2682_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_2683_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_2684_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_2685_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_2686_signed__take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K )
=> ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_2687_signed__take__bit__int__eq__self,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
= K ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_2688_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
= K )
= ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_2689_signed__take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_2690_of__int__round__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% of_int_round_le
thf(fact_2691_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_ge
thf(fact_2692_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_gt
thf(fact_2693_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ X ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_2694_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X )
= N ) ) ) ).
% round_unique'
thf(fact_2695_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X2: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X2 ) ) @ ( archimedean_ceiling @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ) ).
% round_altdef
thf(fact_2696_tanh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( tanh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% tanh_ln_real
thf(fact_2697_log__base__10__eq1,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq1
thf(fact_2698_central__binomial__lower__bound,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ N ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) @ ( semiring_1_of_nat @ real @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ) ) ).
% central_binomial_lower_bound
thf(fact_2699_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_2700_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_2701_log__base__10__eq2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq2
thf(fact_2702_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_2703_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ A3 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_2704_dvd__0__left__iff,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A3 )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left_iff
thf(fact_2705_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_2706_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_2707_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( dvd_dvd @ nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_2708_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_2709_tanh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( tanh @ real @ X ) @ ( tanh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% tanh_real_less_iff
thf(fact_2710_tanh__real__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X ) @ ( tanh @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% tanh_real_le_iff
thf(fact_2711_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_2712_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_2713_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B3 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_2714_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B3 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_2715_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B3 ) @ ( times_times @ A @ A3 @ C3 ) )
= ( dvd_dvd @ A @ B3 @ C3 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_2716_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B3 @ A3 ) @ ( times_times @ A @ C3 @ A3 ) )
= ( dvd_dvd @ A @ B3 @ C3 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_2717_dvd__imp__mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( modulo_modulo @ A @ B3 @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% dvd_imp_mod_0
thf(fact_2718_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
= N ) ).
% binomial_1
thf(fact_2719_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_2720_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( binomial @ N @ K )
= ( zero_zero @ nat ) )
= ( ord_less @ nat @ N @ K ) ) ).
% binomial_eq_0_iff
thf(fact_2721_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_2722_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_2723_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_2724_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).
% semiring_norm(70)
thf(fact_2725_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_2726_tanh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% tanh_real_pos_iff
thf(fact_2727_tanh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( tanh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% tanh_real_neg_iff
thf(fact_2728_tanh__real__nonneg__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% tanh_real_nonneg_iff
thf(fact_2729_tanh__real__nonpos__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% tanh_real_nonpos_iff
thf(fact_2730_dbl__inc__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bit1 @ K ) ) ) ) ).
% dbl_inc_simps(5)
thf(fact_2731_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N: nat,A3: A,B3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_2732_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K ) )
= ( ord_less_eq @ nat @ K @ N ) ) ).
% zero_less_binomial_iff
thf(fact_2733_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_2734_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_2735_dbl__inc__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ).
% dbl_inc_simps(3)
thf(fact_2736_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_2737_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_2738_power__less__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq
thf(fact_2739_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
= N ) ) ).
% odd_Suc_minus_one
thf(fact_2740_even__diff__nat,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) )
= ( ( ord_less @ nat @ M @ N )
| ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ).
% even_diff_nat
thf(fact_2741_dbl__dec__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_2742_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( ( numeral_numeral @ nat @ W2 )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_2743_even__power,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A3 @ N ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% even_power
thf(fact_2744_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_2745_semiring__parity__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_2746_dvd__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B5: A] :
( ( A5
= ( zero_zero @ A ) )
=> ( B5
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_2747_dvd__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A3 )
=> ( A3
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left
thf(fact_2748_gcd__nat_Oextremum,axiom,
! [A3: nat] : ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% gcd_nat.extremum
thf(fact_2749_gcd__nat_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
& ( ( zero_zero @ nat )
!= A3 ) ) ).
% gcd_nat.extremum_strict
thf(fact_2750_gcd__nat_Oextremum__unique,axiom,
! [A3: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
= ( A3
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_unique
thf(fact_2751_gcd__nat_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
= ( ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) )
& ( A3
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_2752_gcd__nat_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( A3
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_2753_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ M )
=> ( ( dvd_dvd @ nat @ K @ N )
=> ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_2754_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( binomial @ N @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_2755_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus @ nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_2756_binomial__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq @ nat @ R2 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ R2 ) @ ( power_power @ nat @ N @ R2 ) ) ) ).
% binomial_le_pow
thf(fact_2757_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_2758_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S3: B] :
? [Z3: B] :
! [X5: B] :
( ( ord_less @ B @ Z3 @ X5 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S3 ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S3 ) ) ) ) ) ).
% pinf(9)
thf(fact_2759_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S3: B] :
? [Z3: B] :
! [X5: B] :
( ( ord_less @ B @ Z3 @ X5 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S3 ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S3 ) ) ) ) ) ) ).
% pinf(10)
thf(fact_2760_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S3: B] :
? [Z3: B] :
! [X5: B] :
( ( ord_less @ B @ X5 @ Z3 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S3 ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S3 ) ) ) ) ) ).
% minf(9)
thf(fact_2761_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S3: B] :
? [Z3: B] :
! [X5: B] :
( ( ord_less @ B @ X5 @ Z3 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S3 ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S3 ) ) ) ) ) ) ).
% minf(10)
thf(fact_2762_dvd__div__eq__0__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_2763_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one2 )
=> ( ! [X24: num] :
( Y
!= ( bit0 @ X24 ) )
=> ~ ! [X32: num] :
( Y
!= ( bit1 @ X32 ) ) ) ) ).
% num.exhaust
thf(fact_2764_mod__eq__0__iff__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B3: A] :
( ( ( modulo_modulo @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ B3 @ A3 ) ) ) ).
% mod_eq_0_iff_dvd
thf(fact_2765_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B5: A] :
( ( modulo_modulo @ A @ B5 @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_2766_mod__0__imp__dvd,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( ( modulo_modulo @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ B3 @ A3 ) ) ) ).
% mod_0_imp_dvd
thf(fact_2767_le__imp__power__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% le_imp_power_dvd
thf(fact_2768_power__le__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat,B3: A,M: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ B3 )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M ) @ B3 ) ) ) ) ).
% power_le_dvd
thf(fact_2769_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A,N: nat,M: nat] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ M ) ) ) ) ) ).
% dvd_power_le
thf(fact_2770_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ M @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) ) ) ).
% dvd_pos_nat
thf(fact_2771_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N )
=> ~ ( dvd_dvd @ nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_2772_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ M ) )
= ( ( ord_less @ nat @ N @ M )
| ( dvd_dvd @ nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_2773_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( dvd_dvd @ int @ M @ N )
=> ( ( dvd_dvd @ int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_2774_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
=> ( ( dvd_dvd @ nat @ K @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_2775_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
=> ( ( dvd_dvd @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_2776_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ M @ N )
= ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_2777_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M )
=> ( ( ord_less @ int @ M @ N )
=> ~ ( dvd_dvd @ int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_2778_fact__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M ) ) ) ) ).
% fact_dvd
thf(fact_2779_dvd__Gcd__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A,B3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( dvd_dvd @ A @ B3 @ ( semiring_gcd_Gcd_fin @ A @ A4 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( dvd_dvd @ A @ B3 @ X2 ) ) ) ) ) ) ).
% dvd_Gcd_fin_iff
thf(fact_2780_Gcd__fin__greatest,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [B7: A] :
( ( member @ A @ B7 @ A4 )
=> ( dvd_dvd @ A @ A3 @ B7 ) )
=> ( dvd_dvd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ A4 ) ) ) ) ) ).
% Gcd_fin_greatest
thf(fact_2781_tanh__real__lt__1,axiom,
! [X: real] : ( ord_less @ real @ ( tanh @ real @ X ) @ ( one_one @ real ) ) ).
% tanh_real_lt_1
thf(fact_2782_nat__dvd__iff,axiom,
! [Z: int,M: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ Z ) @ M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( dvd_dvd @ int @ Z @ ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_dvd_iff
thf(fact_2783_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K ) ) ) ).
% zero_less_binomial
thf(fact_2784_choose__mult,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
= ( times_times @ nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus @ nat @ N @ K ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_2785_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [C5: A] :
( B3
!= ( times_times @ A @ A3 @ C5 ) ) ) ) ) ).
% unit_dvdE
thf(fact_2786_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X2: A] : ( P @ ( times_times @ A @ L @ X2 ) ) )
= ( ? [X2: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X2 @ ( zero_zero @ A ) ) )
& ( P @ X2 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_2787_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C3: A,B3: A,D2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( dvd_dvd @ A @ C3 @ D2 )
=> ( ( ( divide_divide @ A @ B3 @ A3 )
= ( divide_divide @ A @ D2 @ C3 ) )
= ( ( times_times @ A @ B3 @ C3 )
= ( times_times @ A @ A3 @ D2 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_2788_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C3 @ B3 )
=> ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_2789_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C3 )
= ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_2790_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( ( divide_divide @ A @ B3 @ A3 )
= C3 )
= ( B3
= ( times_times @ A @ C3 @ A3 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_2791_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% unit_div_eq_0_iff
thf(fact_2792_numeral__Bit1,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit1 @ N ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_Bit1
thf(fact_2793_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( modulo_modulo @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% unit_imp_mod_eq_0
thf(fact_2794_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_2795_eval__nat__numeral_I3_J,axiom,
! [N: num] :
( ( numeral_numeral @ nat @ ( bit1 @ N ) )
= ( suc @ ( numeral_numeral @ nat @ ( bit0 @ N ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_2796_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_2797_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( dvd_dvd @ nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_2798_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( dvd_dvd @ nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_2799_bezout__add__strong__nat,axiom,
! [A3: nat,B3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [D6: nat,X3: nat,Y2: nat] :
( ( dvd_dvd @ nat @ D6 @ A3 )
& ( dvd_dvd @ nat @ D6 @ B3 )
& ( ( times_times @ nat @ A3 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B3 @ Y2 ) @ D6 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_2800_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( ( dvd_dvd @ int @ Z @ N )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ Z @ N ) ) ) ).
% zdvd_imp_le
thf(fact_2801_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ N ) )
= ( ~ ( dvd_dvd @ nat @ N @ M ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_2802_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_2803_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M: nat,Q5: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( ( modulo_modulo @ nat @ M @ Q5 )
= ( modulo_modulo @ nat @ N @ Q5 ) )
= ( dvd_dvd @ nat @ Q5 @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_2804_dvd__fact,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ nat @ M @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% dvd_fact
thf(fact_2805_even__nat__iff,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat2 @ K ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) ).
% even_nat_iff
thf(fact_2806_tanh__real__gt__neg1,axiom,
! [X: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( tanh @ real @ X ) ) ).
% tanh_real_gt_neg1
thf(fact_2807_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_2808_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B3 @ A3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B3 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_2809_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B3 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_2810_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [B7: A] :
( ( B7
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B7 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= B7 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B7 )
= A3 )
=> ( ( ( times_times @ A @ A3 @ B7 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C3 @ A3 )
!= ( times_times @ A @ C3 @ B7 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_2811_odd__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% odd_mod_4_div_2
thf(fact_2812_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ N ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_2813_binomial__fact__lemma,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
= ( semiring_char_0_fact @ nat @ N ) ) ) ).
% binomial_fact_lemma
thf(fact_2814_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q5: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q5 ) ) )
!= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(3)
thf(fact_2815_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat,X: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
| ( X
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N ) ) ) ) ).
% dvd_power
thf(fact_2816_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_2817_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_2818_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% choose_dvd
thf(fact_2819_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ M @ N ) @ M )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel1
thf(fact_2820_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ N @ M ) @ M )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel2
thf(fact_2821_dvd__minus__add,axiom,
! [Q5: nat,N: nat,R2: nat,M: nat] :
( ( ord_less_eq @ nat @ Q5 @ N )
=> ( ( ord_less_eq @ nat @ Q5 @ ( times_times @ nat @ R2 @ M ) )
=> ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ Q5 ) )
= ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R2 @ M ) @ Q5 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_2822_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_2823_mod__nat__eqI,axiom,
! [R2: nat,N: nat,M: nat] :
( ( ord_less @ nat @ R2 @ N )
=> ( ( ord_less_eq @ nat @ R2 @ M )
=> ( ( dvd_dvd @ nat @ N @ ( minus_minus @ nat @ M @ R2 ) )
=> ( ( modulo_modulo @ nat @ M @ N )
= R2 ) ) ) ) ).
% mod_nat_eqI
thf(fact_2824_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_2825_binomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_2826_binomial__maximum_H,axiom,
! [N: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K ) @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ).
% binomial_maximum'
thf(fact_2827_binomial__mono,axiom,
! [K: nat,K7: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K7 ) ) ) ) ).
% binomial_mono
thf(fact_2828_binomial__maximum,axiom,
! [N: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% binomial_maximum
thf(fact_2829_binomial__antimono,axiom,
! [K: nat,K7: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K )
=> ( ( ord_less_eq @ nat @ K7 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K7 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_antimono
thf(fact_2830_binomial__le__pow2,axiom,
! [N: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% binomial_le_pow2
thf(fact_2831_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_2832_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( times_times @ nat @ K @ ( binomial @ N @ K ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_2833_num_Osize_I6_J,axiom,
! [X33: num] :
( ( size_size @ num @ ( bit1 @ X33 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X33 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(6)
thf(fact_2834_even__iff__mod__2__eq__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_2835_binomial__altdef__nat,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_2836_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q5: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q5 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q5 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q5 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(7)
thf(fact_2837_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q5: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q5 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q5 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q5 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(11)
thf(fact_2838_power__mono__odd,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A,B3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ).
% power_mono_odd
thf(fact_2839_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% odd_pos
thf(fact_2840_card__3__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X2: A,Y3: A,Z6: A] :
( ( S
= ( insert2 @ A @ X2 @ ( insert2 @ A @ Y3 @ ( insert2 @ A @ Z6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X2 != Y3 )
& ( Y3 != Z6 )
& ( X2 != Z6 ) ) ) ) ).
% card_3_iff
thf(fact_2841_odd__card__imp__not__empty,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A4 ) )
=> ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_2842_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ( dvd_dvd @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_2843_even__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_unset_bit_iff
thf(fact_2844_even__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( M
!= ( zero_zero @ nat ) ) ) ) ) ).
% even_set_bit_iff
thf(fact_2845_exp__le,axiom,
ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ).
% exp_le
thf(fact_2846_binomial__less__binomial__Suc,axiom,
! [K: nat,N: nat] :
( ( ord_less @ nat @ K @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_2847_binomial__strict__antimono,axiom,
! [K: nat,K7: nat,N: nat] :
( ( ord_less @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) )
=> ( ( ord_less_eq @ nat @ K7 @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K7 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_2848_binomial__strict__mono,axiom,
! [K: nat,K7: nat,N: nat] :
( ( ord_less @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K7 ) ) ) ) ).
% binomial_strict_mono
thf(fact_2849_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_2850_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_binomial
thf(fact_2851_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_2852_parity__cases,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) ) )
=> ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) ) ) ) ) ).
% parity_cases
thf(fact_2853_mod2__eq__if,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ) ).
% mod2_eq_if
thf(fact_2854_zero__le__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq
thf(fact_2855_zero__le__odd__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% zero_le_odd_power
thf(fact_2856_zero__le__even__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% zero_le_even_power
thf(fact_2857_power__mono__even,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A,B3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ).
% power_mono_even
thf(fact_2858_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_2859_even__set__encode__iff,axiom,
! [A4: set @ nat] :
( ( finite_finite2 @ nat @ A4 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A4 ) )
= ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A4 ) ) ) ) ).
% even_set_encode_iff
thf(fact_2860_zero__less__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq
thf(fact_2861_even__mask__div__iff_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% even_mask_div_iff'
thf(fact_2862_power__le__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_2863_even__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( suc @ ( zero_zero @ nat ) ) )
=> ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% even_mod_4_div_2
thf(fact_2864_even__mask__div__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% even_mask_div_iff
thf(fact_2865_Bernoulli__inequality__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_2866_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ord_less @ nat @ N @ M )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M @ N )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_2867_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N2: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_2868_binomial__code,axiom,
( binomial
= ( ^ [N2: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N2 @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N2 @ ( minus_minus @ nat @ N2 @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N2 @ K3 ) @ ( one_one @ nat ) ) @ N2 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_2869_num_Osize__gen_I3_J,axiom,
! [X33: num] :
( ( size_num @ ( bit1 @ X33 ) )
= ( plus_plus @ nat @ ( size_num @ X33 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(3)
thf(fact_2870_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N2: nat,A5: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_2871_num_Osize__gen_I2_J,axiom,
! [X23: num] :
( ( size_num @ ( bit0 @ X23 ) )
= ( plus_plus @ nat @ ( size_num @ X23 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(2)
thf(fact_2872_set__decode__plus__power__2,axiom,
! [N: nat,Z: nat] :
( ~ ( member @ nat @ N @ ( nat_set_decode @ Z ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z ) )
= ( insert2 @ nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_2873_take__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% take_bit_of_0
thf(fact_2874_take__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% take_bit_0
thf(fact_2875_sin__coeff__0,axiom,
( ( sin_coeff @ ( zero_zero @ nat ) )
= ( zero_zero @ real ) ) ).
% sin_coeff_0
thf(fact_2876_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_2877_set__encode__inverse,axiom,
! [A4: set @ nat] :
( ( finite_finite2 @ nat @ A4 )
=> ( ( nat_set_decode @ ( nat_set_encode @ A4 ) )
= A4 ) ) ).
% set_encode_inverse
thf(fact_2878_take__bit__of__1__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
= ( zero_zero @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_2879_even__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_take_bit_eq
thf(fact_2880_set__decode__0,axiom,
! [X: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% set_decode_0
thf(fact_2881_take__bit__Suc__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ ( zero_zero @ nat ) ) @ A3 )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_0
thf(fact_2882_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_2883_take__bit__tightened,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A,B3: A,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ N @ B3 ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ M @ B3 ) ) ) ) ) ).
% take_bit_tightened
thf(fact_2884_take__bit__nat__less__eq__self,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ M ) ).
% take_bit_nat_less_eq_self
thf(fact_2885_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N: nat,Q5: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M @ Q5 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N @ Q5 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_2886_nat__take__bit__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) ) ) ) ).
% nat_take_bit_eq
thf(fact_2887_take__bit__nat__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_nat_eq
thf(fact_2888_take__bit__tightened__less__eq__int,axiom,
! [M: nat,N: nat,K: int] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_2889_take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% take_bit_int_less_eq_self_iff
thf(fact_2890_take__bit__nonnegative,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ).
% take_bit_nonnegative
thf(fact_2891_not__take__bit__negative,axiom,
! [N: nat,K: int] :
~ ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) ) ).
% not_take_bit_negative
thf(fact_2892_take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% take_bit_int_greater_self_iff
thf(fact_2893_signed__take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N @ M ) @ ( bit_se2584673776208193580ke_bit @ A @ N ) @ ( bit_ri4674362597316999326ke_bit @ A @ M ) @ A3 ) ) ) ).
% signed_take_bit_take_bit
thf(fact_2894_finite__set__decode,axiom,
! [N: nat] : ( finite_finite2 @ nat @ ( nat_set_decode @ N ) ) ).
% finite_set_decode
thf(fact_2895_take__bit__unset__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
= ( bit_se2638667681897837118et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_2896_take__bit__set__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
= ( bit_se5668285175392031749et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_2897_take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ M @ A3 ) ) ) ) ).
% take_bit_signed_take_bit
thf(fact_2898_subset__decode__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% subset_decode_imp_le
thf(fact_2899_take__bit__nat__eq__self,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M )
= M ) ) ).
% take_bit_nat_eq_self
thf(fact_2900_take__bit__nat__less__exp,axiom,
! [N: nat,M: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% take_bit_nat_less_exp
thf(fact_2901_take__bit__nat__eq__self__iff,axiom,
! [N: nat,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M )
= M )
= ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_2902_take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% take_bit_int_less_exp
thf(fact_2903_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa3: nat,Xb3: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa3 @ Xb3 @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb3 @ Xa3 )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb3 @ Xa3 )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa3 @ ( one_one @ nat ) ) @ Xb3 @ ( X @ Xa3 @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_2904_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F2: nat > A > A,A5: nat,B5: nat,Acc: A] : ( if @ A @ ( ord_less @ nat @ B5 @ A5 ) @ Acc @ ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B5 @ ( F2 @ A5 @ Acc ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_2905_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_2906_take__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A3 ) ) ) ).
% take_bit_eq_0_iff
thf(fact_2907_take__bit__nat__less__self__iff,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ M )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M ) ) ).
% take_bit_nat_less_self_iff
thf(fact_2908_take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% take_bit_int_less_self_iff
thf(fact_2909_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_2910_take__bit__int__eq__self,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= K ) ) ) ).
% take_bit_int_eq_self
thf(fact_2911_take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= K )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_2912_take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_2913_take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_int_greater_eq
thf(fact_2914_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_2915_take__bit__minus__small__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_2916_modulo__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N ) ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L )
@ ( minus_minus @ int
@ ( semiring_1_of_nat @ int
@ ( times_times @ nat @ N
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N @ M ) ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_2917_divide__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_2918_sqrt__sum__squares__half__less,axiom,
! [X: real,U: real,Y: real] :
( ( ord_less @ real @ X @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_2919_even__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
!= ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_flip_bit_iff
thf(fact_2920_one__mod__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% one_mod_2_pow_eq
thf(fact_2921_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_2922_flip__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% flip_bit_negative_int_iff
thf(fact_2923_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_2924_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_2925_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_2926_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_2927_real__sqrt__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% real_sqrt_less_iff
thf(fact_2928_real__sqrt__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% real_sqrt_le_iff
thf(fact_2929_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_2930_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_2931_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_2932_real__sqrt__lt__0__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% real_sqrt_lt_0_iff
thf(fact_2933_real__sqrt__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% real_sqrt_le_0_iff
thf(fact_2934_real__sqrt__ge__0__iff,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ Y ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ).
% real_sqrt_ge_0_iff
thf(fact_2935_real__sqrt__lt__1__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ).
% real_sqrt_lt_1_iff
thf(fact_2936_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_2937_real__sqrt__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ).
% real_sqrt_le_1_iff
thf(fact_2938_real__sqrt__ge__1__iff,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ Y ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ).
% real_sqrt_ge_1_iff
thf(fact_2939_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_2940_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_2941_sgn__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_abs
thf(fact_2942_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( abs_abs @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_2943_Suc__0__mod__eq,axiom,
! [N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( zero_neq_one_of_bool @ nat
@ ( N
!= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% Suc_0_mod_eq
thf(fact_2944_take__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% take_bit_of_1
thf(fact_2945_sgn__of__nat,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat] :
( ( sgn_sgn @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sgn_of_nat
thf(fact_2946_take__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_neq_one_of_bool @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% take_bit_of_Suc_0
thf(fact_2947_of__bool__half__eq__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [B3: $o] :
( ( divide_divide @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ).
% of_bool_half_eq_0
thf(fact_2948_real__sqrt__pow2__iff,axiom,
! [X: real] :
( ( ( power_power @ real @ ( sqrt @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% real_sqrt_pow2_iff
thf(fact_2949_real__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( sqrt @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X ) ) ).
% real_sqrt_pow2
thf(fact_2950_bits__1__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% bits_1_div_exp
thf(fact_2951_one__div__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% one_div_2_pow_eq
thf(fact_2952_take__bit__of__exp,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_of_exp
thf(fact_2953_take__bit__of__2,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_of_2
thf(fact_2954_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_2955_real__sqrt__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_less_mono
thf(fact_2956_real__sqrt__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_le_mono
thf(fact_2957_real__sqrt__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_gt_zero
thf(fact_2958_real__sqrt__eq__zero__cancel,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( sqrt @ X )
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ).
% real_sqrt_eq_zero_cancel
thf(fact_2959_real__sqrt__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_zero
thf(fact_2960_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_2961_real__sqrt__ge__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_one
thf(fact_2962_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_2963_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P5: $o] : ( if @ A @ P5 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_2964_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P6: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P6 ) )
= ( ( P6
=> ( P @ ( one_one @ A ) ) )
& ( ~ P6
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_2965_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P6: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P6 ) )
= ( ~ ( ( P6
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P6
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_2966_real__div__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( divide_divide @ real @ X @ ( sqrt @ X ) )
= ( sqrt @ X ) ) ) ).
% real_div_sqrt
thf(fact_2967_sqrt__add__le__add__sqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ X @ Y ) ) @ ( plus_plus @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).
% sqrt_add_le_add_sqrt
thf(fact_2968_le__real__sqrt__sumsq,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( times_times @ real @ X @ X ) @ ( times_times @ real @ Y @ Y ) ) ) ) ).
% le_real_sqrt_sumsq
thf(fact_2969_sqrt2__less__2,axiom,
ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% sqrt2_less_2
thf(fact_2970_sqrt__divide__self__eq,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( divide_divide @ real @ ( sqrt @ X ) @ X )
= ( inverse_inverse @ real @ ( sqrt @ X ) ) ) ) ).
% sqrt_divide_self_eq
thf(fact_2971_take__bit__flip__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
= ( bit_se8732182000553998342ip_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_2972_real__less__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ord_less @ real @ X @ ( sqrt @ Y ) ) ) ).
% real_less_rsqrt
thf(fact_2973_real__le__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ord_less_eq @ real @ X @ ( sqrt @ Y ) ) ) ).
% real_le_rsqrt
thf(fact_2974_sqrt__le__D,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ Y )
=> ( ord_less_eq @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% sqrt_le_D
thf(fact_2975_real__sqrt__unique,axiom,
! [Y: real,X: real] :
( ( ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( sqrt @ X )
= Y ) ) ) ).
% real_sqrt_unique
thf(fact_2976_real__le__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_le_lsqrt
thf(fact_2977_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_2978_real__sqrt__sum__squares__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge1
thf(fact_2979_real__sqrt__sum__squares__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq @ real @ Y @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge2
thf(fact_2980_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A3: real,C3: real,B3: real,D2: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ ( plus_plus @ real @ A3 @ C3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( plus_plus @ real @ B3 @ D2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ C3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_triangle_ineq
thf(fact_2981_sqrt__ge__absD,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( sqrt @ Y ) )
=> ( ord_less_eq @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) ).
% sqrt_ge_absD
thf(fact_2982_real__less__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_less_lsqrt
thf(fact_2983_sqrt__sum__squares__le__sum,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ X @ Y ) ) ) ) ).
% sqrt_sum_squares_le_sum
thf(fact_2984_real__inv__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( inverse_inverse @ real @ X ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_2985_sqrt__sum__squares__le__sum__abs,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( abs_abs @ real @ X ) @ ( abs_abs @ real @ Y ) ) ) ).
% sqrt_sum_squares_le_sum_abs
thf(fact_2986_real__sqrt__ge__abs2,axiom,
! [Y: real,X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs2
thf(fact_2987_real__sqrt__ge__abs1,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs1
thf(fact_2988_ln__sqrt,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( sqrt @ X ) )
= ( divide_divide @ real @ ( ln_ln @ real @ X ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% ln_sqrt
thf(fact_2989_exp__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% exp_mod_exp
thf(fact_2990_arsinh__real__aux,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% arsinh_real_aux
thf(fact_2991_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X: real,Y: real,Xa3: real,Ya: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_ge_zero
thf(fact_2992_real__sqrt__power__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( sqrt @ X ) @ N )
= ( power_power @ real @ X @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_power_even
thf(fact_2993_arith__geo__mean__sqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( times_times @ real @ X @ Y ) ) @ ( divide_divide @ real @ ( plus_plus @ real @ X @ Y ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arith_geo_mean_sqrt
thf(fact_2994_powr__half__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( sqrt @ X ) ) ) ).
% powr_half_sqrt
thf(fact_2995_cos__x__y__le__one,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( divide_divide @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( one_one @ real ) ) ).
% cos_x_y_le_one
thf(fact_2996_real__sqrt__sum__squares__less,axiom,
! [X: real,U: real,Y: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_2997_arcosh__real__def,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( arcosh @ real @ X )
= ( ln_ln @ real @ ( plus_plus @ real @ X @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arcosh_real_def
thf(fact_2998_exp__div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
!= ( zero_zero @ A ) )
& ( ord_less_eq @ nat @ N @ M ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ).
% exp_div_exp_eq
thf(fact_2999_cosh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( cosh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( plus_plus @ real @ X @ ( inverse_inverse @ real @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% cosh_ln_real
thf(fact_3000_Suc__0__xor__eq,axiom,
! [N: nat] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Suc_0_xor_eq
thf(fact_3001_xor__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se5824344971392196577ns_xor @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% xor_Suc_0_eq
thf(fact_3002_cosh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cosh @ A @ X )
= ( zero_zero @ A ) )
= ( ( power_power @ A @ ( exp @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% cosh_zero_iff
thf(fact_3003_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N2
@ ( if @ nat
@ ( N2
= ( zero_zero @ nat ) )
@ M2
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_3004_xor_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% xor.right_neutral
thf(fact_3005_xor_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% xor.left_neutral
thf(fact_3006_xor__self__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% xor_self_eq
thf(fact_3007_bit_Oxor__self,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ X )
= ( zero_zero @ A ) ) ) ).
% bit.xor_self
thf(fact_3008_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_3009_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_3010_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_3011_xor__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% xor_nat_numerals(3)
thf(fact_3012_xor__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ X ) ) ) ).
% xor_nat_numerals(4)
thf(fact_3013_cosh__real__pos,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_pos
thf(fact_3014_cosh__real__nonpos__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less_eq @ real @ Y @ X ) ) ) ) ).
% cosh_real_nonpos_le_iff
thf(fact_3015_cosh__real__nonneg__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% cosh_real_nonneg_le_iff
thf(fact_3016_cosh__real__nonneg,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_nonneg
thf(fact_3017_arcosh__cosh__real,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( arcosh @ real @ ( cosh @ real @ X ) )
= X ) ) ).
% arcosh_cosh_real
thf(fact_3018_cosh__real__ge__1,axiom,
! [X: real] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_ge_1
thf(fact_3019_sinh__less__cosh__real,axiom,
! [X: real] : ( ord_less @ real @ ( sinh @ real @ X ) @ ( cosh @ real @ X ) ) ).
% sinh_less_cosh_real
thf(fact_3020_sinh__le__cosh__real,axiom,
! [X: real] : ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( cosh @ real @ X ) ) ).
% sinh_le_cosh_real
thf(fact_3021_cosh__real__nonpos__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_3022_cosh__real__nonneg__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_3023_cosh__real__strict__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_3024_tanh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cosh @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cosh @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( tanh @ A @ ( plus_plus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tanh @ A @ X ) @ ( tanh @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tanh @ A @ X ) @ ( tanh @ A @ Y ) ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_3025_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N2: nat,A5: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ A5
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_3026_Suc__0__or__eq,axiom,
! [N: nat] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Suc_0_or_eq
thf(fact_3027_or__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se1065995026697491101ons_or @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% or_Suc_0_eq
thf(fact_3028_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( ( M2
= ( zero_zero @ nat ) )
| ( N2
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_3029_is__empty__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( is_empty @ A @ ( set2 @ A @ Xs ) )
= ( null @ A @ Xs ) ) ).
% is_empty_set
thf(fact_3030_and__zero__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_zero_eq
thf(fact_3031_zero__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% zero_and_eq
thf(fact_3032_bit_Oconj__zero__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ X )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_left
thf(fact_3033_bit_Oconj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_right
thf(fact_3034_or_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% or.left_neutral
thf(fact_3035_or_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% or.right_neutral
thf(fact_3036_xor__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% xor_nonnegative_int_iff
thf(fact_3037_drop__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% drop_bit_nonnegative_int_iff
thf(fact_3038_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_3039_drop__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% drop_bit_negative_int_iff
thf(fact_3040_drop__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% drop_bit_of_0
thf(fact_3041_drop__bit__of__bool,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,B3: $o] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( zero_neq_one_of_bool @ A @ B3 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( N
= ( zero_zero @ nat ) )
& B3 ) ) ) ) ).
% drop_bit_of_bool
thf(fact_3042_drop__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se4197421643247451524op_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_neq_one_of_bool @ nat
@ ( N
= ( zero_zero @ nat ) ) ) ) ).
% drop_bit_of_Suc_0
thf(fact_3043_drop__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% drop_bit_of_1
thf(fact_3044_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_3045_and__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(5)
thf(fact_3046_and__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(3)
thf(fact_3047_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_3048_or__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(4)
thf(fact_3049_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_3050_or__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(3)
thf(fact_3051_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_3052_and__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(4)
thf(fact_3053_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_3054_Suc__0__and__eq,axiom,
! [N: nat] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Suc_0_and_eq
thf(fact_3055_and__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se5824344872417868541ns_and @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% and_Suc_0_eq
thf(fact_3056_or__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B3: A] :
( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% or_eq_0_iff
thf(fact_3057_bit_Odisj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( zero_zero @ A ) )
= X ) ) ).
% bit.disj_zero_right
thf(fact_3058_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ X )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X = Y ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_3059_XOR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y ) ) ) ) ).
% XOR_lower
thf(fact_3060_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= A3 )
= ( ( bit_se4197421643247451524op_bit @ A @ N @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_3061_XOR__upper,axiom,
! [X: int,N: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_3062_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N2
@ ( if @ nat
@ ( N2
= ( zero_zero @ nat ) )
@ M2
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_3063_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_3064_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_3065_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_3066_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B3: A,N: nat] :
( ! [J2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ J2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_3067_max_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A] :
( ( ord_max @ A @ A3 @ A3 )
= A3 ) ) ).
% max.idem
thf(fact_3068_max_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_max @ A @ A3 @ ( ord_max @ A @ A3 @ B3 ) )
= ( ord_max @ A @ A3 @ B3 ) ) ) ).
% max.left_idem
thf(fact_3069_max_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B3 ) @ B3 )
= ( ord_max @ A @ A3 @ B3 ) ) ) ).
% max.right_idem
thf(fact_3070_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_max @ A @ A3 @ B3 )
= A3 ) ) ) ).
% max.absorb1
thf(fact_3071_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_max @ A @ A3 @ B3 )
= B3 ) ) ) ).
% max.absorb2
thf(fact_3072_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B3 @ C3 ) @ A3 )
= ( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% max.bounded_iff
thf(fact_3073_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_3074_max_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_max @ A @ A3 @ B3 )
= A3 ) ) ) ).
% max.absorb3
thf(fact_3075_max_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_max @ A @ A3 @ B3 )
= B3 ) ) ) ).
% max.absorb4
thf(fact_3076_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Y @ Z ) ) ) ) ).
% max_less_iff_conj
thf(fact_3077_or__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% or_nonnegative_int_iff
thf(fact_3078_and__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% and_nonnegative_int_iff
thf(fact_3079_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_3080_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_3081_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% max_bot
thf(fact_3082_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% max_bot2
thf(fact_3083_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ M @ N ) ) ) ).
% max_Suc_Suc
thf(fact_3084_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_3085_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_3086_max__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ A3 @ ( zero_zero @ nat ) )
= A3 ) ).
% max_nat.right_neutral
thf(fact_3087_max__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( zero_zero @ nat )
= ( ord_max @ nat @ A3 @ B3 ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B3
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_3088_max__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A3 )
= A3 ) ).
% max_nat.left_neutral
thf(fact_3089_max__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_max @ nat @ A3 @ B3 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B3
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_3090_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_3091_max__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(3)
thf(fact_3092_max__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(4)
thf(fact_3093_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_3094_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_3095_max__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(5)
thf(fact_3096_max__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(6)
thf(fact_3097_signed__take__bit__nonnegative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% signed_take_bit_nonnegative_iff
thf(fact_3098_signed__take__bit__negative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ).
% signed_take_bit_negative_iff
thf(fact_3099_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_3100_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_3101_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_3102_bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( zero_zero @ nat ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% bit_0
thf(fact_3103_bit__mod__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( N
= ( zero_zero @ nat ) )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% bit_mod_2_iff
thf(fact_3104_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B5 ) @ B5 @ A5 ) ) ) ) ).
% max_def
thf(fact_3105_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_max @ A @ X @ Y )
= X ) ) ) ).
% max_absorb1
thf(fact_3106_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_max @ A @ X @ Y )
= Y ) ) ) ).
% max_absorb2
thf(fact_3107_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% max.coboundedI2
thf(fact_3108_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ C3 @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% max.coboundedI1
thf(fact_3109_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_max @ A @ A5 @ B5 )
= B5 ) ) ) ) ).
% max.absorb_iff2
thf(fact_3110_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_max @ A @ A5 @ B5 )
= A5 ) ) ) ) ).
% max.absorb_iff1
thf(fact_3111_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less_eq @ A @ Z @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less_eq @ A @ Z @ X )
| ( ord_less_eq @ A @ Z @ Y ) ) ) ) ).
% le_max_iff_disj
thf(fact_3112_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] : ( ord_less_eq @ A @ B3 @ ( ord_max @ A @ A3 @ B3 ) ) ) ).
% max.cobounded2
thf(fact_3113_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ A3 @ ( ord_max @ A @ A3 @ B3 ) ) ) ).
% max.cobounded1
thf(fact_3114_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( A5
= ( ord_max @ A @ A5 @ B5 ) ) ) ) ) ).
% max.order_iff
thf(fact_3115_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B3 @ C3 ) @ A3 ) ) ) ) ).
% max.boundedI
thf(fact_3116_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B3 @ C3 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B3 @ A3 )
=> ~ ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% max.boundedE
thf(fact_3117_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( ord_max @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% max.orderI
thf(fact_3118_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3
= ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% max.orderE
thf(fact_3119_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A3: A,D2: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ( ord_less_eq @ A @ D2 @ B3 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C3 @ D2 ) @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ) ).
% max.mono
thf(fact_3120_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_3121_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ C3 @ A3 )
=> ( ord_less @ A @ C3 @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_3122_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( A5
= ( ord_max @ A @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_3123_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less @ A @ ( ord_max @ A @ B3 @ C3 ) @ A3 )
=> ~ ( ( ord_less @ A @ B3 @ A3 )
=> ~ ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% max.strict_boundedE
thf(fact_3124_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z @ X )
| ( ord_less @ A @ Z @ Y ) ) ) ) ).
% less_max_iff_disj
thf(fact_3125_of__nat__max,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X @ Y ) )
= ( ord_max @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_max
thf(fact_3126_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_3127_max_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B3 ) @ C3 )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B3 @ C3 ) ) ) ) ).
% max.assoc
thf(fact_3128_max_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B5: A] : ( ord_max @ A @ B5 @ A5 ) ) ) ) ).
% max.commute
thf(fact_3129_max_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_max @ A @ B3 @ ( ord_max @ A @ A3 @ C3 ) )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B3 @ C3 ) ) ) ) ).
% max.left_commute
thf(fact_3130_sup__max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% sup_max
thf(fact_3131_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ord_max @ A @ ( minus_minus @ A @ X @ Z ) @ ( minus_minus @ A @ Y @ Z ) ) ) ) ).
% max_diff_distrib_left
thf(fact_3132_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ X @ ( ord_max @ A @ Y @ Z ) )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z ) ) ) ) ).
% max_add_distrib_right
thf(fact_3133_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Z ) @ ( plus_plus @ A @ Y @ Z ) ) ) ) ).
% max_add_distrib_left
thf(fact_3134_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q5: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M @ N ) @ Q5 )
= ( ord_max @ nat @ ( times_times @ nat @ M @ Q5 ) @ ( times_times @ nat @ N @ Q5 ) ) ) ).
% nat_mult_max_left
thf(fact_3135_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q5: nat] :
( ( times_times @ nat @ M @ ( ord_max @ nat @ N @ Q5 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q5 ) ) ) ).
% nat_mult_max_right
thf(fact_3136_nat__add__max__left,axiom,
! [M: nat,N: nat,Q5: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N ) @ Q5 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q5 ) @ ( plus_plus @ nat @ N @ Q5 ) ) ) ).
% nat_add_max_left
thf(fact_3137_nat__add__max__right,axiom,
! [M: nat,N: nat,Q5: nat] :
( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N @ Q5 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ N ) @ ( plus_plus @ nat @ M @ Q5 ) ) ) ).
% nat_add_max_right
thf(fact_3138_bit__nat__iff,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( nat2 @ K ) @ N )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% bit_nat_iff
thf(fact_3139_bit__1__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% bit_1_iff
thf(fact_3140_not__bit__Suc__0__Suc,axiom,
! [N: nat] :
~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( suc @ N ) ) ).
% not_bit_Suc_0_Suc
thf(fact_3141_bit__Suc__0__iff,axiom,
! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( N
= ( zero_zero @ nat ) ) ) ).
% bit_Suc_0_iff
thf(fact_3142_bit__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M @ A3 ) @ N )
= ( ( ord_less @ nat @ N @ M )
& ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% bit_take_bit_iff
thf(fact_3143_or__greater__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ K @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) ) ) ).
% or_greater_eq
thf(fact_3144_OR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) ) ) ) ).
% OR_lower
thf(fact_3145_bit__of__bool__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [B3: $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ N )
= ( B3
& ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% bit_of_bool_iff
thf(fact_3146_AND__upper2_H,axiom,
! [Y: int,Z: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ Y @ Z )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Z ) ) ) ).
% AND_upper2'
thf(fact_3147_AND__upper1_H,axiom,
! [Y: int,Z: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ Y @ Z )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya ) @ Z ) ) ) ).
% AND_upper1'
thf(fact_3148_AND__upper2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Y ) ) ).
% AND_upper2
thf(fact_3149_AND__upper1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ X ) ) ).
% AND_upper1
thf(fact_3150_AND__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) ) ) ).
% AND_lower
thf(fact_3151_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ N @ M ) @ M )
= ( ord_max @ nat @ N @ M ) ) ).
% nat_minus_add_max
thf(fact_3152_pi__gt__zero,axiom,
ord_less @ real @ ( zero_zero @ real ) @ pi ).
% pi_gt_zero
thf(fact_3153_pi__not__less__zero,axiom,
~ ( ord_less @ real @ pi @ ( zero_zero @ real ) ) ).
% pi_not_less_zero
thf(fact_3154_pi__ge__zero,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ pi ).
% pi_ge_zero
thf(fact_3155_not__bit__Suc__0__numeral,axiom,
! [N: num] :
~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ N ) ) ).
% not_bit_Suc_0_numeral
thf(fact_3156_AND__upper2_H_H,axiom,
! [Y: int,Z: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Z ) ) ) ).
% AND_upper2''
thf(fact_3157_AND__upper1_H_H,axiom,
! [Y: int,Z: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya ) @ Z ) ) ) ).
% AND_upper1''
thf(fact_3158_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_3159_bit__imp__take__bit__positive,axiom,
! [N: nat,M: nat,K: int] :
( ( ord_less @ nat @ N @ M )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_3160_pi__less__4,axiom,
ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ).
% pi_less_4
thf(fact_3161_pi__ge__two,axiom,
ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ).
% pi_ge_two
thf(fact_3162_exp__eq__0__imp__not__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,A3: A] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_3163_int__bit__bound,axiom,
! [K: int] :
~ ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ N3 @ M3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M3 )
= ( 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_3164_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_3165_pi__half__le__two,axiom,
ord_less_eq @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% pi_half_le_two
thf(fact_3166_and__exp__eq__0__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_zero @ A ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_3167_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_3168_pi__half__ge__zero,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% pi_half_ge_zero
thf(fact_3169_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_3170_arctan__ubound,axiom,
! [Y: real] : ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arctan_ubound
thf(fact_3171_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_3172_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_3173_OR__upper,axiom,
! [X: int,N: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_3174_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_3175_cot__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cot @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% cot_less_zero
thf(fact_3176_cos__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
=> ? [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_3177_sin__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ? [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_3178_cot__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cot @ real @ X ) ) ) ) ).
% cot_gt_zero
thf(fact_3179_arcsin__lbound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).
% arcsin_lbound
thf(fact_3180_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_3181_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_3182_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_3183_sin__arcsin,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arcsin @ Y ) )
= Y ) ) ) ).
% sin_arcsin
thf(fact_3184_cos__one__sin__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
= ( one_one @ A ) )
=> ( ( sin @ A @ X )
= ( zero_zero @ A ) ) ) ) ).
% cos_one_sin_zero
thf(fact_3185_sin__zero__norm__cos__one,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( cos @ A @ X ) )
= ( one_one @ real ) ) ) ) ).
% sin_zero_norm_cos_one
thf(fact_3186_sincos__principal__value,axiom,
! [X: real] :
? [Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ Y2 )
& ( ord_less_eq @ real @ Y2 @ pi )
& ( ( sin @ real @ Y2 )
= ( sin @ real @ X ) )
& ( ( cos @ real @ Y2 )
= ( cos @ real @ X ) ) ) ).
% sincos_principal_value
thf(fact_3187_sin__x__le__x,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( sin @ real @ X ) @ X ) ) ).
% sin_x_le_x
thf(fact_3188_sin__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( one_one @ real ) ) ).
% sin_le_one
thf(fact_3189_cos__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( one_one @ real ) ) ).
% cos_le_one
thf(fact_3190_cos__arcsin__nonzero,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X ) )
!= ( zero_zero @ real ) ) ) ) ).
% cos_arcsin_nonzero
thf(fact_3191_abs__sin__x__le__abs__x,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X ) ) @ ( abs_abs @ real @ X ) ) ).
% abs_sin_x_le_abs_x
thf(fact_3192_sin__cos__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( plus_plus @ real @ ( times_times @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) ) @ ( times_times @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) @ ( one_one @ real ) ) ).
% sin_cos_le1
thf(fact_3193_sin__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ pi )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero
thf(fact_3194_sin__x__ge__neg__x,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ X ) @ ( sin @ real @ X ) ) ) ).
% sin_x_ge_neg_x
thf(fact_3195_sin__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_ge_zero
thf(fact_3196_sin__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( sin @ real @ X ) ) ).
% sin_ge_minus_one
thf(fact_3197_cos__inj__pi,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ( cos @ real @ X )
= ( cos @ real @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% cos_inj_pi
thf(fact_3198_cos__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) )
= ( ord_less_eq @ real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_le_eq
thf(fact_3199_cos__monotone__0__pi__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi_le
thf(fact_3200_cos__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( cos @ real @ X ) ) ).
% cos_ge_minus_one
thf(fact_3201_abs__sin__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X ) ) @ ( one_one @ real ) ) ).
% abs_sin_le_one
thf(fact_3202_abs__cos__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( cos @ real @ X ) ) @ ( one_one @ real ) ) ).
% abs_cos_le_one
thf(fact_3203_arcsin__sin,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arcsin @ ( sin @ real @ X ) )
= X ) ) ) ).
% arcsin_sin
thf(fact_3204_arcsin__minus,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( arcsin @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( arcsin @ X ) ) ) ) ) ).
% arcsin_minus
thf(fact_3205_arcsin__le__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_le_arcsin
thf(fact_3206_arcsin__eq__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ( arcsin @ X )
= ( arcsin @ Y ) )
= ( X = Y ) ) ) ) ).
% arcsin_eq_iff
thf(fact_3207_arcsin__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% arcsin_le_mono
thf(fact_3208_cos__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ord_less @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_3209_cos__monotone__0__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_3210_sin__eq__0__pi,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( ord_less @ real @ X @ pi )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% sin_eq_0_pi
thf(fact_3211_sin__zero__pi__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ pi )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% sin_zero_pi_iff
thf(fact_3212_cos__monotone__minus__pi__0_H,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( cos @ real @ Y ) @ ( cos @ real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0'
thf(fact_3213_arcsin,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin
thf(fact_3214_arcsin__pi,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq @ real @ ( arcsin @ Y ) @ pi )
& ( ( sin @ real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin_pi
thf(fact_3215_arcsin__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X ) @ Y )
= ( ord_less_eq @ real @ X @ ( sin @ real @ Y ) ) ) ) ) ) ) ).
% arcsin_le_iff
thf(fact_3216_le__arcsin__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ Y @ ( arcsin @ X ) )
= ( ord_less_eq @ real @ ( sin @ real @ Y ) @ X ) ) ) ) ) ) ).
% le_arcsin_iff
thf(fact_3217_sincos__total__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T6: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less_eq @ real @ T6 @ pi )
& ( X
= ( cos @ real @ T6 ) )
& ( Y
= ( sin @ real @ T6 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_3218_sin__cos__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) )
=> ( ( sin @ real @ X )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( cos @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_cos_sqrt
thf(fact_3219_cos__arcsin,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_arcsin
thf(fact_3220_arcsin__less__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_3221_sin__gt__zero__02,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero_02
thf(fact_3222_arcsin__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% arcsin_less_mono
thf(fact_3223_cos__two__less__zero,axiom,
ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_less_zero
thf(fact_3224_cos__is__zero,axiom,
? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X3 )
= ( zero_zero @ real ) )
& ! [Y5: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y5 )
& ( ord_less_eq @ real @ Y5 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ Y5 )
= ( zero_zero @ real ) ) )
=> ( Y5 = X3 ) ) ) ).
% cos_is_zero
thf(fact_3225_cos__two__le__zero,axiom,
ord_less_eq @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_le_zero
thf(fact_3226_cos__monotone__minus__pi__0,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cos @ real @ Y ) @ ( cos @ real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_3227_cos__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ pi )
& ( ( cos @ real @ X3 )
= Y )
& ! [Y5: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y5 )
& ( ord_less_eq @ real @ Y5 @ pi )
& ( ( cos @ real @ Y5 )
= Y ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% cos_total
thf(fact_3228_sincos__total__pi__half,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T6: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less_eq @ real @ T6 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( X
= ( cos @ real @ T6 ) )
& ( Y
= ( sin @ real @ T6 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_3229_sincos__total__2pi__le,axiom,
! [X: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T6: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less_eq @ real @ T6 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
& ( X
= ( cos @ real @ T6 ) )
& ( Y
= ( sin @ real @ T6 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_3230_sincos__total__2pi,axiom,
! [X: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ~ ! [T6: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T6 )
=> ( ( ord_less @ real @ T6 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ( X
= ( cos @ real @ T6 ) )
=> ( Y
!= ( sin @ real @ T6 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_3231_sin__pi__divide__n__ge__0,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% sin_pi_divide_n_ge_0
thf(fact_3232_sin__gt__zero2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero2
thf(fact_3233_sin__lt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ pi @ X )
=> ( ( ord_less @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_lt_zero
thf(fact_3234_cos__double__less__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) ) @ ( one_one @ real ) ) ) ) ).
% cos_double_less_one
thf(fact_3235_cos__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_gt_zero
thf(fact_3236_sin__monotone__2pi__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X ) ) ) ) ) ).
% sin_monotone_2pi_le
thf(fact_3237_sin__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_le_eq
thf(fact_3238_sin__inj__pi,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( sin @ real @ X )
= ( sin @ real @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% sin_inj_pi
thf(fact_3239_sin__le__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ pi @ X )
=> ( ( ord_less @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_le_zero
thf(fact_3240_sin__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_less_zero
thf(fact_3241_sin__monotone__2pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_3242_sin__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_3243_sin__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X3 )
= Y )
& ! [Y5: real] :
( ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y5 )
& ( ord_less_eq @ real @ Y5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ Y5 )
= Y ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% sin_total
thf(fact_3244_cos__gt__zero__pi,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_gt_zero_pi
thf(fact_3245_cos__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_ge_zero
thf(fact_3246_sin__pi__divide__n__gt__0,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% sin_pi_divide_n_gt_0
thf(fact_3247_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_3248_arcsin__bounded,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_bounded
thf(fact_3249_arcsin__ubound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arcsin_ubound
thf(fact_3250_tan__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( divide_divide @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( tan @ A @ X ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_3251_sin__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( sin @ real @ X )
= ( divide_divide @ real @ ( tan @ real @ X ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_tan
thf(fact_3252_cos__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( cos @ real @ X )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_tan
thf(fact_3253_complex__unimodular__polar,axiom,
! [Z: complex] :
( ( ( real_V7770717601297561774m_norm @ complex @ Z )
= ( one_one @ real ) )
=> ~ ! [T6: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T6 )
=> ( ( ord_less @ real @ T6 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( Z
!= ( complex2 @ ( cos @ real @ T6 ) @ ( sin @ real @ T6 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_3254_sin__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ Y ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arccos_abs
thf(fact_3255_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_3256_cos__arccos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y ) )
= Y ) ) ) ).
% cos_arccos
thf(fact_3257_arccos__le__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_le_arccos
thf(fact_3258_arccos__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less_eq @ real @ Y @ X ) ) ) ) ).
% arccos_le_mono
thf(fact_3259_arccos__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
& ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) ) )
=> ( ( ( arccos @ X )
= ( arccos @ Y ) )
= ( X = Y ) ) ) ).
% arccos_eq_iff
thf(fact_3260_arccos__lbound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) ) ) ) ).
% arccos_lbound
thf(fact_3261_arccos__less__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_3262_arccos__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ).
% arccos_less_mono
thf(fact_3263_arccos__ubound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y ) @ pi ) ) ) ).
% arccos_ubound
thf(fact_3264_arccos__cos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( arccos @ ( cos @ real @ X ) )
= X ) ) ) ).
% arccos_cos
thf(fact_3265_cos__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y ) )
= Y ) ) ).
% cos_arccos_abs
thf(fact_3266_arccos__cos__eq__abs,axiom,
! [Theta: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Theta ) @ pi )
=> ( ( arccos @ ( cos @ real @ Theta ) )
= ( abs_abs @ real @ Theta ) ) ) ).
% arccos_cos_eq_abs
thf(fact_3267_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_3268_arccos__bounded,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) )
& ( ord_less_eq @ real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_bounded
thf(fact_3269_sin__arccos__nonzero,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X ) )
!= ( zero_zero @ real ) ) ) ) ).
% sin_arccos_nonzero
thf(fact_3270_arccos__cos2,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( arccos @ ( cos @ real @ X ) )
= ( uminus_uminus @ real @ X ) ) ) ) ).
% arccos_cos2
thf(fact_3271_arccos__minus,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X ) )
= ( minus_minus @ real @ pi @ ( arccos @ X ) ) ) ) ) ).
% arccos_minus
thf(fact_3272_tan__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( tan @ real @ X ) ) ) ) ).
% tan_gt_zero
thf(fact_3273_lemma__tan__total,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ? [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 @ Y @ ( tan @ real @ X3 ) ) ) ) ).
% lemma_tan_total
thf(fact_3274_lemma__tan__total1,axiom,
! [Y: real] :
? [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y ) ) ).
% lemma_tan_total1
thf(fact_3275_tan__mono__lt__eq,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_3276_tan__monotone_H,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y @ X )
= ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_3277_tan__monotone,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X ) ) ) ) ) ).
% tan_monotone
thf(fact_3278_tan__total,axiom,
! [Y: real] :
? [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= 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 = X3 ) ) ) ).
% tan_total
thf(fact_3279_arccos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) )
& ( ord_less_eq @ real @ ( arccos @ Y ) @ pi )
& ( ( cos @ real @ ( arccos @ Y ) )
= Y ) ) ) ) ).
% arccos
thf(fact_3280_arccos__minus__abs,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X ) )
= ( minus_minus @ real @ pi @ ( arccos @ X ) ) ) ) ).
% arccos_minus_abs
thf(fact_3281_add__tan__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) )
= ( divide_divide @ A @ ( sin @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_3282_tan__total__pos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y ) ) ) ).
% tan_total_pos
thf(fact_3283_tan__pos__pi2__le,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tan @ real @ X ) ) ) ) ).
% tan_pos_pi2_le
thf(fact_3284_tan__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( tan @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% tan_less_zero
thf(fact_3285_tan__mono__le,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) ) ) ) ) ).
% tan_mono_le
thf(fact_3286_tan__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_3287_tan__bound__pi2,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( abs_abs @ real @ ( tan @ real @ X ) ) @ ( one_one @ real ) ) ) ).
% tan_bound_pi2
thf(fact_3288_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_3289_arctan__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arctan @ ( tan @ real @ X ) )
= X ) ) ) ).
% arctan_tan
thf(fact_3290_arctan__unique,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( tan @ real @ X )
= Y )
=> ( ( arctan @ Y )
= X ) ) ) ) ).
% arctan_unique
thf(fact_3291_lemma__tan__add1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) )
= ( divide_divide @ A @ ( cos @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_3292_tan__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( minus_minus @ A @ X @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( minus_minus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_3293_tan__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( plus_plus @ A @ X @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( plus_plus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_3294_tan__total__pi4,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ? [Z3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) @ Z3 )
& ( ord_less @ real @ Z3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
& ( ( tan @ real @ Z3 )
= X ) ) ) ).
% tan_total_pi4
thf(fact_3295_arccos__le__pi2,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arccos_le_pi2
thf(fact_3296_tan__sec,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ ( inverse_inverse @ A @ ( cos @ A @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tan_sec
thf(fact_3297_sin__arccos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_arccos
thf(fact_3298_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_3299_complete__linorder__sup__max,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% complete_linorder_sup_max
thf(fact_3300_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_3301_push__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% push_bit_of_Suc_0
thf(fact_3302_set__removeAll,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( set2 @ A @ ( removeAll @ A @ X @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_removeAll
thf(fact_3303_push__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% push_bit_of_0
thf(fact_3304_push__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A3: A] :
( ( ( bit_se4730199178511100633sh_bit @ A @ N @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% push_bit_eq_0_iff
thf(fact_3305_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_3306_of__real__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real] :
( ( ( real_Vector_of_real @ A @ X )
= ( zero_zero @ A ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% of_real_eq_0_iff
thf(fact_3307_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_3308_even__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) )
= ( ( N
!= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_push_bit_iff
thf(fact_3309_length__removeAll__less__eq,axiom,
! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_3310_bit__push__bit__iff__nat,axiom,
! [M: nat,Q5: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M @ Q5 ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ Q5 @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_3311_norm__less__p1,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ ( real_V7770717601297561774m_norm @ A @ X ) ) @ ( one_one @ A ) ) ) ) ) ).
% norm_less_p1
thf(fact_3312_length__removeAll__less,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_3313_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N2: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_3314_norm__of__real__diff,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [B3: real,A3: real] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( real_Vector_of_real @ A @ B3 ) @ ( real_Vector_of_real @ A @ A3 ) ) ) @ ( abs_abs @ real @ ( minus_minus @ real @ B3 @ A3 ) ) ) ) ).
% norm_of_real_diff
thf(fact_3315_bit__horner__sum__bit__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Bs: list @ $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( list @ $o ) @ Bs ) )
& ( nth @ $o @ Bs @ N ) ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_3316_inthall,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,N: nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_3317_both__member__options__ding,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_ding
thf(fact_3318_sub__num__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [L: num] :
( ( neg_numeral_sub @ A @ one2 @ ( bit1 @ L ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ L ) ) ) ) ) ).
% sub_num_simps(3)
thf(fact_3319_deg__deg__n,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( Deg = N ) ) ).
% deg_deg_n
thf(fact_3320_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
=> ? [Info2: option @ ( product_prod @ nat @ nat ),TreeList2: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S4 ) ) ) ).
% deg_SUcn_Node
thf(fact_3321_push__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% push_bit_nonnegative_int_iff
thf(fact_3322_push__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% push_bit_negative_int_iff
thf(fact_3323_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_3324_diff__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ M @ N ) ) ) ).
% diff_numeral_simps(1)
thf(fact_3325_sub__num__simps_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(6)
thf(fact_3326_sub__num__simps_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(9)
thf(fact_3327_add__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ M @ N ) ) ) ).
% add_neg_numeral_simps(1)
thf(fact_3328_add__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ N @ M ) ) ) ).
% add_neg_numeral_simps(2)
thf(fact_3329_diff__numeral__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ M ) ) ) ).
% diff_numeral_simps(4)
thf(fact_3330_sub__num__simps_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl_inc @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(8)
thf(fact_3331_sub__num__simps_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl_dec @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(7)
thf(fact_3332_diff__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ M @ one2 ) ) ) ).
% diff_numeral_special(2)
thf(fact_3333_diff__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ one2 @ N ) ) ) ).
% diff_numeral_special(1)
thf(fact_3334_sub__num__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ one2 )
= ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% sub_num_simps(5)
thf(fact_3335_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_3336_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_3337_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_3338_add__neg__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% add_neg_numeral_special(4)
thf(fact_3339_minus__sub__one__diff__one,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( neg_numeral_sub @ A @ M @ one2 ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% minus_sub_one_diff_one
thf(fact_3340_diff__numeral__special_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% diff_numeral_special(7)
thf(fact_3341_diff__numeral__special_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% diff_numeral_special(8)
thf(fact_3342_VEBT_Odistinct_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,X21: $o,X22: $o] :
( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leaf @ X21 @ X22 ) ) ).
% VEBT.distinct(1)
thf(fact_3343_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,X222: $o] :
( Y
!= ( vEBT_Leaf @ X212 @ X222 ) ) ) ).
% VEBT.exhaust
thf(fact_3344_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y4: list @ A,Z2: list @ A] : Y4 = Z2 )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( nth @ A @ Xs3 @ I4 )
= ( nth @ A @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_3345_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ? [X8: A] : ( P @ I4 @ X8 ) ) )
= ( ? [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ( P @ I4 @ ( nth @ A @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_3346_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_3347_vebt__insert_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S3 ) @ X )
= ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S3 ) ) ).
% vebt_insert.simps(2)
thf(fact_3348_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_3349_neg__numeral__class_Osub__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_sub @ A )
= ( ^ [K3: num,L2: num] : ( minus_minus @ A @ ( numeral_numeral @ A @ K3 ) @ ( numeral_numeral @ A @ L2 ) ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_3350_nth__mem,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ ( nth @ A @ Xs @ N ) @ ( set2 @ A @ Xs ) ) ) ).
% nth_mem
thf(fact_3351_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_3352_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_3353_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,X: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I2 ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_3354_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_3355_bit__push__bit__iff__int,axiom,
! [M: nat,K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M @ K ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_3356_vebt__insert_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) @ X )
= ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) ) ).
% vebt_insert.simps(3)
thf(fact_3357_sub__non__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M ) )
= ( ord_less_eq @ num @ M @ N ) ) ) ).
% sub_non_negative
thf(fact_3358_sub__non__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M: num] :
( ( ord_less_eq @ A @ ( neg_numeral_sub @ A @ N @ M ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ num @ N @ M ) ) ) ).
% sub_non_positive
thf(fact_3359_sub__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M ) )
= ( ord_less @ num @ M @ N ) ) ) ).
% sub_positive
thf(fact_3360_sub__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M: num] :
( ( ord_less @ A @ ( neg_numeral_sub @ A @ N @ M ) @ ( zero_zero @ A ) )
= ( ord_less @ num @ N @ M ) ) ) ).
% sub_negative
thf(fact_3361_nth__rotate1,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate1 @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_3362_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N2: nat,TreeList3: list @ vEBT_VEBT,X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X2 @ N2 ) ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) ) ) ).
% in_children_def
thf(fact_3363_invar__vebt_Ointros_I3_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_3364_invar__vebt_Ointros_I2_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_3365_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_3366_member__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
& ( ( X = Mi )
| ( X = Ma )
| ( ( ord_less @ nat @ X @ Ma )
& ( ord_less @ nat @ Mi @ X )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_3367_mi__eq__ma__no__ch,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
=> ( ( Mi = Ma )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
& ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_3368_insert__simp__mima,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
| ( X = Ma ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_3369_mi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
& ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_3370_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
| ( X = Mi )
| ( X = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_3371_VEBT__internal_OminNull_Ocases,axiom,
! [X: vEBT_VEBT] :
( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy ) )
=> ~ ! [Uz: product_prod @ nat @ nat,Va2: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va2 @ Vb @ Vc ) ) ) ) ) ) ).
% VEBT_internal.minNull.cases
thf(fact_3372_vebt__insert_Osimps_I4_J,axiom,
! [V2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ X ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary ) ) ).
% vebt_insert.simps(4)
thf(fact_3373_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ S3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ).
% subrelI
thf(fact_3374_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S3: B,R: set @ ( product_prod @ A @ B ),S5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R )
=> ( ( S5 = S3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S5 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_3375_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ X )
= ( ( X = Mi )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_3376_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y )
=> ( ( ? [Uv2: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> Y )
=> ( ( ? [Uu2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> Y )
=> ( ( ? [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy ) )
=> ~ Y )
=> ~ ( ? [Uz: product_prod @ nat @ nat,Va2: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va2 @ Vb @ Vc ) )
=> Y ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_3377_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ).
% VEBT_internal.minNull.simps(5)
thf(fact_3378_vebt__member_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X ) ).
% vebt_member.simps(2)
thf(fact_3379_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw: nat,Ux: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy2 ) ) ).
% VEBT_internal.minNull.simps(4)
thf(fact_3380_vebt__member_Osimps_I3_J,axiom,
! [V2: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ X ) ).
% vebt_member.simps(3)
thf(fact_3381_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ! [Uz: product_prod @ nat @ nat,Va2: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va2 @ Vb @ Vc ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_3382_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux: list @ vEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy2 ) @ Uz2 ) ).
% VEBT_internal.membermima.simps(2)
thf(fact_3383_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_3384_vebt__member_Osimps_I4_J,axiom,
! [V2: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ X ) ).
% vebt_member.simps(4)
thf(fact_3385_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A12: vEBT_VEBT,A23: nat] :
( ( ? [A5: $o,B5: $o] :
( A12
= ( vEBT_Leaf @ A5 @ B5 ) )
& ( A23
= ( suc @ ( zero_zero @ nat ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A23 @ TreeList3 @ Summary2 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary2 @ N2 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
& ( A23
= ( plus_plus @ nat @ N2 @ N2 ) )
& ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A23 @ TreeList3 @ Summary2 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary2 @ ( suc @ N2 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
& ( A23
= ( plus_plus @ nat @ N2 @ ( suc @ N2 ) ) )
& ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ A23 @ TreeList3 @ Summary2 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary2 @ N2 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
& ( A23
= ( plus_plus @ nat @ N2 @ N2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi2 != Ma2 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ A23 @ TreeList3 @ Summary2 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary2 @ ( suc @ N2 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
& ( A23
= ( plus_plus @ nat @ N2 @ ( suc @ N2 ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi2 != Ma2 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_3386_invar__vebt_Ocases,axiom,
! [A13: vEBT_VEBT,A24: nat] :
( ( vEBT_invar_vebt @ A13 @ A24 )
=> ( ( ? [A7: $o,B7: $o] :
( A13
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ( A24
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M4: nat,Deg2: nat] :
( ( A13
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( A24 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary3 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4 = N3 )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M4 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_12 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M4: nat,Deg2: nat] :
( ( A13
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( A24 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary3 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M4 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_12 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M4: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
( ( A13
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( A24 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary3 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4 = N3 )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M4 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
=> ( ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
=> ( ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N3 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList2: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M4: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
( ( A13
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( A24 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary3 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M4 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
=> ( ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
=> ( ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N3 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_3387_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_3388_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_3389_insert__simp__norm,axiom,
! [X: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less @ nat @ Mi @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( ord_max @ nat @ X @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_norm
thf(fact_3390_insert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList: list @ vEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less @ nat @ X @ Mi )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ ( ord_max @ nat @ Mi @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_excp
thf(fact_3391_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R2: A,Q5: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q5 @ R2 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( 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 @ Q5 @ R2 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R2 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_3392_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q5: A,R2: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q5 @ R2 ) )
= ( R2
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_3393_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys2: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys2 ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) @ ( nth @ B @ Ys2 @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_3394_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( list_update @ A @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_3395_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_3396_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_3397_less__by__empty,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A )] :
( ( A4
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B2 ) ) ).
% less_by_empty
thf(fact_3398_set__update__subsetI,axiom,
! [A: $tType,Xs: list @ A,A4: set @ A,X: A,I: nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_3399_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,D6: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D6 ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT,Deg3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) @ Deg3 ) ) ) ).
% VEBT_internal.valid'.cases
thf(fact_3400_set__update__subset__insert,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X ) ) @ ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_3401_set__update__memI,axiom,
! [A: $tType,N: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_3402_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I @ X )
= Xs )
= ( ( nth @ A @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_3403_nth__list__update,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_3404_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A7: $o,B7: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A7 @ B7 ) @ X3 ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Ux2 ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S4: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S4 ) @ X3 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_3405_vebt__insert_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A7: $o,B7: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A7 @ B7 ) @ X3 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S4: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S4 ) @ X3 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S4: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S4 ) @ X3 ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary3 ) @ X3 ) )
=> ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) @ X3 ) ) ) ) ) ) ).
% vebt_insert.cases
thf(fact_3406_VEBT__internal_Omembermima_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy ) @ Uz ) )
=> ( ! [Mi3: nat,Ma3: nat,Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) @ X3 ) )
=> ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ X3 ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ X3 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_3407_vebt__member_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A7: $o,B7: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A7 @ B7 ) @ X3 ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X3 ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X3 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X3 ) )
=> ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) @ X3 ) ) ) ) ) ) ).
% vebt_member.cases
thf(fact_3408_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( ( ord_less @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_3409_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( ( ord_less_eq @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_3410_neg__eucl__rel__int__mult__2,axiom,
! [B3: int,A3: int,Q5: int,R2: int] :
( ( ord_less_eq @ int @ B3 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A3 @ ( one_one @ int ) ) @ B3 @ ( product_Pair @ int @ int @ Q5 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( product_Pair @ int @ int @ Q5 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_3411_pos__eucl__rel__int__mult__2,axiom,
! [B3: int,A3: int,Q5: int,R2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( eucl_rel_int @ A3 @ B3 @ ( product_Pair @ int @ int @ Q5 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( product_Pair @ int @ int @ Q5 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_3412_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_3413_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_3414_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_3415_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_3416_option_Osize__neq,axiom,
! [A: $tType,X: option @ A] :
( ( size_size @ ( option @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_3417_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q5: int,R2: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q5 @ R2 ) )
= ( ( K
= ( plus_plus @ int @ ( times_times @ int @ L @ Q5 ) @ 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 ) )
=> ( Q5
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_3418_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K: int,Q5: 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 @ Q5 @ L ) @ R2 ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q5 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_3419_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A32: product_prod @ int @ int] :
( ? [K3: int] :
( ( A12 = K3 )
& ( A23
= ( zero_zero @ int ) )
& ( A32
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
| ? [L2: int,K3: int,Q6: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q6 @ ( zero_zero @ int ) ) )
& ( L2
!= ( zero_zero @ int ) )
& ( K3
= ( times_times @ int @ Q6 @ L2 ) ) )
| ? [R5: int,L2: int,K3: int,Q6: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q6 @ 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 @ Q6 @ L2 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_3420_eucl__rel__int_Ocases,axiom,
! [A13: int,A24: int,A33: product_prod @ int @ int] :
( ( eucl_rel_int @ A13 @ A24 @ A33 )
=> ( ( ( A24
= ( zero_zero @ int ) )
=> ( A33
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A13 ) ) )
=> ( ! [Q3: int] :
( ( A33
= ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) )
=> ( ( A24
!= ( zero_zero @ int ) )
=> ( A13
!= ( times_times @ int @ Q3 @ A24 ) ) ) )
=> ~ ! [R3: int,Q3: int] :
( ( A33
= ( product_Pair @ int @ int @ Q3 @ R3 ) )
=> ( ( ( sgn_sgn @ int @ R3 )
= ( sgn_sgn @ int @ A24 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ A24 ) )
=> ( A13
!= ( plus_plus @ int @ ( times_times @ int @ Q3 @ A24 ) @ R3 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_3421_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M2: num,N2: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M2 @ N2 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M2 ) ) @ ( unique1321980374590559556d_step @ A @ N2 @ ( unique8689654367752047608divmod @ A @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_3422_option_Osize_I4_J,axiom,
! [A: $tType,X23: A] :
( ( size_size @ ( option @ A ) @ ( some @ A @ X23 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(4)
thf(fact_3423_option_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X23: A] :
( ( size_option @ A @ X @ ( some @ A @ X23 ) )
= ( plus_plus @ nat @ ( X @ X23 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_3424_vebt__insert_Oelims,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa3 )
= Y )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ~ ( ( ( Xa3
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B7 ) ) )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( ( ( Xa3
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A7 @ $true ) ) )
& ( ( Xa3
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A7 @ B7 ) ) ) ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S4 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S4 ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S4 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S4 ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary3 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa3 @ Xa3 ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary3 ) ) )
=> ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) )
=> ( Y
!= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Mi3 @ Xa3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa3 = Mi3 )
| ( Xa3 = Ma3 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Xa3 @ Mi3 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Mi3 @ Xa3 ) @ Ma3 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Mi3 @ Xa3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Mi3 @ Xa3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Mi3 @ Xa3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Mi3 @ Xa3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Mi3 @ Xa3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary3 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_3425_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa3 )
= Y )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ( Y
= ( ~ ( ( ( Xa3
= ( zero_zero @ nat ) )
=> A7 )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( ( ( Xa3
= ( one_one @ nat ) )
=> B7 )
& ( Xa3
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> Y )
=> ( ( ? [V3: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy @ Uz ) )
=> Y )
=> ( ( ? [V3: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) )
=> Y )
=> ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary3: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) )
=> ( Y
= ( ~ ( ( Xa3 != Mi3 )
=> ( ( Xa3 != Ma3 )
=> ( ~ ( ord_less @ nat @ Xa3 @ Mi3 )
& ( ~ ( ord_less @ nat @ Xa3 @ Mi3 )
=> ( ~ ( ord_less @ nat @ Ma3 @ Xa3 )
& ( ~ ( ord_less @ nat @ Ma3 @ Xa3 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( 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 @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_3426_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa3 )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ( ( ( Xa3
= ( zero_zero @ nat ) )
=> A7 )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( ( ( Xa3
= ( one_one @ nat ) )
=> B7 )
& ( Xa3
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy @ Uz ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) )
=> ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary3: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) )
=> ( ( Xa3 != Mi3 )
=> ( ( Xa3 != Ma3 )
=> ( ~ ( ord_less @ nat @ Xa3 @ Mi3 )
& ( ~ ( ord_less @ nat @ Xa3 @ Mi3 )
=> ( ~ ( ord_less @ nat @ Ma3 @ Xa3 )
& ( ~ ( ord_less @ nat @ Ma3 @ Xa3 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( 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 @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_3427_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va3: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_3428_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% set_vebt'_def
thf(fact_3429_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3430_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
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3431_finite__Collect__subsets,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B6 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_3432_finite__nth__roots,axiom,
! [N: nat,C3: complex] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= C3 ) ) ) ) ).
% finite_nth_roots
thf(fact_3433_singleton__conv,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ^ [X2: A] : X2 = A3 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_3434_singleton__conv2,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ( ^ [Y4: A,Z2: A] : Y4 = Z2
@ A3 ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_3435_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less @ nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_3436_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_3437_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_3438_finite__interval__int1,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less_eq @ int @ A3 @ I4 )
& ( ord_less_eq @ int @ I4 @ B3 ) ) ) ) ).
% finite_interval_int1
thf(fact_3439_finite__interval__int4,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less @ int @ A3 @ I4 )
& ( ord_less @ int @ I4 @ B3 ) ) ) ) ).
% finite_interval_int4
thf(fact_3440_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less_eq @ nat @ I4 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_3441_finite__interval__int3,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less @ int @ A3 @ I4 )
& ( ord_less_eq @ int @ I4 @ B3 ) ) ) ) ).
% finite_interval_int3
thf(fact_3442_finite__interval__int2,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less_eq @ int @ A3 @ I4 )
& ( ord_less @ int @ I4 @ B3 ) ) ) ) ).
% finite_interval_int2
thf(fact_3443_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_3444_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_3445_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_3446_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_3447_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_3448_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( X2 = A3 )
& ( P @ X2 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( X2 = A3 )
& ( P @ X2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_3449_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( A3 = X2 )
& ( P @ X2 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( A3 = X2 )
& ( P @ X2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_3450_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X2: A] : $false ) ) ).
% empty_def
thf(fact_3451_Collect__imp__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_3452_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_3453_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A6 )
@ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_3454_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A6 )
| ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_3455_sup__Un__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R )
@ ^ [X2: A] : ( member @ A @ X2 @ S ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_3456_insert__def,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A5: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] : X2 = A5 ) ) ) ) ).
% insert_def
thf(fact_3457_insert__Collect,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( insert2 @ A @ A3 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U2: A] :
( ( U2 != A3 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_3458_insert__compr,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A5: A,B6: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( X2 = A5 )
| ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_3459_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H2: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_3460_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= ( one_one @ complex ) ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_3461_card__nth__roots,axiom,
! [C3: complex,N: nat] :
( ( C3
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= C3 ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_3462_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A6 )
& ~ ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_3463_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A6 )
@ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_3464_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,R: A > B > $o] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ? [Xa: B] :
( ( member @ B @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ B2 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A4 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_3465_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_3466_inf__Int__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( inf_inf @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R )
@ ^ [X2: A] : ( member @ A @ X2 @ S ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( inf_inf @ ( set @ A ) @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_3467_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A6 )
& ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_3468_Int__Collect,axiom,
! [A: $tType,X: A,A4: set @ A,P: A > $o] :
( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X @ A4 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_3469_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A6 )
@ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_3470_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_3471_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A6: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A6 ) ) ) ) ) ).
% uminus_set_def
thf(fact_3472_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_3473_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A6: set @ A] :
( collect @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ A6 ) ) ) ) ).
% Compl_eq
thf(fact_3474_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_3475_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A6 )
@ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_3476_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ A3 ) )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ B3 ) ) )
= ( ( dvd_dvd @ A @ A3 @ B3 )
& ~ ( dvd_dvd @ A @ B3 @ A3 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_3477_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ A3 ) )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ B3 ) ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ).
% subset_divisors_dvd
thf(fact_3478_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A6 )
@ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_3479_Collect__restrict,axiom,
! [A: $tType,X4: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ X4 )
& ( P @ X2 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_3480_prop__restrict,axiom,
! [A: $tType,X: A,Z7: set @ A,X4: set @ A,P: A > $o] :
( ( member @ A @ X @ Z7 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z7
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ X4 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_3481_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R )
@ ^ [X2: A] : ( member @ A @ X2 @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% pred_subset_eq
thf(fact_3482_Collect__subset,axiom,
! [A: $tType,A4: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( P @ X2 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_3483_finite__less__ub,axiom,
! [F3: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F3 @ N3 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ ( F3 @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_3484_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B5 ) @ B5 @ A5 ) ) ) ) ).
% max_def_raw
thf(fact_3485_numeral__code_I2_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit0 @ N ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_code(2)
thf(fact_3486_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% set_vebt_def
thf(fact_3487_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A3 @ X2 )
& ( ord_less_eq @ A @ X2 @ B3 ) ) ) ) ) ).
% finite_int_segment
thf(fact_3488_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_less_as_int
thf(fact_3489_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_leq_as_int
thf(fact_3490_numeral__code_I3_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit1 @ N ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_code(3)
thf(fact_3491_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [K3: A] :
( ( member @ A @ K3 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( abs_abs @ A @ K3 ) @ A3 ) ) ) ) ) ).
% finite_abs_int_segment
thf(fact_3492_n__subsets,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
& ( ( finite_card @ A @ B6 )
= K ) ) ) )
= ( binomial @ ( finite_card @ A @ A4 ) @ K ) ) ) ).
% n_subsets
thf(fact_3493_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_3494_card__less,axiom,
! [M5: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M5 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_3495_card__less__Suc,axiom,
! [M5: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M5 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M5 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_3496_card__less__Suc2,axiom,
! [M5: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M5 )
& ( ord_less @ nat @ K3 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M5 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_3497_finite__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( power_power @ A @ Z6 @ N )
= ( one_one @ A ) ) ) ) ) ) ).
% finite_roots_unity
thf(fact_3498_card__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( power_power @ A @ Z6 @ N )
= ( one_one @ A ) ) ) )
@ N ) ) ) ).
% card_roots_unity
thf(fact_3499_finite__lists__length__eq,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3500_card__lists__length__eq,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A4 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_3501_diff__nat__eq__if,axiom,
! [Z5: int,Z: int] :
( ( ( ord_less @ int @ Z5 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less @ int @ Z5 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z @ Z5 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z @ Z5 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_3502_finite__lists__length__le,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3503_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] :
( if @ A
@ ( K3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L2: nat] : ( times_times @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ L2 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_3504_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy2: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList: list @ vEBT_VEBT,S3: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_3505_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V2: nat,TreeList: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_3506_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va3: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList @ Summary ) @ X )
= ( ( X != Mi )
=> ( ( X != Ma )
=> ( ~ ( ord_less @ nat @ X @ Mi )
& ( ~ ( ord_less @ nat @ X @ Mi )
=> ( ~ ( ord_less @ nat @ Ma @ X )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_3507_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V2: nat,TreeList: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ X )
= ( ( X = Mi )
| ( X = Ma )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_3508_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa3 )
= Y )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ( Y
= ( ~ ( ( ( Xa3
= ( zero_zero @ nat ) )
=> A7 )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( ( ( Xa3
= ( one_one @ nat ) )
=> B7 )
& ( Xa3
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> Y )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S4: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S4 ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_3509_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa3 )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ~ ( ( ( Xa3
= ( zero_zero @ nat ) )
=> A7 )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( ( ( Xa3
= ( one_one @ nat ) )
=> B7 )
& ( Xa3
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S4: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S4 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_3510_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa3 )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ( ( ( Xa3
= ( zero_zero @ nat ) )
=> A7 )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( ( ( Xa3
= ( one_one @ nat ) )
=> B7 )
& ( Xa3
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S4: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S4 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_3511_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa3 )
=> ( ! [Mi3: nat,Ma3: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) )
=> ~ ( ( Xa3 = Mi3 )
| ( Xa3 = Ma3 ) ) )
=> ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ~ ( ( Xa3 = Mi3 )
| ( Xa3 = Ma3 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_3512_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_vebt_member @ X @ Xa3 )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ~ ( ( ( Xa3
= ( zero_zero @ nat ) )
=> A7 )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( ( ( Xa3
= ( one_one @ nat ) )
=> B7 )
& ( Xa3
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary3: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) )
=> ~ ( ( Xa3 != Mi3 )
=> ( ( Xa3 != Ma3 )
=> ( ~ ( ord_less @ nat @ Xa3 @ Mi3 )
& ( ~ ( ord_less @ nat @ Xa3 @ Mi3 )
=> ( ~ ( ord_less @ nat @ Ma3 @ Xa3 )
& ( ~ ( ord_less @ nat @ Ma3 @ Xa3 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( 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 @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_3513_option_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_option @ A @ X @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size_gen(1)
thf(fact_3514_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa3 )
=> ( ! [Uu2: $o,Uv2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy ) )
=> ( ! [Mi3: nat,Ma3: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) )
=> ( ( Xa3 = Mi3 )
| ( Xa3 = Ma3 ) ) )
=> ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ( ( Xa3 = Mi3 )
| ( Xa3 = Ma3 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_3515_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa3 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> Y )
=> ( ( ? [Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy ) )
=> Y )
=> ( ! [Mi3: nat,Ma3: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) )
=> ( Y
= ( ~ ( ( Xa3 = Mi3 )
| ( Xa3 = Ma3 ) ) ) ) )
=> ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ( Y
= ( ~ ( ( Xa3 = Mi3 )
| ( Xa3 = Ma3 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_3516_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_3517_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( topological_monoseq @ real
@ ^ [N2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_3518_vebt__insert_Opelims,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa3 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa3 ) )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ( ( ( ( Xa3
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B7 ) ) )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( ( ( Xa3
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A7 @ $true ) ) )
& ( ( Xa3
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A7 @ B7 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A7 @ B7 ) @ Xa3 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S4 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S4 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S4 ) @ Xa3 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S4 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S4 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S4 ) @ Xa3 ) ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary3 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa3 @ Xa3 ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary3 ) @ Xa3 ) ) ) )
=> ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) )
=> ( ( Y
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Mi3 @ Xa3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa3 = Mi3 )
| ( Xa3 = Ma3 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Xa3 @ Mi3 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Mi3 @ Xa3 ) @ Ma3 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Mi3 @ Xa3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Mi3 @ Xa3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Mi3 @ Xa3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Mi3 @ Xa3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa3 @ Mi3 ) @ Mi3 @ Xa3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary3 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_3519_ln__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ( ln_ln @ real @ X )
= ( suminf @ real
@ ^ [N2: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ ( one_one @ real ) ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_3520_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( arctan @ X )
= ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_3521_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X3: A,Y2: B] :
( ( P @ X3 @ Y2 )
=> ( Q @ X3 @ Y2 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_3522_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_3523_inf1I,axiom,
! [A: $tType,A4: A > $o,X: A,B2: A > $o] :
( ( A4 @ X )
=> ( ( B2 @ X )
=> ( inf_inf @ ( A > $o ) @ A4 @ B2 @ X ) ) ) ).
% inf1I
thf(fact_3524_inf2I,axiom,
! [A: $tType,B: $tType,A4: A > B > $o,X: A,Y: B,B2: A > B > $o] :
( ( A4 @ X @ Y )
=> ( ( B2 @ X @ Y )
=> ( inf_inf @ ( A > B > $o ) @ A4 @ B2 @ X @ Y ) ) ) ).
% inf2I
thf(fact_3525_powser__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F3: nat > A] :
( ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) ) )
= ( F3 @ ( zero_zero @ nat ) ) ) ) ).
% powser_zero
thf(fact_3526_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_3527_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_3528_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_3529_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_3530_inf2D2,axiom,
! [A: $tType,B: $tType,A4: A > B > $o,B2: A > B > $o,X: A,Y: B] :
( ( inf_inf @ ( A > B > $o ) @ A4 @ B2 @ X @ Y )
=> ( B2 @ X @ Y ) ) ).
% inf2D2
thf(fact_3531_inf2D1,axiom,
! [A: $tType,B: $tType,A4: A > B > $o,B2: A > B > $o,X: A,Y: B] :
( ( inf_inf @ ( A > B > $o ) @ A4 @ B2 @ X @ Y )
=> ( A4 @ X @ Y ) ) ).
% inf2D1
thf(fact_3532_inf1D2,axiom,
! [A: $tType,A4: A > $o,B2: A > $o,X: A] :
( ( inf_inf @ ( A > $o ) @ A4 @ B2 @ X )
=> ( B2 @ X ) ) ).
% inf1D2
thf(fact_3533_inf1D1,axiom,
! [A: $tType,A4: A > $o,B2: A > $o,X: A] :
( ( inf_inf @ ( A > $o ) @ A4 @ B2 @ X )
=> ( A4 @ X ) ) ).
% inf1D1
thf(fact_3534_inf2E,axiom,
! [A: $tType,B: $tType,A4: A > B > $o,B2: A > B > $o,X: A,Y: B] :
( ( inf_inf @ ( A > B > $o ) @ A4 @ B2 @ X @ Y )
=> ~ ( ( A4 @ X @ Y )
=> ~ ( B2 @ X @ Y ) ) ) ).
% inf2E
thf(fact_3535_inf1E,axiom,
! [A: $tType,A4: A > $o,B2: A > $o,X: A] :
( ( inf_inf @ ( A > $o ) @ A4 @ B2 @ X )
=> ~ ( ( A4 @ X )
=> ~ ( B2 @ X ) ) ) ).
% inf1E
thf(fact_3536_monoseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( topological_monoseq @ real @ ( power_power @ real @ X ) ) ) ) ).
% monoseq_realpow
thf(fact_3537_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa3 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa3 ) )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ( ( Y
= ( ( ( Xa3
= ( zero_zero @ nat ) )
=> A7 )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( ( ( Xa3
= ( one_one @ nat ) )
=> B7 )
& ( Xa3
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A7 @ B7 ) @ Xa3 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa3 ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy @ Uz ) )
=> ( ~ 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 ) @ Uy @ Uz ) @ Xa3 ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) )
=> ( ~ 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 ) ) @ Vb @ Vc ) @ Xa3 ) ) ) )
=> ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) )
=> ( ( Y
= ( ( Xa3 != Mi3 )
=> ( ( Xa3 != Ma3 )
=> ( ~ ( ord_less @ nat @ Xa3 @ Mi3 )
& ( ~ ( ord_less @ nat @ Xa3 @ Mi3 )
=> ( ~ ( ord_less @ nat @ Ma3 @ Xa3 )
& ( ~ ( ord_less @ nat @ Ma3 @ Xa3 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( 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 @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_3538_vebt__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa3 ) )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A7 @ B7 ) @ Xa3 ) )
=> ( ( ( Xa3
= ( zero_zero @ nat ) )
=> A7 )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( ( ( Xa3
= ( one_one @ nat ) )
=> B7 )
& ( Xa3
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa3 ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy @ Uz ) )
=> ~ ( 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 ) @ Uy @ Uz ) @ Xa3 ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) )
=> ~ ( 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 ) ) @ Vb @ Vc ) @ Xa3 ) ) )
=> ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) )
=> ( ( 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 @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) @ Xa3 ) )
=> ( ( Xa3 != Mi3 )
=> ( ( Xa3 != Ma3 )
=> ( ~ ( ord_less @ nat @ Xa3 @ Mi3 )
& ( ~ ( ord_less @ nat @ Xa3 @ Mi3 )
=> ( ~ ( ord_less @ nat @ Ma3 @ Xa3 )
& ( ~ ( ord_less @ nat @ Ma3 @ Xa3 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( 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 @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_3539_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa3 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa3 ) )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ( ( Y
= ( ( ( Xa3
= ( zero_zero @ nat ) )
=> A7 )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( ( ( Xa3
= ( one_one @ nat ) )
=> B7 )
& ( Xa3
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A7 @ B7 ) @ Xa3 ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa3 ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S4 ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S4 ) @ Xa3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_3540_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa3 ) )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A7 @ B7 ) @ Xa3 ) )
=> ~ ( ( ( Xa3
= ( zero_zero @ nat ) )
=> A7 )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( ( ( Xa3
= ( one_one @ nat ) )
=> B7 )
& ( Xa3
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S4 ) @ Xa3 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_3541_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa3 ) )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A7 @ B7 ) @ Xa3 ) )
=> ( ( ( Xa3
= ( zero_zero @ nat ) )
=> A7 )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( ( ( Xa3
= ( one_one @ nat ) )
=> B7 )
& ( Xa3
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa3 ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S4 ) @ Xa3 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_3542_vebt__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_vebt_member @ X @ Xa3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa3 ) )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A7 @ B7 ) @ Xa3 ) )
=> ~ ( ( ( Xa3
= ( zero_zero @ nat ) )
=> A7 )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( ( ( Xa3
= ( one_one @ nat ) )
=> B7 )
& ( Xa3
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) )
=> ( ( 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 @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary3 ) @ Xa3 ) )
=> ~ ( ( Xa3 != Mi3 )
=> ( ( Xa3 != Ma3 )
=> ( ~ ( ord_less @ nat @ Xa3 @ Mi3 )
& ( ~ ( ord_less @ nat @ Xa3 @ Mi3 )
=> ( ~ ( ord_less @ nat @ Ma3 @ Xa3 )
& ( ~ ( ord_less @ nat @ Ma3 @ Xa3 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( 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 @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_3543_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa3 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa3 ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy ) @ Xa3 ) ) )
=> ( ! [Mi3: nat,Ma3: nat,Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) @ Xa3 ) )
=> ( ( Xa3 = Mi3 )
| ( Xa3 = Ma3 ) ) ) )
=> ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa3 ) )
=> ( ( Xa3 = Mi3 )
| ( Xa3 = Ma3 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ Xa3 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_3544_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa3 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa3 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa3 ) ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy ) )
=> ( ~ 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 @ Uy ) @ Xa3 ) ) ) )
=> ( ! [Mi3: nat,Ma3: nat,Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) )
=> ( ( Y
= ( ( Xa3 = Mi3 )
| ( Xa3 = Ma3 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) @ Xa3 ) ) ) )
=> ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ( ( Y
= ( ( Xa3 = Mi3 )
| ( Xa3 = Ma3 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa3 ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_3545_suminf__geometric,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C3 ) @ ( one_one @ real ) )
=> ( ( suminf @ A @ ( power_power @ A @ C3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C3 ) ) ) ) ) ).
% suminf_geometric
thf(fact_3546_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa3 ) )
=> ( ! [Mi3: nat,Ma3: nat,Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) @ Xa3 ) )
=> ~ ( ( Xa3 = Mi3 )
| ( Xa3 = Ma3 ) ) ) )
=> ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa3 ) )
=> ~ ( ( Xa3 = Mi3 )
| ( Xa3 = Ma3 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ Xa3 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_3547_suminf__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% suminf_zero
thf(fact_3548_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_3549_vebt__buildup_Oelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( 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 ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.elims
thf(fact_3550_accp__subset,axiom,
! [A: $tType,R1: A > A > $o,R22: A > A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ R1 @ R22 )
=> ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R22 ) @ ( accp @ A @ R1 ) ) ) ).
% accp_subset
thf(fact_3551_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M: nat,X: A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_3552_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I5 )
& ( ( X @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I5 )
& ( ( Y @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I5 )
& ( ( times_times @ A @ ( X @ I4 ) @ ( Y @ I4 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3553_intind,axiom,
! [A: $tType,I: nat,N: nat,P: A > $o,X: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( P @ X )
=> ( P @ ( nth @ A @ ( replicate @ A @ N @ X ) @ I ) ) ) ) ).
% intind
thf(fact_3554_replicate__eq__replicate,axiom,
! [A: $tType,M: nat,X: A,N: nat,Y: A] :
( ( ( replicate @ A @ M @ X )
= ( replicate @ A @ N @ Y ) )
= ( ( M = N )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_3555_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu3: B] : ( zero_zero @ A )
@ A4 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_3556_summable__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( summable @ A
@ ^ [N2: nat] : ( zero_zero @ A ) ) ) ).
% summable_zero
thf(fact_3557_summable__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F3: nat > A] :
( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) ) ) ) ).
% summable_single
thf(fact_3558_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > A] :
( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty
thf(fact_3559_sum__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [F4: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ F4 )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ F4 )
= ( zero_zero @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ F4 )
=> ( ( F3 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_3560_sum_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G2: B > A] :
( ~ ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 )
= ( zero_zero @ A ) ) ) ) ).
% sum.infinite
thf(fact_3561_in__set__replicate,axiom,
! [A: $tType,X: A,N: nat,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ Y ) ) )
= ( ( X = Y )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_3562_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
& ( P @ X2 ) ) )
= ( ( P @ A3 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_3563_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
=> ( P @ X2 ) ) )
= ( ( P @ A3 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_3564_nth__replicate,axiom,
! [A: $tType,I: nat,N: nat,X: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( replicate @ A @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_3565_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_3566_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B3 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B3 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_3567_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A4: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A4 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F3 @ I4 ) )
@ A4 ) ) ) ).
% sum_abs
thf(fact_3568_summable__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ C3 @ ( F3 @ N2 ) ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( summable @ A @ F3 ) ) ) ) ).
% summable_cmult_iff
thf(fact_3569_summable__divide__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,C3: A] :
( ( summable @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( F3 @ N2 ) @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( summable @ A @ F3 ) ) ) ) ).
% summable_divide_iff
thf(fact_3570_summable__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F3: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite
thf(fact_3571_summable__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A4: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ A4 )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A4 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite_set
thf(fact_3572_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ~ ( member @ B @ X @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( insert2 @ B @ X @ A4 ) )
= ( plus_plus @ A @ ( G2 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) ) ) ) ) ) ).
% sum.insert
thf(fact_3573_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A4: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F3 @ I4 ) )
@ A4 ) ) ) ).
% sum_abs_ge_zero
thf(fact_3574_set__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_3575_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A4: set @ B,F3: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F3 @ X2 ) @ ( zero_neq_one_of_bool @ A @ ( P @ X2 ) ) )
@ A4 )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_3576_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A4: set @ B,P: B > $o,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X2 ) ) @ ( F3 @ X2 ) )
@ A4 )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_3577_summable__geometric__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A] :
( ( summable @ A @ ( power_power @ A @ C3 ) )
= ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C3 ) @ ( one_one @ real ) ) ) ) ).
% summable_geometric_iff
thf(fact_3578_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G2: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_3579_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: set @ nat,C3: nat > A] :
( ( ( ( finite_finite2 @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A4 )
= ( C3 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_3580_sum__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A4: set @ B,P: B > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( zero_neq_one_of_bool @ A @ ( P @ X2 ) )
@ A4 )
= ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_3581_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: set @ nat,C3: nat > A,D2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D2 @ I4 ) )
@ A4 )
= ( divide_divide @ A @ ( C3 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D2 @ I4 ) )
@ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_3582_summable__const__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C3: A] :
( ( summable @ A
@ ^ [Uu3: nat] : C3 )
= ( C3
= ( zero_zero @ A ) ) ) ) ).
% summable_const_iff
thf(fact_3583_sum_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B2: set @ C,G2: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ C @ B2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ C @ A @ ( G2 @ X2 )
@ ( collect @ C
@ ^ [Y3: C] :
( ( member @ C @ Y3 @ B2 )
& ( R @ X2 @ Y3 ) ) ) )
@ A4 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y3: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( G2 @ X2 @ Y3 )
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( R @ X2 @ Y3 ) ) ) )
@ B2 ) ) ) ) ) ).
% sum.swap_restrict
thf(fact_3584_norm__sum,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,A4: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 ) )
@ ( groups7311177749621191930dd_sum @ B @ real
@ ^ [I4: B] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ I4 ) )
@ A4 ) ) ) ).
% norm_sum
thf(fact_3585_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K4: set @ B,F3: B > A,G2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ K4 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ K4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ K4 ) ) ) ) ).
% sum_mono
thf(fact_3586_sum__cong__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ nat,F3: nat > A,G2: nat > A] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ! [X3: nat] :
( ( member @ nat @ ( suc @ X3 ) @ A4 )
=> ( ( F3 @ ( suc @ X3 ) )
= ( G2 @ ( suc @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ A4 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ A4 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_3587_sum__mono__inv,axiom,
! [A: $tType,I6: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [F3: I6 > A,I5: set @ I6,G2: I6 > A,I: I6] :
( ( ( groups7311177749621191930dd_sum @ I6 @ A @ F3 @ I5 )
= ( groups7311177749621191930dd_sum @ I6 @ A @ G2 @ I5 ) )
=> ( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ( member @ I6 @ I @ I5 )
=> ( ( finite_finite2 @ I6 @ I5 )
=> ( ( F3 @ I )
= ( G2 @ I ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_3588_sum__norm__le,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S: set @ B,F3: B > A,G2: B > real] :
( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X3 ) ) @ ( G2 @ X3 ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S ) ) @ ( groups7311177749621191930dd_sum @ B @ real @ G2 @ S ) ) ) ) ).
% sum_norm_le
thf(fact_3589_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_3590_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 ) ) ) ) ).
% sum_nonneg
thf(fact_3591_sum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > A,A4: set @ B] :
( ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 )
!= ( zero_zero @ A ) )
=> ~ ! [A7: B] :
( ( member @ B @ A7 @ A4 )
=> ( ( G2 @ A7 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_3592_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 )
= ( zero_zero @ A ) ) ) ) ).
% sum.neutral
thf(fact_3593_summable__comparison__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G2: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( G2 @ N3 ) ) )
=> ( ( summable @ real @ G2 )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test
thf(fact_3594_summable__comparison__test_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G2: nat > real,N6: nat,F3: nat > A] :
( ( summable @ real @ G2 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( G2 @ N3 ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test'
thf(fact_3595_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G2: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( P @ X2 ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G2 @ X2 ) @ ( zero_zero @ A ) )
@ A4 ) ) ) ) ).
% sum.inter_filter
thf(fact_3596_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,I5: set @ nat] :
( ( summable @ A @ F3 )
=> ( ( finite_finite2 @ nat @ I5 )
=> ( ! [N3: nat] :
( ( member @ nat @ N3 @ ( uminus_uminus @ ( set @ nat ) @ I5 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ I5 ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ).
% sum_le_suminf
thf(fact_3597_powser__insidea,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,X: A,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) ) ) ) ) ) ).
% powser_insidea
thf(fact_3598_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S3: set @ B,T2: set @ C,G2: C > A,I: C > B,F3: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G2 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ? [Xa: C] :
( ( member @ C @ Xa @ T2 )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G2 @ Xa ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S3 ) @ ( groups7311177749621191930dd_sum @ C @ A @ G2 @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_3599_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 )
= ( zero_zero @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ( F3 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_3600_sum__strict__mono__ex1,axiom,
! [A: $tType,I6: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A4: set @ I6,F3: I6 > A,G2: I6 > A] :
( ( finite_finite2 @ I6 @ A4 )
=> ( ! [X3: I6] :
( ( member @ I6 @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ? [X5: I6] :
( ( member @ I6 @ X5 @ A4 )
& ( ord_less @ A @ ( F3 @ X5 ) @ ( G2 @ X5 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I6 @ A @ F3 @ A4 ) @ ( groups7311177749621191930dd_sum @ I6 @ A @ G2 @ A4 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_3601_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R: A > A > $o,S: set @ B,H: B > A,G2: B > A] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X1: A,Y1: A,X24: A,Y24: A] :
( ( ( R @ X1 @ X24 )
& ( R @ Y1 @ Y24 ) )
=> ( R @ ( plus_plus @ A @ X1 @ Y1 ) @ ( plus_plus @ A @ X24 @ Y24 ) ) )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( R @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
=> ( R @ ( groups7311177749621191930dd_sum @ B @ A @ H @ S ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ S ) ) ) ) ) ) ) ).
% sum.related
thf(fact_3602_suminf__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,G2: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G2 @ N3 ) )
=> ( ( summable @ A @ F3 )
=> ( ( summable @ A @ G2 )
=> ( ord_less_eq @ A @ ( suminf @ A @ F3 ) @ ( suminf @ A @ G2 ) ) ) ) ) ) ).
% suminf_le
thf(fact_3603_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A4: set @ B,F3: B > A,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_3604_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ( member @ B @ X @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( insert2 @ B @ X @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) ) )
& ( ~ ( member @ B @ X @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( insert2 @ B @ X @ A4 ) )
= ( plus_plus @ A @ ( G2 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_3605_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S6: set @ B,T7: set @ C,S: set @ B,I: C > B,J: B > C,T4: set @ C,G2: B > A,H: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T7 )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ ( minus_minus @ ( set @ B ) @ S @ S6 ) )
=> ( ( I @ ( J @ A7 ) )
= A7 ) )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ ( minus_minus @ ( set @ B ) @ S @ S6 ) )
=> ( member @ C @ ( J @ A7 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T7 ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ ( minus_minus @ ( set @ C ) @ T4 @ T7 ) )
=> ( ( J @ ( I @ B7 ) )
= B7 ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ ( minus_minus @ ( set @ C ) @ T4 @ T7 ) )
=> ( member @ B @ ( I @ B7 ) @ ( minus_minus @ ( set @ B ) @ S @ S6 ) ) )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ S6 )
=> ( ( G2 @ A7 )
= ( zero_zero @ A ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ T7 )
=> ( ( H @ B7 )
= ( zero_zero @ A ) ) )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ S )
=> ( ( H @ ( J @ A7 ) )
= ( G2 @ A7 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ S )
= ( groups7311177749621191930dd_sum @ C @ A @ H @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_3606_summable__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N6: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N6 )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_finite
thf(fact_3607_summable__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ C3 @ ( F3 @ N2 ) ) )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_mult_D
thf(fact_3608_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_3609_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S3: set @ B,F3: B > A,B2: A,I: B] :
( ( finite_finite2 @ B @ S3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S3 )
= B2 )
=> ( ( member @ B @ I @ S3 )
=> ( ord_less_eq @ A @ ( F3 @ I ) @ B2 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_3610_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S3: set @ B,F3: B > A,I: B] :
( ( finite_finite2 @ B @ S3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S3 )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S3 )
=> ( ( F3 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_3611_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G2: B > A,B2: set @ B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B2 ) @ ( G2 @ X2 ) @ ( zero_zero @ A ) )
@ A4 ) ) ) ) ).
% sum.inter_restrict
thf(fact_3612_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2
@ ( minus_minus @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [X2: B] :
( ( G2 @ X2 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_3613_summable__partial__sum__bound,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,E2: real] :
( ( summable @ A @ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ~ ! [N9: nat] :
~ ! [M3: nat] :
( ( ord_less_eq @ nat @ N9 @ M3 )
=> ! [N4: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ M3 @ N4 ) ) ) @ E2 ) ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_3614_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I5: set @ B,I: B,F3: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( member @ B @ I @ I5 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I5 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_3615_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I5 ) ) ) ) ) ) ).
% sum_pos
thf(fact_3616_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A4: set @ B,F3: B > A,K4: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ K4 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ K4 ) ) ) ) ).
% sum_bounded_above
thf(fact_3617_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A4: set @ B,K4: A,F3: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ K4 @ ( F3 @ I2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ K4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 ) ) ) ) ).
% sum_bounded_below
thf(fact_3618_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C2: set @ B,A4: set @ B,B2: set @ B,G2: B > A,H: B > A] :
( ( finite_finite2 @ B @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C2 )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ ( minus_minus @ ( set @ B ) @ C2 @ A4 ) )
=> ( ( G2 @ A7 )
= ( zero_zero @ A ) ) )
=> ( ! [B7: B] :
( ( member @ B @ B7 @ ( minus_minus @ ( set @ B ) @ C2 @ B2 ) )
=> ( ( H @ B7 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B2 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ C2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ C2 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_3619_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C2: set @ B,A4: set @ B,B2: set @ B,G2: B > A,H: B > A] :
( ( finite_finite2 @ B @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C2 )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ ( minus_minus @ ( set @ B ) @ C2 @ A4 ) )
=> ( ( G2 @ A7 )
= ( zero_zero @ A ) ) )
=> ( ! [B7: B] :
( ( member @ B @ B7 @ ( minus_minus @ ( set @ B ) @ C2 @ B2 ) )
=> ( ( H @ B7 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ C2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ C2 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_3620_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ G2 @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_3621_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ G2 @ S ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_3622_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S: set @ B,H: B > A,G2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T4 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_3623_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S: set @ B,G2: B > A,H: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ S ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_3624_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B2: set @ B,A4: set @ B,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B2 @ A4 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ B2 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_3625_sum__diff,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A4: set @ B,B2: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B2 ) ) ) ) ) ) ).
% sum_diff
thf(fact_3626_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S: set @ B,H: B > A,G2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ S @ T4 ) )
=> ( ( G2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S @ T4 ) )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T4 ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_3627_sum_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B2: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_inter
thf(fact_3628_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G2: B > A,B2: set @ B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_3629_suminf__eq__zero__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ( ( suminf @ A @ F3 )
= ( zero_zero @ A ) )
= ( ! [N2: nat] :
( ( F3 @ N2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_3630_suminf__nonneg,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_nonneg
thf(fact_3631_suminf__pos,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_pos
thf(fact_3632_summable__0__powser,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) ) ) ) ).
% summable_0_powser
thf(fact_3633_summable__zero__power_H,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) ) ) ) ).
% summable_zero_power'
thf(fact_3634_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,P: B > $o,H: B > A,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H @ X2 ) @ ( G2 @ X2 ) )
@ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_3635_summable__norm__comparison__test,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G2: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( G2 @ N3 ) ) )
=> ( ( summable @ real @ G2 )
=> ( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_3636_summable__rabs__comparison__test,axiom,
! [F3: nat > real,G2: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( F3 @ N3 ) ) @ ( G2 @ N3 ) ) )
=> ( ( summable @ real @ G2 )
=> ( summable @ real
@ ^ [N2: nat] : ( abs_abs @ real @ ( F3 @ N2 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_3637_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B2: set @ B,A4: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B2 )
=> ( ! [B7: B] :
( ( member @ B @ B7 @ ( minus_minus @ ( set @ B ) @ B2 @ A4 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ B7 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B2 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_3638_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B2: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ B2 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_3639_summable__rabs,axiom,
! [F3: nat > real] :
( ( summable @ real
@ ^ [N2: nat] : ( abs_abs @ real @ ( F3 @ N2 ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( suminf @ real @ F3 ) )
@ ( suminf @ real
@ ^ [N2: nat] : ( abs_abs @ real @ ( F3 @ N2 ) ) ) ) ) ).
% summable_rabs
thf(fact_3640_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G2: B > A,X: B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( insert2 @ B @ X @ A4 ) )
= ( plus_plus @ A @ ( G2 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_3641_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( member @ B @ X @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 )
= ( plus_plus @ A @ ( G2 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_3642_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A4: set @ B,A3: B,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ( member @ B @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 ) @ ( F3 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_3643_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A4: set @ B,B2: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) ) ) ) ) ) ).
% sum_Un
thf(fact_3644_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B2: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ( inf_inf @ ( set @ B ) @ A4 @ B2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ B2 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_3645_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A4: set @ A,B2: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ) ).
% sum_Un2
thf(fact_3646_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B2: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ B2 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_3647_suminf__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [N6: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N6 )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A @ F3 )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ N6 ) ) ) ) ) ).
% suminf_finite
thf(fact_3648_suminf__pos__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) )
= ( ? [I4: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I4 ) ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_3649_suminf__pos2,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,I: nat] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ).
% suminf_pos2
thf(fact_3650_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B3: B > A,C3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C3 @ K3 ) )
@ S )
= ( plus_plus @ A @ ( B3 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C3 @ K3 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_3651_sum__shift__lb__Suc0__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,K: nat] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_3652_sum_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_3653_sum_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G2 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_3654_sum_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G2 @ ( suc @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_3655_powser__inside,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,X: A,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) ) ) ) ) ).
% powser_inside
thf(fact_3656_summable__geometric,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C3 ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ C3 ) ) ) ) ).
% summable_geometric
thf(fact_3657_complete__algebra__summable__geometric,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ X ) ) ) ) ).
% complete_algebra_summable_geometric
thf(fact_3658_suminf__split__head,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F3 ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% suminf_split_head
thf(fact_3659_set__replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N ) @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_3660_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_3661_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B2: set @ A,A4: set @ A,B3: A,F3: A > B] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F3 @ B3 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X3 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ B2 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_3662_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A4: set @ C,F3: C > B] :
( ( member @ C @ I @ A4 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ ( minus_minus @ ( set @ C ) @ A4 @ ( insert2 @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X3 ) ) )
=> ( ( finite_finite2 @ C @ A4 )
=> ( ord_less_eq @ B @ ( F3 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F3 @ A4 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_3663_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A4: set @ B,F3: B > A,K4: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less @ A @ ( F3 @ I2 ) @ K4 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A4 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ K4 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_3664_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A4: set @ B,F3: B > A,K4: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( divide_divide @ A @ K4 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 ) @ K4 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_3665_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G2 @ M )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_3666_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I4 ) ) @ ( F3 @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F3 @ ( suc @ N ) ) @ ( F3 @ M ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_3667_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,X: A > B,A3: A > B,B3: B,Delta: B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I5 )
= ( one_one @ B ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( times_times @ B @ ( A3 @ I4 ) @ ( X @ I4 ) )
@ I5 )
@ B3 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_3668_summable__norm,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A] :
( ( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( suminf @ A @ F3 ) )
@ ( suminf @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) ) ) ) ) ).
% summable_norm
thf(fact_3669_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,A3: nat,B3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B3 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A5: nat] : ( plus_plus @ A @ ( F3 @ A5 ) )
@ A3
@ B3
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_3670_sum__norm__bound,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S: set @ B,F3: B > A,K4: real] :
( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X3 ) ) @ K4 ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( finite_card @ B @ S ) ) @ K4 ) ) ) ) ).
% sum_norm_bound
thf(fact_3671_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G2: nat > A,P6: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P6 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P6 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_3672_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: set @ B,F3: B > A,B3: A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 ) @ B3 )
= ( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A5: B] : ( divide_divide @ A @ ( F3 @ A5 ) @ B3 )
@ ( inf_inf @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [A5: B] : ( dvd_dvd @ A @ B3 @ ( F3 @ A5 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F3
@ ( inf_inf @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [A5: B] :
~ ( dvd_dvd @ A @ B3 @ ( F3 @ A5 ) ) ) ) )
@ B3 ) ) ) ) ) ).
% sum_div_partition
thf(fact_3673_powser__split__head_I1_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
= ( plus_plus @ A @ ( F3 @ ( zero_zero @ nat ) )
@ ( times_times @ A
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ ( suc @ N2 ) ) @ ( power_power @ A @ Z @ N2 ) ) )
@ Z ) ) ) ) ) ).
% powser_split_head(1)
thf(fact_3674_powser__split__head_I2_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
=> ( ( times_times @ A
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ ( suc @ N2 ) ) @ ( power_power @ A @ Z @ N2 ) ) )
@ Z )
= ( minus_minus @ A
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
@ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% powser_split_head(2)
thf(fact_3675_suminf__exist__split,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,F3: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( summable @ A @ F3 )
=> ? [N9: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ N9 @ N4 )
=> ( ord_less @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( suminf @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N4 ) ) ) )
@ R2 ) ) ) ) ) ).
% suminf_exist_split
thf(fact_3676_summable__power__series,axiom,
! [F3: nat > real,Z: real] :
( ! [I2: nat] : ( ord_less_eq @ real @ ( F3 @ I2 ) @ ( one_one @ real ) )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ I2 ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Z )
=> ( ( ord_less @ real @ Z @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [I4: nat] : ( times_times @ real @ ( F3 @ I4 ) @ ( power_power @ real @ Z @ I4 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_3677_Abel__lemma,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,R0: real,A3: nat > A,M5: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( ord_less @ real @ R2 @ R0 )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A3 @ N3 ) ) @ ( power_power @ real @ R0 @ N3 ) ) @ M5 )
=> ( summable @ real
@ ^ [N2: nat] : ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A3 @ N2 ) ) @ ( power_power @ real @ R2 @ N2 ) ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_3678_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F3 @ M ) @ ( F3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_3679_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ M ) ) ) ) ) ).
% sum_telescope''
thf(fact_3680_summable__ratio__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [C3: real,N6: nat,F3: nat > A] :
( ( ord_less @ real @ C3 @ ( one_one @ real ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ ( suc @ N3 ) ) ) @ ( times_times @ real @ C3 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_ratio_test
thf(fact_3681_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q6: nat] : ( ord_less @ nat @ Q6 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_3682_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_3683_even__sum__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A4: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A4 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( finite_card @ B
@ ( collect @ B
@ ^ [A5: B] :
( ( member @ B @ A5 @ A4 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F3 @ A5 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_3684_accp__subset__induct,axiom,
! [A: $tType,D3: A > $o,R: A > A > $o,X: A,P: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ D3 @ ( accp @ A @ R ) )
=> ( ! [X3: A,Z3: A] :
( ( D3 @ X3 )
=> ( ( R @ Z3 @ X3 )
=> ( D3 @ Z3 ) ) )
=> ( ( D3 @ X )
=> ( ! [X3: A] :
( ( D3 @ X3 )
=> ( ! [Z4: A] :
( ( R @ Z4 @ X3 )
=> ( P @ Z4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_3685_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q6: nat] : ( ord_less @ nat @ Q6 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_3686_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_3687_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_3688_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,D2: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) ) ) ).
% double_arith_series
thf(fact_3689_double__gauss__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum
thf(fact_3690_arith__series__nat,axiom,
! [A3: nat,D2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I4 @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ nat @ N @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_3691_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,D2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_3692_gauss__sum,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum
thf(fact_3693_double__gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_3694_vebt__buildup_Osimps_I3_J,axiom,
! [Va3: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va3 ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va3 ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.simps(3)
thf(fact_3695_gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_3696_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_3697_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I5 )
& ( ( X @ I4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I5 )
& ( ( Y @ I4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I5 )
& ( ( plus_plus @ A @ ( X @ I4 ) @ ( Y @ I4 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3698_vebt__buildup_Opelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( 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 ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_3699_Maclaurin__sin__expansion3,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less @ real @ T6 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( sin_coeff @ M2 ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T6 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_3700_Maclaurin__sin__expansion4,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less_eq @ real @ T6 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( sin_coeff @ M2 ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T6 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_3701_Maclaurin__sin__expansion2,axiom,
! [X: real,N: nat] :
? [T6: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X ) )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( sin_coeff @ M2 ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T6 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_3702_lemma__termdiff2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [H: A,Z: A,N: nat] :
( ( H
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H ) @ N ) @ ( power_power @ A @ Z @ N ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
= ( times_times @ A @ H
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [Q6: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H ) @ Q6 ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q6 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ P5 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_3703_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_3704_finite__lessThan,axiom,
! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_lessThan @ nat @ K ) ) ).
% finite_lessThan
thf(fact_3705_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X ) @ ( set_ord_lessThan @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% lessThan_subset_iff
thf(fact_3706_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_3707_single__Diff__lessThan,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A] :
( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ K @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K ) )
= ( insert2 @ A @ K @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% single_Diff_lessThan
thf(fact_3708_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P: A > nat,N: A] :
( ! [X3: A] : ( ord_less_eq @ nat @ ( Q @ X3 ) @ ( P @ X3 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( minus_minus @ nat @ ( P @ X2 ) @ ( Q @ X2 ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_3709_lessThan__non__empty,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
( ( set_ord_lessThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% lessThan_non_empty
thf(fact_3710_infinite__Iio,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_lessThan @ A @ A3 ) ) ) ).
% infinite_Iio
thf(fact_3711_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ U2 ) ) ) ) ) ).
% lessThan_def
thf(fact_3712_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( order_bot @ A ) )
=> ! [N: A] :
( ( ( set_ord_lessThan @ A @ N )
= ( bot_bot @ ( set @ A ) ) )
= ( N
= ( bot_bot @ A ) ) ) ) ).
% Iio_eq_empty_iff
thf(fact_3713_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
= ( ord_less @ A @ M @ N ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_3714_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( insert2 @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_3715_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_3716_sum__multicount__gen,axiom,
! [A: $tType,B: $tType,S3: set @ A,T2: set @ B,R: A > B > $o,K: B > nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T2 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ S3 )
& ( R @ I4 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T2 )
& ( R @ I4 @ J3 ) ) ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ nat @ K @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_3717_sum__subtractf__nat,axiom,
! [A: $tType,A4: set @ A,G2: A > nat,F3: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ nat @ ( G2 @ X3 ) @ ( F3 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( minus_minus @ nat @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ A4 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G2 @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_3718_finite__nat__bounded,axiom,
! [S: set @ nat] :
( ( finite_finite2 @ nat @ S )
=> ? [K2: nat] : ( ord_less_eq @ ( set @ nat ) @ S @ ( set_ord_lessThan @ nat @ K2 ) ) ) ).
% finite_nat_bounded
thf(fact_3719_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_3720_sum__eq__Suc0__iff,axiom,
! [A: $tType,A4: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A4 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ( F3 @ X2 )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A4 )
=> ( ( X2 != Y3 )
=> ( ( F3 @ Y3 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_3721_sum__SucD,axiom,
! [A: $tType,F3: A > nat,A4: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A4 )
= ( suc @ N ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_3722_sum__eq__1__iff,axiom,
! [A: $tType,A4: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A4 )
= ( one_one @ nat ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ( F3 @ X2 )
= ( one_one @ nat ) )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A4 )
=> ( ( X2 != Y3 )
=> ( ( F3 @ Y3 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_3723_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_3724_sum__multicount,axiom,
! [A: $tType,B: $tType,S: set @ A,T4: set @ B,R: A > B > $o,K: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T4 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ S )
& ( R @ I4 @ X3 ) ) ) )
= K ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T4 )
& ( R @ I4 @ J3 ) ) ) )
@ S )
= ( times_times @ nat @ K @ ( finite_card @ B @ T4 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_3725_sum__diff__nat,axiom,
! [A: $tType,B2: set @ A,A4: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ B2 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_3726_sum__diff1__nat,axiom,
! [A: $tType,A3: A,A4: set @ A,F3: A > nat] :
( ( ( member @ A @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A4 ) @ ( F3 @ A3 ) ) ) )
& ( ~ ( member @ A @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A4 ) ) ) ) ).
% sum_diff1_nat
thf(fact_3727_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,K: A] :
( ( ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_3728_suminf__le__const,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,X: A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X )
=> ( ord_less_eq @ A @ ( suminf @ A @ F3 ) @ X ) ) ) ) ).
% suminf_le_const
thf(fact_3729_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_3730_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ M ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_3731_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F3 @ M ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_3732_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,X: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X )
=> ( summable @ A @ F3 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_3733_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( G2 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_3734_sum__nth__roots,axiom,
! [N: nat,C3: complex] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X2: complex] : X2
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= C3 ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_nth_roots
thf(fact_3735_sum__Un__nat,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_3736_finite__enumerate__initial__segment,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat,S3: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ S @ ( set_ord_lessThan @ A @ S3 ) ) ) )
=> ( ( infini527867602293511546merate @ A @ ( inf_inf @ ( set @ A ) @ S @ ( set_ord_lessThan @ A @ S3 ) ) @ N )
= ( infini527867602293511546merate @ A @ S @ N ) ) ) ) ) ).
% finite_enumerate_initial_segment
thf(fact_3737_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X2: complex] : X2
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= ( one_one @ complex ) ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_roots_unity
thf(fact_3738_sum__less__suminf,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,N: nat] :
( ( summable @ A @ F3 )
=> ( ! [M4: nat] :
( ( ord_less_eq @ nat @ N @ M4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ M4 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% sum_less_suminf
thf(fact_3739_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,F3: nat > A,K4: A,K: nat] :
( ! [P7: nat] :
( ( ord_less @ nat @ P7 @ N )
=> ( ord_less_eq @ A @ ( F3 @ P7 ) @ K4 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K4 )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ K4 ) ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_3740_sum__less__suminf2,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,N: nat,I: nat] :
( ( summable @ A @ F3 )
=> ( ! [M4: nat] :
( ( ord_less_eq @ nat @ N @ M4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ M4 ) ) )
=> ( ( ord_less_eq @ nat @ N @ I )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_3741_Maclaurin__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: real,N: nat,Diff: nat > A > real] :
( ( X
= ( zero_zero @ real ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_3742_Maclaurin__lemma,axiom,
! [H: real,F3: real > real,J: nat > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H )
=> ? [B4: real] :
( ( F3 @ H )
= ( 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 @ H @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ B4 @ ( divide_divide @ real @ ( power_power @ real @ H @ N ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_3743_Maclaurin__exp__le,axiom,
! [X: real,N: nat] :
? [T6: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M2 ) @ ( semiring_char_0_fact @ real @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_3744_Maclaurin__sin__bound,axiom,
! [X: real,N: nat] :
( ord_less_eq @ real
@ ( abs_abs @ real
@ ( minus_minus @ real @ ( sin @ real @ X )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( sin_coeff @ M2 ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) )
@ ( times_times @ real @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( abs_abs @ real @ X ) @ N ) ) ) ).
% Maclaurin_sin_bound
thf(fact_3745_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X2: int] : X2
@ ( set_or1337092689740270186AtMost @ int @ M @ N ) )
= ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N @ ( plus_plus @ int @ N @ ( one_one @ int ) ) ) @ ( times_times @ int @ M @ ( minus_minus @ int @ M @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_3746_sum__pos__lt__pair,axiom,
! [F3: nat > real,K: nat] :
( ( summable @ real @ F3 )
=> ( ! [D6: nat] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( F3 @ ( plus_plus @ nat @ K @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D6 ) ) ) @ ( F3 @ ( plus_plus @ nat @ K @ ( plus_plus @ nat @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D6 ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ord_less @ real @ ( groups7311177749621191930dd_sum @ nat @ real @ F3 @ ( set_ord_lessThan @ nat @ K ) ) @ ( suminf @ real @ F3 ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_3747_Maclaurin__exp__lt,axiom,
! [X: real,N: nat] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T6 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M2 ) @ ( semiring_char_0_fact @ real @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_3748_Maclaurin__cos__expansion2,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less @ real @ T6 @ X )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T6 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_3749_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ? [T6: real] :
( ( ord_less @ real @ X @ T6 )
& ( ord_less @ real @ T6 @ ( zero_zero @ real ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T6 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_3750_Maclaurin__cos__expansion,axiom,
! [X: real,N: nat] :
? [T6: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T6 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_3751_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: nat > A > A,Xa3: nat,Xb3: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa3 @ Xb3 @ Xc )
= Y )
=> ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa3 @ ( product_Pair @ nat @ A @ Xb3 @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less @ nat @ Xb3 @ Xa3 )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb3 @ Xa3 )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa3 @ ( one_one @ nat ) ) @ Xb3 @ ( X @ Xa3 @ Xc ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa3 @ ( product_Pair @ nat @ A @ Xb3 @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_3752_cos__coeff__0,axiom,
( ( cos_coeff @ ( zero_zero @ nat ) )
= ( one_one @ real ) ) ).
% cos_coeff_0
thf(fact_3753_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: nat > A > A,A13: nat,A24: nat,A33: A,P: ( nat > A > A ) > nat > nat > A > $o] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A13 @ ( product_Pair @ nat @ A @ A24 @ A33 ) ) ) )
=> ( ! [F6: nat > A > A,A7: nat,B7: nat,Acc2: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F6 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A7 @ ( product_Pair @ nat @ A @ B7 @ Acc2 ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B7 @ A7 )
=> ( P @ F6 @ ( plus_plus @ nat @ A7 @ ( one_one @ nat ) ) @ B7 @ ( F6 @ A7 @ Acc2 ) ) )
=> ( P @ F6 @ A7 @ B7 @ Acc2 ) ) )
=> ( P @ A0 @ A13 @ A24 @ A33 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_3754_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F3: nat > A > A,A3: nat,B3: nat,Acc3: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A3 @ ( product_Pair @ nat @ A @ B3 @ Acc3 ) ) ) )
=> ( ( ( ord_less @ nat @ B3 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F3 @ A3 @ B3 @ Acc3 )
= Acc3 ) )
& ( ~ ( ord_less @ nat @ B3 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F3 @ A3 @ B3 @ Acc3 )
= ( set_fo6178422350223883121st_nat @ A @ F3 @ ( plus_plus @ nat @ A3 @ ( one_one @ nat ) ) @ B3 @ ( F3 @ A3 @ Acc3 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_3755_choose__even__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_3756_choose__odd__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_3757_in__measure,axiom,
! [A: $tType,X: A,Y: A,F3: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measure @ A @ F3 ) )
= ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ).
% in_measure
thf(fact_3758_in__finite__psubset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A4 @ B2 ) @ ( finite_psubset @ A ) )
= ( ( ord_less @ ( set @ A ) @ A4 @ B2 )
& ( finite_finite2 @ A @ B2 ) ) ) ).
% in_finite_psubset
thf(fact_3759_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_3760_finite__atMost,axiom,
! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_atMost @ nat @ K ) ) ).
% finite_atMost
thf(fact_3761_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% atMost_subset_iff
thf(fact_3762_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H: A,H3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H ) @ ( set_ord_atMost @ A @ H3 ) )
= ( ~ ( ord_less_eq @ A @ L @ H )
| ( ord_less_eq @ A @ H @ H3 ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_3763_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_3764_not__empty__eq__Iic__eq__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [H: A] :
( ( bot_bot @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H ) ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_3765_infinite__Iic,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_atMost @ A @ A3 ) ) ) ).
% infinite_Iic
thf(fact_3766_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ U2 ) ) ) ) ) ).
% atMost_def
thf(fact_3767_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_3768_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K ) )
= ( insert2 @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% atMost_Suc
thf(fact_3769_not__Iic__le__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H: A,L3: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ).
% not_Iic_le_Icc
thf(fact_3770_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_3771_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_lessThan @ A @ B3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% Iic_subset_Iio_iff
thf(fact_3772_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_3773_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ I4 ) @ ( F3 @ ( suc @ I4 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_3774_polyfun__eq__coeffs,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat,D2: nat > A] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( D2 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
=> ( ( C3 @ I4 )
= ( D2 @ I4 ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_3775_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A3: nat > A,B2: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A3 @ ( set_ord_atMost @ nat @ N3 ) ) @ B2 )
=> ( summable @ A @ A3 ) ) ) ) ).
% bounded_imp_summable
thf(fact_3776_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_3777_ivl__disj__un__singleton_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [U: A] :
( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_ord_atMost @ A @ U ) ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_3778_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
=> ( ( C3 @ I4 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_0
thf(fact_3779_zero__polynom__imp__zero__coeffs,axiom,
! [A: $tType] :
( ( ( ab_semigroup_mult @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [C3: nat > A,N: nat,K: nat] :
( ! [W: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ W @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( C3 @ K )
= ( zero_zero @ A ) ) ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_3780_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_3781_sum__choose__diagonal,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( minus_minus @ nat @ N @ K3 ) @ ( minus_minus @ nat @ M @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( suc @ N ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_3782_polyfun__roots__finite,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,K: nat,N: nat] :
( ( ( C3 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z6 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_3783_polyfun__finite__roots,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
= ( ? [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
& ( ( C3 @ I4 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_finite_roots
thf(fact_3784_polyfun__roots__card,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,K: nat,N: nat] :
( ( ( C3 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z6 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ).
% polyfun_roots_card
thf(fact_3785_polyfun__linear__factor__root,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: nat > A,A3: A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ A3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ~ ! [B7: nat > A] :
~ ! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ Z4 @ A3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( B7 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_3786_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_3787_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_3788_polyfun__rootbound,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,K: nat,N: nat] :
( ( ( C3 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z6 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
& ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z6 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ) ).
% polyfun_rootbound
thf(fact_3789_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M: nat,A3: nat > A,N: nat,B3: nat > A,X: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ M @ I2 )
=> ( ( A3 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B3 @ J2 )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( times_times @ A @ ( B3 @ J3 ) @ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [R5: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A3 @ K3 ) @ ( B3 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ A @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_3790_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat,K: A] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= K ) )
= ( ( ( C3 @ ( zero_zero @ nat ) )
= K )
& ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) )
=> ( ( C3 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_const
thf(fact_3791_polynomial__product__nat,axiom,
! [M: nat,A3: nat > nat,N: nat,B3: nat > nat,X: nat] :
( ! [I2: nat] :
( ( ord_less @ nat @ M @ I2 )
=> ( ( A3 @ I2 )
= ( zero_zero @ nat ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B3 @ J2 )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ ( A3 @ I4 ) @ ( power_power @ nat @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( times_times @ nat @ ( B3 @ J3 ) @ ( power_power @ nat @ X @ J3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [R5: nat] :
( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( A3 @ K3 ) @ ( B3 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ nat @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_3792_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P6: nat,K: nat,G2: nat > A,H: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P6 )
=> ( ( ord_less_eq @ nat @ K @ P6 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P6 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P6 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_3793_root__polyfun,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,Z: A,A3: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( ( power_power @ A @ Z @ N )
= A3 )
= ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] :
( times_times @ A
@ ( if @ A
@ ( I4
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ A3 )
@ ( if @ A @ ( I4 = N ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) )
@ ( power_power @ A @ Z @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_3794_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_3795_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A3: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ Y @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( A3 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J3 @ K3 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y @ K3 ) ) @ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ J3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_3796_card__lists__length__le,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A4 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_3797_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_3798_polyfun__extremal__lemma,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [E2: real,C3: nat > A,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M9: real] :
! [Z4: A] :
( ( ord_less_eq @ real @ M9 @ ( real_V7770717601297561774m_norm @ A @ Z4 ) )
=> ( ord_less_eq @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
@ ( times_times @ real @ E2 @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z4 ) @ ( suc @ N ) ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_3799_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A3: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ Y @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ ( minus_minus @ nat @ I4 @ J3 ) @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_3800_geometric__deriv__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Z: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( one_one @ real ) )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) @ ( power_power @ A @ Z @ N2 ) )
@ ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( minus_minus @ A @ ( one_one @ A ) @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_3801_monoI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: nat > A] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq @ nat @ M4 @ N3 )
=> ( ord_less_eq @ A @ ( X4 @ M4 ) @ ( X4 @ N3 ) ) )
=> ( topological_monoseq @ A @ X4 ) ) ) ).
% monoI1
thf(fact_3802_monoI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: nat > A] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq @ nat @ M4 @ N3 )
=> ( ord_less_eq @ A @ ( X4 @ N3 ) @ ( X4 @ M4 ) ) )
=> ( topological_monoseq @ A @ X4 ) ) ) ).
% monoI2
thf(fact_3803_monoseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X8: nat > A] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ A @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
| ! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ A @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ) ).
% monoseq_def
thf(fact_3804_sums__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( sums @ A
@ ^ [N2: nat] : ( zero_zero @ A )
@ ( zero_zero @ A ) ) ) ).
% sums_zero
thf(fact_3805_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: nat > A,X: A] :
( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A3 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) )
@ X )
= ( ( A3 @ ( zero_zero @ nat ) )
= X ) ) ) ).
% powser_sums_zero_iff
thf(fact_3806_sums__0,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A] :
( ! [N3: nat] :
( ( F3 @ N3 )
= ( zero_zero @ A ) )
=> ( sums @ A @ F3 @ ( zero_zero @ A ) ) ) ) ).
% sums_0
thf(fact_3807_sums__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,G2: nat > A,S3: A,T2: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G2 @ N3 ) )
=> ( ( sums @ A @ F3 @ S3 )
=> ( ( sums @ A @ G2 @ T2 )
=> ( ord_less_eq @ A @ S3 @ T2 ) ) ) ) ) ).
% sums_le
thf(fact_3808_sums__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F3: nat > A] :
( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) )
@ ( F3 @ I ) ) ) ).
% sums_single
thf(fact_3809_sums__mult2__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C3: A,F3: nat > A,D2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ C3 )
@ ( times_times @ A @ D2 @ C3 ) )
= ( sums @ A @ F3 @ D2 ) ) ) ) ).
% sums_mult2_iff
thf(fact_3810_sums__mult__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C3: A,F3: nat > A,D2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ C3 @ ( F3 @ N2 ) )
@ ( times_times @ A @ C3 @ D2 ) )
= ( sums @ A @ F3 @ D2 ) ) ) ) ).
% sums_mult_iff
thf(fact_3811_sums__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A,A3: A] :
( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ C3 @ ( F3 @ N2 ) )
@ A3 )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( sums @ A @ F3 @ ( divide_divide @ A @ A3 @ C3 ) ) ) ) ) ).
% sums_mult_D
thf(fact_3812_sums__Suc__imp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S3: A] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) )
@ S3 )
=> ( sums @ A @ F3 @ S3 ) ) ) ) ).
% sums_Suc_imp
thf(fact_3813_sums__Suc,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,L: A] :
( ( sums @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) )
@ L )
=> ( sums @ A @ F3 @ ( plus_plus @ A @ L @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc
thf(fact_3814_sums__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S3: A] :
( ( sums @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) )
@ S3 )
= ( sums @ A @ F3 @ ( plus_plus @ A @ S3 @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc_iff
thf(fact_3815_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N: nat,F3: nat > A,S3: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ( F3 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N ) )
@ S3 )
= ( sums @ A @ F3 @ S3 ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_3816_sums__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A4: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ A4 )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A4 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ A4 ) ) ) ) ).
% sums_If_finite_set
thf(fact_3817_sums__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F3: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( collect @ nat @ P ) ) ) ) ) ).
% sums_If_finite
thf(fact_3818_sums__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N6: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N6 )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( sums @ A @ F3 @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ N6 ) ) ) ) ) ).
% sums_finite
thf(fact_3819_powser__sums__if,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [M: nat,Z: A] :
( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( if @ A @ ( N2 = M ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z @ N2 ) )
@ ( power_power @ A @ Z @ M ) ) ) ).
% powser_sums_if
thf(fact_3820_powser__sums__zero,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: nat > A] :
( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A3 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) )
@ ( A3 @ ( zero_zero @ nat ) ) ) ) ).
% powser_sums_zero
thf(fact_3821_sums__If__finite__set_H,axiom,
! [A: $tType] :
( ( ( topolo1287966508704411220up_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G2: nat > A,S: A,A4: set @ nat,S6: A,F3: nat > A] :
( ( sums @ A @ G2 @ S )
=> ( ( finite_finite2 @ nat @ A4 )
=> ( ( S6
= ( plus_plus @ A @ S
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ N2 ) @ ( G2 @ N2 ) )
@ A4 ) ) )
=> ( sums @ A
@ ^ [N2: nat] : ( if @ A @ ( member @ nat @ N2 @ A4 ) @ ( F3 @ N2 ) @ ( G2 @ N2 ) )
@ S6 ) ) ) ) ) ).
% sums_If_finite_set'
thf(fact_3822_geometric__sums,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C3 ) @ ( one_one @ real ) )
=> ( sums @ A @ ( power_power @ A @ C3 ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C3 ) ) ) ) ) ).
% geometric_sums
thf(fact_3823_mono__SucI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
=> ( topological_monoseq @ A @ X4 ) ) ) ).
% mono_SucI1
thf(fact_3824_mono__SucI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) )
=> ( topological_monoseq @ A @ X4 ) ) ) ).
% mono_SucI2
thf(fact_3825_monoseq__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X8: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ) ).
% monoseq_Suc
thf(fact_3826_sin__x__sin__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P5 @ N2 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ).
% sin_x_sin_y
thf(fact_3827_sums__cos__x__plus__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 ) @ ( 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 @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P5 @ N2 ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( cos @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_3828_cos__x__cos__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 )
& ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P5 @ N2 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ).
% cos_x_cos_y
thf(fact_3829_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C3: nat > A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( C3 @ N2 ) ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_3830_scaleR__zero__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real] :
( ( real_V8093663219630862766scaleR @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_right
thf(fact_3831_scaleR__cancel__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X: A,B3: real] :
( ( ( real_V8093663219630862766scaleR @ A @ A3 @ X )
= ( real_V8093663219630862766scaleR @ A @ B3 @ X ) )
= ( ( A3 = B3 )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_cancel_right
thf(fact_3832_scaleR__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A3 @ X )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ real ) )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_eq_0_iff
thf(fact_3833_scaleR__zero__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( zero_zero @ real ) @ X )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_left
thf(fact_3834_scaleR__right__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,A3: real,B3: real] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A3 @ X )
= ( real_V8093663219630862766scaleR @ A @ B3 @ X ) )
=> ( A3 = B3 ) ) ) ) ).
% scaleR_right_imp_eq
thf(fact_3835_scaleR__right__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B3: real,A3: real,C3: A] :
( ( ord_less_eq @ real @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ C3 ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ C3 ) ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_3836_scaleR__right__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: real,X: A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ X ) ) ) ) ) ).
% scaleR_right_mono
thf(fact_3837_scaleR__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_3838_scaleR__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_3839_scaleR__le__cancel__left,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B3 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) )
& ( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_3840_scaleR__left__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B3: A,A3: A,C3: real] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ real @ C3 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B3 ) ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_3841_scaleR__left__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,Y: A,A3: real] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y ) ) ) ) ) ).
% scaleR_left_mono
thf(fact_3842_eq__vector__fraction__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,U: real,V2: real,A3: A] :
( ( X
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V2 ) @ A3 ) )
= ( ( ( V2
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V2
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ V2 @ X )
= ( real_V8093663219630862766scaleR @ A @ U @ A3 ) ) ) ) ) ) ).
% eq_vector_fraction_iff
thf(fact_3843_vector__fraction__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,V2: real,A3: A,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V2 ) @ A3 )
= X )
= ( ( ( V2
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V2
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ U @ A3 )
= ( real_V8093663219630862766scaleR @ A @ V2 @ X ) ) ) ) ) ) ).
% vector_fraction_eq_iff
thf(fact_3844_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,E2: A,C3: A,B3: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ E2 ) @ C3 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B3 @ E2 ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A3 @ B3 ) @ E2 ) @ C3 ) @ D2 ) ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_3845_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,E2: A,C3: A,B3: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ E2 ) @ C3 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B3 @ E2 ) @ D2 ) )
= ( ord_less_eq @ A @ C3 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B3 @ A3 ) @ E2 ) @ D2 ) ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_3846_zero__le__scaleR__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ B3 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( A3
= ( zero_zero @ real ) ) ) ) ) ).
% zero_le_scaleR_iff
thf(fact_3847_scaleR__le__0__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( A3
= ( zero_zero @ real ) ) ) ) ) ).
% scaleR_le_0_iff
thf(fact_3848_scaleR__nonpos__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: A] :
( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ B3 ) ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_3849_scaleR__nonpos__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_3850_scaleR__nonneg__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_3851_scaleR__nonneg__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_3852_split__scaleR__pos__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ B3 ) ) ) ) ).
% split_scaleR_pos_le
thf(fact_3853_split__scaleR__neg__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( zero_zero @ A ) ) ) ) ).
% split_scaleR_neg_le
thf(fact_3854_scaleR__mono_H,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: real,C3: A,D2: A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ C3 ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_3855_scaleR__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: real,X: A,Y: A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ Y ) ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_3856_scaleR__left__le__one__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,A3: real] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ real @ A3 @ ( one_one @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ X ) ) ) ) ).
% scaleR_left_le_one_le
thf(fact_3857_neg__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) )
= ( ord_less_eq @ A @ B3 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% neg_le_divideR_eq
thf(fact_3858_neg__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) @ A3 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ B3 ) ) ) ) ).
% neg_divideR_le_eq
thf(fact_3859_pos__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ B3 ) ) ) ) ).
% pos_le_divideR_eq
thf(fact_3860_pos__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) @ A3 )
= ( ord_less_eq @ A @ B3 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% pos_divideR_le_eq
thf(fact_3861_pos__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) @ A3 )
= ( ord_less @ A @ B3 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% pos_divideR_less_eq
thf(fact_3862_pos__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ B3 ) ) ) ) ).
% pos_less_divideR_eq
thf(fact_3863_neg__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) @ A3 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ B3 ) ) ) ) ).
% neg_divideR_less_eq
thf(fact_3864_neg__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) )
= ( ord_less @ A @ B3 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% neg_less_divideR_eq
thf(fact_3865_nonzero__inverse__scaleR__distrib,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A3: real,X: A] :
( ( A3
!= ( zero_zero @ real ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) )
= ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ A3 ) @ ( inverse_inverse @ A @ X ) ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_3866_neg__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) @ A3 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divideR_le_eq
thf(fact_3867_neg__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% neg_le_minus_divideR_eq
thf(fact_3868_pos__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) @ A3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% pos_minus_divideR_le_eq
thf(fact_3869_pos__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_le_minus_divideR_eq
thf(fact_3870_neg__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) @ A3 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divideR_less_eq
thf(fact_3871_neg__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% neg_less_minus_divideR_eq
thf(fact_3872_pos__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) @ A3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% pos_minus_divideR_less_eq
thf(fact_3873_pos__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_less_minus_divideR_eq
thf(fact_3874_termdiff__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,K4: real,C3: nat > A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ K4 )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K4 )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ X3 @ N2 ) ) ) )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ) ).
% termdiff_converges
thf(fact_3875_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X ) ) ) ).
% cosh_converges
thf(fact_3876_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X @ N2 ) ) )
@ ( sinh @ A @ X ) ) ) ).
% sinh_converges
thf(fact_3877_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N2: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I4: A] : ( plus_plus @ A @ I4 @ ( one_one @ A ) )
@ N2
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_3878_upto_Opinduct,axiom,
! [A0: int,A13: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A13 ) )
=> ( ! [I2: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I2 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I2 @ J2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J2 ) )
=> ( P @ I2 @ J2 ) ) )
=> ( P @ A0 @ A13 ) ) ) ).
% upto.pinduct
thf(fact_3879_arctan__def,axiom,
( arctan
= ( ^ [Y3: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
& ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X2 )
= Y3 ) ) ) ) ) ).
% arctan_def
thf(fact_3880_arcsin__def,axiom,
( arcsin
= ( ^ [Y3: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X2 )
= Y3 ) ) ) ) ) ).
% arcsin_def
thf(fact_3881_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,N: nat,I: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N ) @ I )
= ( semiri8178284476397505188at_aux @ A @ Inc @ N @ ( Inc @ I ) ) ) ) ).
% of_nat_aux.simps(2)
thf(fact_3882_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_3883_ln__neg__is__const,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ X )
= ( the @ real
@ ^ [X2: real] : $false ) ) ) ).
% ln_neg_is_const
thf(fact_3884_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X8: set @ A] :
( the @ A
@ ^ [X2: A] :
( X8
= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_3885_arccos__def,axiom,
( arccos
= ( ^ [Y3: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ pi )
& ( ( cos @ real @ X2 )
= Y3 ) ) ) ) ) ).
% arccos_def
thf(fact_3886_pi__half,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
= ( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X2 )
= ( zero_zero @ real ) ) ) ) ) ).
% pi_half
thf(fact_3887_pi__def,axiom,
( pi
= ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) )
@ ( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X2 )
= ( zero_zero @ real ) ) ) ) ) ) ).
% pi_def
thf(fact_3888_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L2: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: 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 ) ) @ Q6 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q6 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_3889_sum__count__set,axiom,
! [A: $tType,Xs: list @ A,X4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X4 )
=> ( ( finite_finite2 @ A @ X4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs ) @ X4 )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_3890_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv2: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
=> ( ! [Uu2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
=> ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy ) )
=> ( Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy ) ) ) )
=> ~ ! [Uz: product_prod @ nat @ nat,Va2: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va2 @ Vb @ Vc ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va2 @ Vb @ Vc ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
thf(fact_3891_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N2: nat] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_3892_pochhammer__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% pochhammer_0
thf(fact_3893_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% pochhammer_Suc0
thf(fact_3894_count__notin,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( count_list @ A @ Xs @ X )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_3895_pochhammer__pos,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% pochhammer_pos
thf(fact_3896_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ M )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_3897_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_3898_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% pochhammer_nonneg
thf(fact_3899_pochhammer__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_0_left
thf(fact_3900_count__le__length,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% count_le_length
thf(fact_3901_pochhammer__of__nat__eq__0__lemma,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_3902_pochhammer__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [N: nat,K: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
= ( zero_zero @ A ) )
= ( ord_less @ nat @ N @ K ) ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_3903_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
= ( zero_zero @ A ) )
= ( ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N )
& ( A3
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_3904_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
!= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_3905_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,Z: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ Z @ N )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_3906_floor__real__def,axiom,
( ( archim6421214686448440834_floor @ real )
= ( ^ [X2: real] :
( the @ int
@ ^ [Z6: int] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ Z6 ) @ X2 )
& ( ord_less @ real @ X2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ Z6 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_3907_VEBT__internal_OminNull_Opelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ! [Uv2: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
=> ( ! [Uu2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
=> ~ ! [Uz: product_prod @ nat @ nat,Va2: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va2 @ Vb @ Vc ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va2 @ Vb @ Vc ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(3)
thf(fact_3908_VEBT__internal_OminNull_Opelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
thf(fact_3909_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L2: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q6: 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 ) ) @ Q6 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_3910_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L2: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q6: 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 ) ) @ Q6 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_3911_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z: A,N: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_3912_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M2: nat,N2: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N2
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M2 @ N2 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M2 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q6: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q6 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_3913_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X2: rat] :
( the @ int
@ ^ [Z6: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z6 ) @ X2 )
& ( ord_less @ rat @ X2 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z6 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_3914_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F4: set @ A,I5: set @ A,F3: A > B,I: A] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I5 )
& ( ( F3 @ I4 )
!= ( zero_zero @ B ) ) ) )
@ F4 )
=> ( ( ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I5 ) @ ( F3 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I5 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_3915_prod__zero__iff,axiom,
! [A: $tType,B: $tType] :
( ( semidom @ A )
=> ! [A4: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 )
= ( zero_zero @ A ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ( F3 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% prod_zero_iff
thf(fact_3916_prod_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A] :
( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty
thf(fact_3917_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G2: B > A] :
( ~ ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_3918_dvd__prod__eqI,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: set @ B,A3: B,B3: A,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( member @ B @ A3 @ A4 )
=> ( ( B3
= ( F3 @ A3 ) )
=> ( dvd_dvd @ A @ B3 @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_3919_dvd__prodI,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: set @ B,A3: B,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( member @ B @ A3 @ A4 )
=> ( dvd_dvd @ A @ ( F3 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) ) ) ) ) ).
% dvd_prodI
thf(fact_3920_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P6: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P6 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_3921_sum_Oeq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,P6: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P6 @ I5 )
= ( groups7311177749621191930dd_sum @ B @ A @ P6 @ I5 ) ) ) ) ).
% sum.eq_sum
thf(fact_3922_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B3 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B3 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_3923_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_3924_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ~ ( member @ B @ X @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert2 @ B @ X @ A4 ) )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) ) ) ) ) ) ).
% prod.insert
thf(fact_3925_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,P6: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( P6 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P6 @ ( insert2 @ B @ I @ I5 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P6 @ I5 ) ) )
& ( ~ ( member @ B @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P6 @ ( insert2 @ B @ I @ I5 ) )
= ( plus_plus @ A @ ( P6 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P6 @ I5 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_3926_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G2: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_3927_abs__rat__def,axiom,
( ( abs_abs @ rat )
= ( ^ [A5: rat] : ( if @ rat @ ( ord_less @ rat @ A5 @ ( zero_zero @ rat ) ) @ ( uminus_uminus @ rat @ A5 ) @ A5 ) ) ) ).
% abs_rat_def
thf(fact_3928_sgn__rat__def,axiom,
( ( sgn_sgn @ rat )
= ( ^ [A5: rat] :
( if @ rat
@ ( A5
= ( zero_zero @ rat ) )
@ ( zero_zero @ rat )
@ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A5 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ).
% sgn_rat_def
thf(fact_3929_norm__prod__le,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [F3: B > A,A4: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) )
@ ( groups7121269368397514597t_prod @ B @ real
@ ^ [A5: B] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ A5 ) )
@ A4 ) ) ) ).
% norm_prod_le
thf(fact_3930_less__eq__rat__def,axiom,
( ( ord_less_eq @ rat )
= ( ^ [X2: rat,Y3: rat] :
( ( ord_less @ rat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% less_eq_rat_def
thf(fact_3931_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ~ ! [S4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ S4 )
=> ! [T6: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ T6 )
=> ( R2
!= ( plus_plus @ rat @ S4 @ T6 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_3932_prod_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B2: set @ C,G2: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ C @ B2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] :
( groups7121269368397514597t_prod @ C @ A @ ( G2 @ X2 )
@ ( collect @ C
@ ^ [Y3: C] :
( ( member @ C @ Y3 @ B2 )
& ( R @ X2 @ Y3 ) ) ) )
@ A4 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y3: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G2 @ X2 @ Y3 )
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( R @ X2 @ Y3 ) ) ) )
@ B2 ) ) ) ) ) ).
% prod.swap_restrict
thf(fact_3933_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > A,I5: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( G2 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G2 @ I5 ) ) ) ).
% sum.non_neutral'
thf(fact_3934_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) ) ) ) ).
% prod_nonneg
thf(fact_3935_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F3: B > A,G2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
& ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( G2 @ I2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) ) ) ) ).
% prod_mono
thf(fact_3936_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) ) ) ) ).
% prod_pos
thf(fact_3937_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) ) ) ) ).
% prod_ge_1
thf(fact_3938_prod__zero,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ? [X5: B] :
( ( member @ B @ X5 @ A4 )
& ( ( F3 @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% prod_zero
thf(fact_3939_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G2: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( P @ X2 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G2 @ X2 ) @ ( one_one @ A ) )
@ A4 ) ) ) ) ).
% prod.inter_filter
thf(fact_3940_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G2 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_3941_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) )
& ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_3942_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R: A > A > $o,S: set @ B,H: B > A,G2: B > A] :
( ( R @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X1: A,Y1: A,X24: A,Y24: A] :
( ( ( R @ X1 @ X24 )
& ( R @ Y1 @ Y24 ) )
=> ( R @ ( times_times @ A @ X1 @ Y1 ) @ ( times_times @ A @ X24 @ Y24 ) ) )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( R @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
=> ( R @ ( groups7121269368397514597t_prod @ B @ A @ H @ S ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ S ) ) ) ) ) ) ) ).
% prod.related
thf(fact_3943_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ( member @ B @ X @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert2 @ B @ X @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) ) )
& ( ~ ( member @ B @ X @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert2 @ B @ X @ A4 ) )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_3944_prod__dvd__prod__subset2,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B2: set @ B,A4: set @ B,F3: B > A,G2: B > A] :
( ( finite_finite2 @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B2 )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ A4 )
=> ( dvd_dvd @ A @ ( F3 @ A7 ) @ ( G2 @ A7 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ B2 ) ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_3945_prod__dvd__prod__subset,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B2: set @ B,A4: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B2 )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ B2 ) ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_3946_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S6: set @ B,T7: set @ C,S: set @ B,I: C > B,J: B > C,T4: set @ C,G2: B > A,H: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T7 )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ ( minus_minus @ ( set @ B ) @ S @ S6 ) )
=> ( ( I @ ( J @ A7 ) )
= A7 ) )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ ( minus_minus @ ( set @ B ) @ S @ S6 ) )
=> ( member @ C @ ( J @ A7 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T7 ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ ( minus_minus @ ( set @ C ) @ T4 @ T7 ) )
=> ( ( J @ ( I @ B7 ) )
= B7 ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ ( minus_minus @ ( set @ C ) @ T4 @ T7 ) )
=> ( member @ B @ ( I @ B7 ) @ ( minus_minus @ ( set @ B ) @ S @ S6 ) ) )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ S6 )
=> ( ( G2 @ A7 )
= ( one_one @ A ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ T7 )
=> ( ( H @ B7 )
= ( one_one @ A ) ) )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ S )
=> ( ( H @ ( J @ A7 ) )
= ( G2 @ A7 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ S )
= ( groups7121269368397514597t_prod @ C @ A @ H @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_3947_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,G2: B > A,H: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I4: B] : ( plus_plus @ A @ ( G2 @ I4 ) @ ( H @ I4 ) )
@ I5 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G2 @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H @ I5 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_3948_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G2 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_3949_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G2: B > A,B2: set @ B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B2 ) @ ( G2 @ X2 ) @ ( one_one @ A ) )
@ A4 ) ) ) ) ).
% prod.inter_restrict
thf(fact_3950_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2
@ ( minus_minus @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [X2: B] :
( ( G2 @ X2 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_3951_exp__sum,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ B )
& ( real_Vector_banach @ B )
& ( real_V2822296259951069270ebra_1 @ B ) )
=> ! [I5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ I5 )
=> ( ( exp @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ I5 ) )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X2: A] : ( exp @ B @ ( F3 @ X2 ) )
@ I5 ) ) ) ) ).
% exp_sum
thf(fact_3952_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,I: A,F3: A > B] :
( ( finite_finite2 @ A @ I5 )
=> ( ( member @ A @ I @ I5 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I5 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_3953_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T4: set @ B,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G2 @ S )
= ( groups1027152243600224163dd_sum @ B @ A @ G2 @ T4 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_3954_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T4: set @ B,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G2 @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ G2 @ S ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_3955_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T4: set @ B,H: B > A,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G2 @ S )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_3956_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T4: set @ B,G2: B > A,H: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G2 @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_3957_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ I5 )
=> ( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I5 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_3958_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B2: set @ B,A4: set @ B,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B2 @ A4 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ B2 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_3959_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S: set @ B,G2: B > A,H: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ S ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_3960_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S: set @ B,H: B > A,G2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H @ T4 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_3961_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_3962_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ S )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ T4 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_3963_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: set @ B,A4: set @ B,B2: set @ B,G2: B > A,H: B > A] :
( ( finite_finite2 @ B @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C2 )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ ( minus_minus @ ( set @ B ) @ C2 @ A4 ) )
=> ( ( G2 @ A7 )
= ( one_one @ A ) ) )
=> ( ! [B7: B] :
( ( member @ B @ B7 @ ( minus_minus @ ( set @ B ) @ C2 @ B2 ) )
=> ( ( H @ B7 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ C2 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ C2 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_3964_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: set @ B,A4: set @ B,B2: set @ B,G2: B > A,H: B > A] :
( ( finite_finite2 @ B @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C2 )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ ( minus_minus @ ( set @ B ) @ C2 @ A4 ) )
=> ( ( G2 @ A7 )
= ( one_one @ A ) ) )
=> ( ! [B7: B] :
( ( member @ B @ B7 @ ( minus_minus @ ( set @ B ) @ C2 @ B2 ) )
=> ( ( H @ B7 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ B2 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ C2 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ C2 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_3965_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B2: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_inter
thf(fact_3966_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G2: B > A,B2: set @ B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_3967_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S: set @ B,H: B > A,G2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H @ I2 )
= ( one_one @ A ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ S @ T4 ) )
=> ( ( G2 @ I2 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S @ T4 ) )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H @ T4 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_3968_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_3969_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G2 @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_3970_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_3971_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,G2: B > A,H: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( G2 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( H @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I4: B] : ( plus_plus @ A @ ( G2 @ I4 ) @ ( H @ I4 ) )
@ I5 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G2 @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H @ I5 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_3972_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P5: B > A,I7: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I7 )
& ( ( P5 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P5
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I7 )
& ( ( P5 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_3973_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,P: B > $o,H: B > A,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H @ X2 ) @ ( G2 @ X2 ) )
@ A4 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_3974_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_3975_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ M )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_3976_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_3977_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( G2 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_3978_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F3: B > A,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
& ( ord_less @ A @ ( F3 @ I2 ) @ ( G2 @ I2 ) ) ) )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_3979_even__prod__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A4: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F3 @ X2 ) ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_3980_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G2: B > A,X: B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert2 @ B @ X @ A4 ) )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_3981_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( member @ B @ X @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_3982_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B2: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ B2 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_3983_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B2: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ( inf_inf @ ( set @ B ) @ A4 @ B2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ B2 ) ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_3984_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B2: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ B2 @ A4 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_3985_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A,P6: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P6 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P6 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_3986_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B3: B > A,C3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C3 @ K3 ) )
@ S )
= ( times_times @ A @ ( B3 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C3 @ K3 ) )
@ S )
= ( groups7121269368397514597t_prod @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_3987_norm__prod__diff,axiom,
! [A: $tType,I6: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [I5: set @ I6,Z: I6 > A,W2: I6 > A] :
( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( Z @ I2 ) ) @ ( one_one @ real ) ) )
=> ( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( W2 @ I2 ) ) @ ( one_one @ real ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( groups7121269368397514597t_prod @ I6 @ A @ Z @ I5 ) @ ( groups7121269368397514597t_prod @ I6 @ A @ W2 @ I5 ) ) )
@ ( groups7311177749621191930dd_sum @ I6 @ real
@ ^ [I4: I6] : ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( Z @ I4 ) @ ( W2 @ I4 ) ) )
@ I5 ) ) ) ) ) ).
% norm_prod_diff
thf(fact_3988_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_3989_fact__eq__fact__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( semiring_char_0_fact @ nat @ M )
= ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N )
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_3990_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B2: set @ A,A4: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ! [B7: A] :
( ( member @ A @ B7 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ B7 ) ) )
=> ( ! [A7: A] :
( ( member @ A @ A7 @ A4 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ A7 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A4 ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ B2 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_3991_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F3: B > A,N: A,K: nat] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
& ( ord_less_eq @ A @ ( F3 @ I2 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A4 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_3992_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A4: set @ B,B2: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) )
=> ( ( F3 @ X3 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_3993_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A4: set @ B,F3: B > A,A3: B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A3 @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) @ ( F3 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F3 @ A4 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_3994_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B3: B > A,C3: A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ C3 )
@ S )
= ( times_times @ A @ ( B3 @ A3 ) @ ( power_power @ A @ C3 @ ( minus_minus @ nat @ ( finite_card @ B @ S ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ C3 )
@ S )
= ( power_power @ A @ C3 @ ( finite_card @ B @ S ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_3995_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_3996_fact__div__fact,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_3997_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_3998_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I5: set @ A,F3: A > B,I: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I5 )
& ( ( F3 @ I4 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I5 ) @ ( F3 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I5 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_3999_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P6: nat,K: nat,G2: nat > A,H: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P6 )
=> ( ( ord_less_eq @ nat @ K @ P6 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P6 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P6 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_4000_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K ) )
= ( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_4001_of__nat__code__if,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N2: nat] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [M2: nat,Q6: nat] :
( if @ A
@ ( Q6
= ( zero_zero @ nat ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( one_one @ A ) ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_4002_card__lists__distinct__length__eq,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A4 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_4003_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A4: set @ A] :
( ( ord_less @ nat @ K @ ( finite_card @ A @ A4 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_4004_normalize__negative,axiom,
! [Q5: int,P6: int] :
( ( ord_less @ int @ Q5 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P6 @ Q5 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P6 ) @ ( uminus_uminus @ int @ Q5 ) ) ) ) ) ).
% normalize_negative
thf(fact_4005_prod__eq__1__iff,axiom,
! [A: $tType,A4: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( groups7121269368397514597t_prod @ A @ nat @ F3 @ A4 )
= ( one_one @ nat ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( F3 @ X2 )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_4006_prod__pos__nat__iff,axiom,
! [A: $tType,A4: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F3 @ A4 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X2 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_4007_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_4008_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_4009_finite__distinct__list,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [Xs2: list @ A] :
( ( ( set2 @ A @ Xs2 )
= A4 )
& ( distinct @ A @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_4010_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_4011_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs3: list @ A] :
! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( I4 != J3 )
=> ( ( nth @ A @ Xs3 @ I4 )
!= ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_4012_distinct__Ex1,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [X3: nat] :
( ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ X3 )
= X )
& ! [Y5: nat] :
( ( ( ord_less @ nat @ Y5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ Y5 )
= X ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_4013_normalize__denom__pos,axiom,
! [R2: product_prod @ int @ int,P6: int,Q5: int] :
( ( ( normalize @ R2 )
= ( product_Pair @ int @ int @ P6 @ Q5 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q5 ) ) ).
% normalize_denom_pos
thf(fact_4014_ln__prod,axiom,
! [A: $tType,I5: set @ A,F3: A > real] :
( ( finite_finite2 @ A @ I5 )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ I2 ) ) )
=> ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F3 @ I5 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X2: A] : ( ln_ln @ real @ ( F3 @ X2 ) )
@ I5 ) ) ) ) ).
% ln_prod
thf(fact_4015_distinct__list__update,axiom,
! [A: $tType,Xs: list @ A,A3: A,I: nat] :
( ( distinct @ A @ Xs )
=> ( ~ ( member @ A @ A3 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ ( nth @ A @ Xs @ I ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs @ I @ A3 ) ) ) ) ).
% distinct_list_update
thf(fact_4016_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G2 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_4017_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G2 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_4018_set__update__distinct,axiom,
! [A: $tType,Xs: list @ A,N: nat,X: A] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs @ N @ X ) )
= ( insert2 @ A @ X @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ ( nth @ A @ Xs @ N ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_4019_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A4: A > B > $o,B2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A4 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B2 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_4020_finite__enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S ) )
=> ( ( infini527867602293511546merate @ A @ S @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S8: A] :
( ( member @ A @ S8 @ S )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ N ) @ S8 ) ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_4021_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_4022_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_4023_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_4024_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2
thf(fact_4025_LeastI__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI_ex
thf(fact_4026_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_ex
thf(fact_4027_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI
thf(fact_4028_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [A7: A] :
( ( P @ A7 )
=> ( ! [B11: A] :
( ( P @ B11 )
=> ( ord_less_eq @ A @ A7 @ B11 ) )
=> ( Q @ A7 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_4029_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [A7: A] :
( ( P @ A7 )
=> ( ! [B11: A] :
( ( P @ B11 )
=> ( ord_less_eq @ A @ A7 @ B11 ) )
=> ( Q @ A7 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_4030_Least__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) )
=> ( ( ord_Least @ A @ P )
= X ) ) ) ) ).
% Least_equality
thf(fact_4031_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ X3 @ Y5 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_4032_Least1__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,Z: A] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X5 @ Y2 ) )
& ! [Y2: A] :
( ( ( P @ Y2 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y2 @ Ya2 ) ) )
=> ( Y2 = X5 ) ) )
=> ( ( P @ Z )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z ) ) ) ) ).
% Least1_le
thf(fact_4033_Least1I,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X5 @ Y2 ) )
& ! [Y2: A] :
( ( ( P @ Y2 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y2 @ Ya2 ) ) )
=> ( Y2 = X5 ) ) )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% Least1I
thf(fact_4034_Least__le,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ K ) ) ) ).
% Least_le
thf(fact_4035_not__less__Least,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [K: A,P: A > $o] :
( ( ord_less @ A @ K @ ( ord_Least @ A @ P ) )
=> ~ ( P @ K ) ) ) ).
% not_less_Least
thf(fact_4036_Least__Suc2,axiom,
! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
( ( P @ N )
=> ( ( Q @ M )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ! [K2: nat] :
( ( P @ ( suc @ K2 ) )
= ( Q @ K2 ) )
=> ( ( ord_Least @ nat @ P )
= ( suc @ ( ord_Least @ nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_4037_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( suc
@ ( ord_Least @ nat
@ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_4038_size__list__estimation,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: nat,F3: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less @ nat @ Y @ ( F3 @ X ) )
=> ( ord_less @ nat @ Y @ ( size_list @ A @ F3 @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_4039_size__list__pointwise,axiom,
! [A: $tType,Xs: list @ A,F3: A > nat,G2: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F3 @ Xs ) @ ( size_list @ A @ G2 @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_4040_size__list__estimation_H,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: nat,F3: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ Y @ ( F3 @ X ) )
=> ( ord_less_eq @ nat @ Y @ ( size_list @ A @ F3 @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_4041_enumerate__0,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A] :
( ( infini527867602293511546merate @ A @ S @ ( zero_zero @ nat ) )
= ( ord_Least @ A
@ ^ [N2: A] : ( member @ A @ N2 @ S ) ) ) ) ).
% enumerate_0
thf(fact_4042_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_4043_enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S )
=> ( ( infini527867602293511546merate @ A @ S @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S8: A] :
( ( member @ A @ S8 @ S )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ N ) @ S8 ) ) ) ) ) ) ).
% enumerate_Suc''
thf(fact_4044_enumerate__Suc,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S @ ( suc @ N ) )
= ( infini527867602293511546merate @ A
@ ( minus_minus @ ( set @ A ) @ S
@ ( insert2 @ A
@ ( ord_Least @ A
@ ^ [N2: A] : ( member @ A @ N2 @ S ) )
@ ( bot_bot @ ( set @ A ) ) ) )
@ N ) ) ) ).
% enumerate_Suc
thf(fact_4045_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z8: int,Z6: int] :
( ( ord_less_eq @ int @ D5 @ Z6 )
& ( ord_less @ int @ Z8 @ Z6 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_4046_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z8: int,Z6: int] :
( ( ord_less_eq @ int @ D5 @ Z8 )
& ( ord_less @ int @ Z8 @ Z6 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_4047_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_4048_card__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys2 ) )
= ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% card_disjoint_shuffles
thf(fact_4049_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_4050_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_4051_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( set_or7035219750837199246ssThan @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_4052_ivl__subset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,J: A,M: A,N: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M @ N ) )
= ( ( ord_less_eq @ A @ J @ I )
| ( ( ord_less_eq @ A @ M @ I )
& ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% ivl_subset
thf(fact_4053_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_4054_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) )
= ( ~ ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_4055_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% infinite_Ico_iff
thf(fact_4056_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,N: A,M: A] :
( ( ord_less_eq @ A @ I @ N )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M ) @ ( set_or7035219750837199246ssThan @ A @ I @ N ) )
= ( set_or7035219750837199246ssThan @ A @ N @ M ) ) ) ) ).
% ivl_diff
thf(fact_4057_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
= ( insert2 @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_4058_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G2: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_4059_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G2: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_4060_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B3 )
= ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
= ( ( A3 = C3 )
& ( B3 = D2 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_4061_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B3 )
= ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( A3 = C3 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_4062_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B3 )
= ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( B3 = D2 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_4063_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
| ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_4064_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) ) ) ) ).
% infinite_Ico
thf(fact_4065_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_4066_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_4067_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_4068_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_4069_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_4070_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_4071_set__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys2: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys2 ) )
=> ( ( set2 @ A @ Zs )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) ) ) ) ).
% set_shuffles
thf(fact_4072_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A3: B,C3: B,B3: B,D2: B,G2: B > A,H: B > A] :
( ( A3 = C3 )
=> ( ( B3 = D2 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C3 @ X3 )
=> ( ( ord_less @ B @ X3 @ D2 )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( set_or7035219750837199246ssThan @ B @ A3 @ B3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ ( set_or7035219750837199246ssThan @ B @ C3 @ D2 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_4073_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A3: B,C3: B,B3: B,D2: B,G2: B > A,H: B > A] :
( ( A3 = C3 )
=> ( ( B3 = D2 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C3 @ X3 )
=> ( ( ord_less @ B @ X3 @ D2 )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( set_or7035219750837199246ssThan @ B @ A3 @ B3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H @ ( set_or7035219750837199246ssThan @ B @ C3 @ D2 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_4074_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_4075_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_4076_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,P6: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P6 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ N @ P6 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ P6 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_4077_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,P6: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P6 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M @ P6 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P6 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_4078_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_4079_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_4080_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,P6: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P6 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ N @ P6 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ P6 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_4081_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert2 @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_4082_subset__eq__atLeast0__lessThan__finite,axiom,
! [N6: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N6 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_4083_subset__card__intvl__is__intvl,axiom,
! [A4: set @ nat,K: nat] :
( ( ord_less_eq @ ( set @ nat ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A4 ) ) ) )
=> ( A4
= ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A4 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_4084_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
= ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less @ A @ B3 @ D2 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4085_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_4086_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_4087_sum__shift__lb__Suc0__0__upt,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,K: nat] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_4088_sum_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G2 @ N ) ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_4089_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G2 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_4090_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat,B3: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B3 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B3 ) ) @ ( G2 @ B3 ) ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_4091_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A5: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A5 @ B5 ) @ ( insert2 @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_4092_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G2 @ N ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_4093_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( times_times @ A @ ( G2 @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_4094_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat,B3: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B3 ) ) @ ( G2 @ B3 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_4095_sum_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G2 @ N ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% sum.last_plus
thf(fact_4096_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G2 @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_4097_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_4098_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I4 ) ) @ ( F3 @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ M ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_4099_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
= ( insert2 @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_4100_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I4 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_4101_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I4 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_4102_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( distinct @ A @ Ys2 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys2 ) )
=> ( distinct @ A @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_4103_prod__Suc__Suc__fact,axiom,
! [N: nat] :
( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( semiring_char_0_fact @ nat @ N ) ) ).
% prod_Suc_Suc_fact
thf(fact_4104_prod__Suc__fact,axiom,
! [N: nat] :
( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
= ( semiring_char_0_fact @ nat @ N ) ) ).
% prod_Suc_fact
thf(fact_4105_subset__eq__atLeast0__lessThan__card,axiom,
! [N6: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N6 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_4106_card__sum__le__nat__sum,axiom,
! [S: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ S ) ) ).
% card_sum_le_nat_sum
thf(fact_4107_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 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_4108_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G2: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ) ).
% sum.head_if
thf(fact_4109_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G2: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ) ).
% prod.head_if
thf(fact_4110_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N2: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_4111_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_4112_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N2: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_4113_summable__Cauchy,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ( ( summable @ A )
= ( ^ [F2: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [N5: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ N5 @ M2 )
=> ! [N2: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ).
% summable_Cauchy
thf(fact_4114_sums__group,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A,S3: A,K: nat] :
( ( sums @ A @ F3 @ S3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( sums @ A
@ ^ [N2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ K ) @ K ) ) )
@ S3 ) ) ) ) ).
% sums_group
thf(fact_4115_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N2: nat,A5: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ K3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% take_bit_sum
thf(fact_4116_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_4117_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N @ K ) @ N ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_split
thf(fact_4118_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_4119_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_4120_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_4121_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_4122_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_4123_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F2: B > A,A5: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ ( nth @ B @ Xs3 @ N2 ) ) @ ( power_power @ A @ A5 @ N2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_4124_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: nat > A,B3: nat > A] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( A3 @ I2 ) @ ( A3 @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( B3 @ J2 ) @ ( B3 @ I2 ) ) ) )
=> ( ord_less_eq @ A
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A3 @ K3 ) @ ( B3 @ K3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_4125_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A3: nat > nat,B3: nat > nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( A3 @ I2 ) @ ( A3 @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( B3 @ J2 ) @ ( B3 @ I2 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ ( A3 @ I4 ) @ ( B3 @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ nat @ ( groups7311177749621191930dd_sum @ nat @ nat @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ nat @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_4126_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L2: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q6: 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 ) ) @ Q6 ) @ ( 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 ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_4127_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_4128_less__eq__integer__code_I1_J,axiom,
ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ).
% less_eq_integer_code(1)
thf(fact_4129_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_4130_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_4131_card__Pow,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ ( set @ A ) @ ( pow2 @ A @ A4 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A4 ) ) ) ) ).
% card_Pow
thf(fact_4132_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_4133_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa3: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa3 )
= Y )
=> ( ( ( ord_less_eq @ nat @ Xa3 @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa3 @ ( minus_minus @ nat @ X @ Xa3 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa3 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa3 @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_4134_Pow__empty,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_empty
thf(fact_4135_Pow__singleton__iff,axiom,
! [A: $tType,X4: set @ A,Y6: set @ A] :
( ( ( pow2 @ A @ X4 )
= ( insert2 @ ( set @ A ) @ Y6 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X4
= ( bot_bot @ ( set @ A ) ) )
& ( Y6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_4136_Pow__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A4 @ ( pow2 @ A @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% Pow_iff
thf(fact_4137_PowI,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( member @ ( set @ A ) @ A4 @ ( pow2 @ A @ B2 ) ) ) ).
% PowI
thf(fact_4138_Pow__Int__eq,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( pow2 @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A4 ) @ ( pow2 @ A @ B2 ) ) ) ).
% Pow_Int_eq
thf(fact_4139_finite__Pow__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ ( set @ A ) @ ( pow2 @ A @ A4 ) )
= ( finite_finite2 @ A @ A4 ) ) ).
% finite_Pow_iff
thf(fact_4140_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_4141_less__integer__code_I1_J,axiom,
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(1)
thf(fact_4142_Un__Pow__subset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A4 ) @ ( pow2 @ A @ B2 ) ) @ ( pow2 @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% Un_Pow_subset
thf(fact_4143_less__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( ord_less @ int @ Xa3 @ X ) ) ).
% less_integer.abs_eq
thf(fact_4144_PowD,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A4 @ ( pow2 @ A @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% PowD
thf(fact_4145_Pow__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A4 ) @ ( pow2 @ A @ B2 ) ) ) ).
% Pow_mono
thf(fact_4146_Pow__top,axiom,
! [A: $tType,A4: set @ A] : ( member @ ( set @ A ) @ A4 @ ( pow2 @ A @ A4 ) ) ).
% Pow_top
thf(fact_4147_Pow__not__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( pow2 @ A @ A4 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_4148_Pow__bottom,axiom,
! [A: $tType,B2: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow2 @ A @ B2 ) ) ).
% Pow_bottom
thf(fact_4149_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_4150_less__eq__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( ord_less_eq @ int @ Xa3 @ X ) ) ).
% less_eq_integer.abs_eq
thf(fact_4151_binomial__def,axiom,
( binomial
= ( ^ [N2: nat,K3: nat] :
( finite_card @ ( set @ nat )
@ ( collect @ ( set @ nat )
@ ^ [K5: set @ nat] :
( ( member @ ( set @ nat ) @ K5 @ ( pow2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
& ( ( finite_card @ nat @ K5 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_4152_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_4153_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa3: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa3 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa3 @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa3 @ ( minus_minus @ nat @ X @ Xa3 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa3 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa3 @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_4154_card__partition,axiom,
! [A: $tType,C2: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ ( set @ A ) @ C2 )
=> ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
=> ( ! [C5: set @ A] :
( ( member @ ( set @ A ) @ C5 @ C2 )
=> ( ( finite_card @ A @ C5 )
= K ) )
=> ( ! [C1: set @ A,C22: set @ A] :
( ( member @ ( set @ A ) @ C1 @ C2 )
=> ( ( member @ ( set @ A ) @ C22 @ C2 )
=> ( ( C1 != C22 )
=> ( ( inf_inf @ ( set @ A ) @ C1 @ C22 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( times_times @ nat @ K @ ( finite_card @ ( set @ A ) @ C2 ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_4155_finite__enumerate,axiom,
! [S: set @ nat] :
( ( finite_finite2 @ nat @ S )
=> ? [R3: nat > nat] :
( ( strict_mono_on @ nat @ nat @ R3 @ ( set_ord_lessThan @ nat @ ( finite_card @ nat @ S ) ) )
& ! [N4: nat] :
( ( ord_less @ nat @ N4 @ ( finite_card @ nat @ S ) )
=> ( member @ nat @ ( R3 @ N4 ) @ S ) ) ) ) ).
% finite_enumerate
thf(fact_4156_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_4157_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ A4 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( X2
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_4158_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( ( complete_Sup_Sup @ A @ A4 )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( X2
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_4159_Sup__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_4160_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastAtMost
thf(fact_4161_cSup__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_atLeastAtMost
thf(fact_4162_cSup__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% cSup_singleton
thf(fact_4163_cSup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_atLeastLessThan
thf(fact_4164_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastLessThan
thf(fact_4165_Sup__insert,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: A,A4: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ).
% Sup_insert
thf(fact_4166_finite__Union,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A4 )
=> ( ! [M9: set @ A] :
( ( member @ ( set @ A ) @ M9 @ A4 )
=> ( finite_finite2 @ A @ M9 ) )
=> ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) ) ) ) ).
% finite_Union
thf(fact_4167_Union__Un__distrib,axiom,
! [A: $tType,A4: set @ ( set @ A ),B2: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% Union_Un_distrib
thf(fact_4168_subset__Pow__Union,axiom,
! [A: $tType,A4: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ A4 @ ( pow2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) ) ) ).
% subset_Pow_Union
thf(fact_4169_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: A,S: set @ A] :
( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ S ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S )
& ( ord_less @ A @ A3 @ X2 ) ) ) ) ) ).
% less_Sup_iff
thf(fact_4170_Union__insert,axiom,
! [A: $tType,A3: set @ A,B2: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( insert2 @ ( set @ A ) @ A3 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% Union_insert
thf(fact_4171_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_4172_Union__empty__conv,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ A4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A4 )
=> ( X2
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_empty_conv
thf(fact_4173_empty__Union__conv,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
= ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A4 )
=> ( X2
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% empty_Union_conv
thf(fact_4174_Union__subsetI,axiom,
! [A: $tType,A4: set @ ( set @ A ),B2: set @ ( set @ A )] :
( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
=> ? [Y5: set @ A] :
( ( member @ ( set @ A ) @ Y5 @ B2 )
& ( ord_less_eq @ ( set @ A ) @ X3 @ Y5 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% Union_subsetI
thf(fact_4175_Union__upper,axiom,
! [A: $tType,B2: set @ A,A4: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B2 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) ) ) ).
% Union_upper
thf(fact_4176_Union__least,axiom,
! [A: $tType,A4: set @ ( set @ A ),C2: set @ A] :
( ! [X9: set @ A] :
( ( member @ ( set @ A ) @ X9 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ X9 @ C2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) @ C2 ) ) ).
% Union_least
thf(fact_4177_Union__mono,axiom,
! [A: $tType,A4: set @ ( set @ A ),B2: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% Union_mono
thf(fact_4178_Sup__upper2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A4: set @ A,V2: A] :
( ( member @ A @ U @ A4 )
=> ( ( ord_less_eq @ A @ V2 @ U )
=> ( ord_less_eq @ A @ V2 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% Sup_upper2
thf(fact_4179_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B3: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ B3 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ A @ X2 @ B3 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_4180_Sup__upper,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ).
% Sup_upper
thf(fact_4181_Sup__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,Z: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ Z ) ) ) ).
% Sup_least
thf(fact_4182_Sup__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ! [A7: A] :
( ( member @ A @ A7 @ A4 )
=> ? [X5: A] :
( ( member @ A @ X5 @ B2 )
& ( ord_less_eq @ A @ A7 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% Sup_mono
thf(fact_4183_Sup__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,X: A] :
( ! [Y2: A] :
( ( member @ A @ Y2 @ A4 )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ! [Y2: A] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ A4 )
=> ( ord_less_eq @ A @ Z4 @ Y2 ) )
=> ( ord_less_eq @ A @ X @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ A4 )
= X ) ) ) ) ).
% Sup_eqI
thf(fact_4184_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z: A,X4: set @ A] :
( ( member @ A @ Z @ X4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ( complete_Sup_Sup @ A @ X4 )
= Z ) ) ) ) ).
% cSup_eq_maximum
thf(fact_4185_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_bot @ A ) )
=> ! [X4: set @ A,A3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ord_less_eq @ A @ X3 @ A3 ) )
=> ( ! [Y2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X4 )
=> ( ord_less_eq @ A @ X5 @ Y2 ) )
=> ( ord_less_eq @ A @ A3 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ X4 )
= A3 ) ) ) ) ).
% cSup_eq
thf(fact_4186_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,A4: set @ A] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A4 ) )
= ( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X )
=> ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ord_less @ A @ Y3 @ X2 ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_4187_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,Z: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X4 ) @ Z ) ) ) ) ).
% cSup_least
thf(fact_4188_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,A3: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ord_less_eq @ A @ X3 @ A3 ) )
=> ( ! [Y2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X4 )
=> ( ord_less_eq @ A @ X5 @ Y2 ) )
=> ( ord_less_eq @ A @ A3 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ X4 )
= A3 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_4189_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A4 )
=> ( ord_less_eq @ A @ U @ V3 ) )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_4190_le__cSup__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,X: A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( member @ A @ X @ X4 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X4 ) ) ) ) ) ).
% le_cSup_finite
thf(fact_4191_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% Sup_subset_mono
thf(fact_4192_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y: A,X4: set @ A] :
( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X4 ) )
=> ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ) ) ).
% less_cSupE
thf(fact_4193_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A,Z: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z @ ( complete_Sup_Sup @ A @ X4 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ X4 )
& ( ord_less @ A @ Z @ X3 ) ) ) ) ) ).
% less_cSupD
thf(fact_4194_finite__imp__Sup__less,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A,X: A,A3: A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( member @ A @ X @ X4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ord_less @ A @ X3 @ A3 ) )
=> ( ord_less @ A @ ( complete_Sup_Sup @ A @ X4 ) @ A3 ) ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_4195_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B2: set @ A,A3: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B2 ) @ A3 )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( ( inf_inf @ A @ X2 @ A3 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_4196_Sup__union__distrib,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% Sup_union_distrib
thf(fact_4197_insert__partition,axiom,
! [A: $tType,X: set @ A,F4: set @ ( set @ A )] :
( ~ ( member @ ( set @ A ) @ X @ F4 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ ( insert2 @ ( set @ A ) @ X @ F4 ) )
=> ! [Xa2: set @ A] :
( ( member @ ( set @ A ) @ Xa2 @ ( insert2 @ ( set @ A ) @ X @ F4 ) )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ Xa2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ X @ ( complete_Sup_Sup @ ( set @ A ) @ F4 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_partition
thf(fact_4198_Union__disjoint,axiom,
! [A: $tType,C2: set @ ( set @ A ),A4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) @ A4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C2 )
=> ( ( inf_inf @ ( set @ A ) @ X2 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_disjoint
thf(fact_4199_Union__Int__subset,axiom,
! [A: $tType,A4: set @ ( set @ A ),B2: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A4 @ B2 ) ) @ ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% Union_Int_subset
thf(fact_4200_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_4201_finite__UnionD,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
=> ( finite_finite2 @ ( set @ A ) @ A4 ) ) ).
% finite_UnionD
thf(fact_4202_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A,A3: A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X4 ) @ A3 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ X4 )
=> ( ord_less @ A @ X2 @ A3 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_4203_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,A3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S ) ) @ A3 ) ) ) ) ).
% cSup_abs_le
thf(fact_4204_Sup__inter__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B2: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% Sup_inter_less_eq
thf(fact_4205_finite__Sup__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( member @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ A4 ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A4 ) @ A4 ) ) ) ) ) ).
% finite_Sup_in
thf(fact_4206_card__Union__le__sum__card__weak,axiom,
! [A: $tType,U3: set @ ( set @ A )] :
( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ U3 )
=> ( finite_finite2 @ A @ X3 ) )
=> ( 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_4207_Sup__fin__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% Sup_fin_Sup
thf(fact_4208_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X4 )
= ( lattic5882676163264333800up_fin @ A @ X4 ) ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_4209_finite__subset__Union,axiom,
! [A: $tType,A4: set @ A,B12: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( 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 ) @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ F7 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_4210_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,L: A,E2: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S ) @ L ) ) @ E2 ) ) ) ) ).
% cSup_asclose
thf(fact_4211_Sup__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A,X: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( S
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X @ S ) )
= X ) )
& ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X @ S ) )
= ( ord_max @ A @ X @ ( complete_Sup_Sup @ A @ S ) ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_4212_dvd__partition,axiom,
! [A: $tType,C2: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C2 )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ X3 ) ) )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C2 )
=> ! [Xa2: set @ A] :
( ( member @ ( set @ A ) @ Xa2 @ C2 )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ Xa2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_4213_ccpo__Sup__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% ccpo_Sup_singleton
thf(fact_4214_ccSup__empty,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% ccSup_empty
thf(fact_4215_card__UNION,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A4 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
=> ( finite_finite2 @ A @ X3 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
= ( nat2
@ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
@ ^ [I7: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I7 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I7 ) ) ) )
@ ( collect @ ( set @ ( set @ A ) )
@ ^ [I7: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I7 @ A4 )
& ( I7
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_4216_Sup__finite__insert,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ! [A3: A,A4: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ).
% Sup_finite_insert
thf(fact_4217_finite__Inter,axiom,
! [A: $tType,M5: set @ ( set @ A )] :
( ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ M5 )
& ( finite_finite2 @ A @ X5 ) )
=> ( finite_finite2 @ A @ ( complete_Inf_Inf @ ( set @ A ) @ M5 ) ) ) ).
% finite_Inter
thf(fact_4218_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_4219_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( ( complete_Inf_Inf @ A @ A4 )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A4 )
& ( ord_less @ A @ Y3 @ X2 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_4220_cInf__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_atLeastAtMost
thf(fact_4221_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastAtMost
thf(fact_4222_cInf__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% cInf_singleton
thf(fact_4223_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastLessThan
thf(fact_4224_cInf__atLeastLessThan,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_atLeastLessThan
thf(fact_4225_Inf__insert,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: A,A4: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ A4 ) ) ) ) ).
% Inf_insert
thf(fact_4226_Inf__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% Inf_atMost
thf(fact_4227_Inf__finite__insert,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ! [A3: A,A4: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ A4 ) ) ) ) ).
% Inf_finite_insert
thf(fact_4228_Inf__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,Z: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ Z @ X3 ) )
=> ( ord_less_eq @ A @ Z @ ( complete_Inf_Inf @ A @ A4 ) ) ) ) ).
% Inf_greatest
thf(fact_4229_le__Inf__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: A,A4: set @ A] :
( ( ord_less_eq @ A @ B3 @ ( complete_Inf_Inf @ A @ A4 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ A @ B3 @ X2 ) ) ) ) ) ).
% le_Inf_iff
thf(fact_4230_Inf__lower2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A4: set @ A,V2: A] :
( ( member @ A @ U @ A4 )
=> ( ( ord_less_eq @ A @ U @ V2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ V2 ) ) ) ) ).
% Inf_lower2
thf(fact_4231_Inf__lower,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ X ) ) ) ).
% Inf_lower
thf(fact_4232_Inf__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ! [B7: A] :
( ( member @ A @ B7 @ B2 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A4 )
& ( ord_less_eq @ A @ X5 @ B7 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ).
% Inf_mono
thf(fact_4233_Inf__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,X: A] :
( ! [I2: A] :
( ( member @ A @ I2 @ A4 )
=> ( ord_less_eq @ A @ X @ I2 ) )
=> ( ! [Y2: A] :
( ! [I3: A] :
( ( member @ A @ I3 @ A4 )
=> ( ord_less_eq @ A @ Y2 @ I3 ) )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( complete_Inf_Inf @ A @ A4 )
= X ) ) ) ) ).
% Inf_eqI
thf(fact_4234_Inter__greatest,axiom,
! [A: $tType,A4: set @ ( set @ A ),C2: set @ A] :
( ! [X9: set @ A] :
( ( member @ ( set @ A ) @ X9 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ C2 @ X9 ) )
=> ( ord_less_eq @ ( set @ A ) @ C2 @ ( complete_Inf_Inf @ ( set @ A ) @ A4 ) ) ) ).
% Inter_greatest
thf(fact_4235_Inter__lower,axiom,
! [A: $tType,B2: set @ A,A4: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B2 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A4 ) @ B2 ) ) ).
% Inter_lower
thf(fact_4236_cInf__eq__minimum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z: A,X4: set @ A] :
( ( member @ A @ Z @ X4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ord_less_eq @ A @ Z @ X3 ) )
=> ( ( complete_Inf_Inf @ A @ X4 )
= Z ) ) ) ) ).
% cInf_eq_minimum
thf(fact_4237_cInf__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_top @ A ) )
=> ! [X4: set @ A,A3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ord_less_eq @ A @ A3 @ X3 ) )
=> ( ! [Y2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X4 )
=> ( ord_less_eq @ A @ Y2 @ X5 ) )
=> ( ord_less_eq @ A @ Y2 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ X4 )
= A3 ) ) ) ) ).
% cInf_eq
thf(fact_4238_Inf__less__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [S: set @ A,A3: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S ) @ A3 )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S )
& ( ord_less @ A @ X2 @ A3 ) ) ) ) ) ).
% Inf_less_iff
thf(fact_4239_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ X )
= ( ! [Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ord_less @ A @ X2 @ Y3 ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_4240_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A4 )
=> ( ord_less_eq @ A @ V3 @ U ) )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_4241_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,A3: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ord_less_eq @ A @ A3 @ X3 ) )
=> ( ! [Y2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X4 )
=> ( ord_less_eq @ A @ Y2 @ X5 ) )
=> ( ord_less_eq @ A @ Y2 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ X4 )
= A3 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_4242_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,Z: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ord_less_eq @ A @ Z @ X3 ) )
=> ( ord_less_eq @ A @ Z @ ( complete_Inf_Inf @ A @ X4 ) ) ) ) ) ).
% cInf_greatest
thf(fact_4243_cInf__le__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,X: A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( member @ A @ X @ X4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X4 ) @ X ) ) ) ) ).
% cInf_le_finite
thf(fact_4244_Inf__superset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ).
% Inf_superset_mono
thf(fact_4245_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A,Z: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X4 ) @ Z )
=> ? [X3: A] :
( ( member @ A @ X3 @ X4 )
& ( ord_less @ A @ X3 @ Z ) ) ) ) ) ).
% cInf_lessD
thf(fact_4246_finite__imp__less__Inf,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A,X: A,A3: A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( member @ A @ X @ X4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ord_less @ A @ A3 @ X3 ) )
=> ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X4 ) ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_4247_Inf__union__distrib,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ).
% Inf_union_distrib
thf(fact_4248_Inter__anti__mono,axiom,
! [A: $tType,B2: set @ ( set @ A ),A4: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B2 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A4 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_4249_Inter__subset,axiom,
! [A: $tType,A4: set @ ( set @ A ),B2: set @ A] :
( ! [X9: set @ A] :
( ( member @ ( set @ A ) @ X9 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ X9 @ B2 ) )
=> ( ( A4
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A4 ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_4250_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A,A3: A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X4 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ X4 )
=> ( ord_less @ A @ A3 @ X2 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_4251_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ).
% Inf_le_Sup
thf(fact_4252_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,A3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S ) ) @ A3 ) ) ) ) ).
% cInf_abs_ge
thf(fact_4253_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B2: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_4254_finite__Inf__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( member @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ A4 ) ) )
=> ( member @ A @ ( complete_Inf_Inf @ A @ A4 ) @ A4 ) ) ) ) ) ).
% finite_Inf_in
thf(fact_4255_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X4 )
= ( lattic7752659483105999362nf_fin @ A @ X4 ) ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_4256_Inf__fin__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ).
% Inf_fin_Inf
thf(fact_4257_Inter__Un__subset,axiom,
! [A: $tType,A4: set @ ( set @ A ),B2: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A4 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B2 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A4 @ B2 ) ) ) ).
% Inter_Un_subset
thf(fact_4258_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,L: A,E2: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S ) @ L ) ) @ E2 ) ) ) ) ).
% cInf_asclose
thf(fact_4259_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F2: A > nat,R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y3: A] :
( ( ord_less @ nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) )
| ( ( ord_less_eq @ nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R6 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_4260_strict__mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [F3: A > B,A4: set @ A,R2: A,S3: A] :
( ( strict_mono_on @ A @ B @ F3 @ A4 )
=> ( ( member @ A @ R2 @ A4 )
=> ( ( member @ A @ S3 @ A4 )
=> ( ( ord_less @ A @ R2 @ S3 )
=> ( ord_less @ B @ ( F3 @ R2 ) @ ( F3 @ S3 ) ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_4261_strict__mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [A4: set @ A,F3: A > B] :
( ! [R3: A,S4: A] :
( ( member @ A @ R3 @ A4 )
=> ( ( member @ A @ S4 @ A4 )
=> ( ( ord_less @ A @ R3 @ S4 )
=> ( ord_less @ B @ ( F3 @ R3 ) @ ( F3 @ S4 ) ) ) ) )
=> ( strict_mono_on @ A @ B @ F3 @ A4 ) ) ) ).
% strict_mono_onI
thf(fact_4262_strict__mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( strict_mono_on @ A @ B )
= ( ^ [F2: 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 @ ( F2 @ R5 ) @ ( F2 @ S8 ) ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_4263_Inf__nat__def1,axiom,
! [K4: set @ nat] :
( ( K4
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( complete_Inf_Inf @ nat @ K4 ) @ K4 ) ) ).
% Inf_nat_def1
thf(fact_4264_mlex__less,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F3 @ R ) ) ) ).
% mlex_less
thf(fact_4265_mlex__iff,axiom,
! [A: $tType,X: A,Y: A,F3: A > nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F3 @ R ) )
= ( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X )
= ( F3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% mlex_iff
thf(fact_4266_mlex__leq,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F3 @ R ) ) ) ) ).
% mlex_leq
thf(fact_4267_strict__mono__on__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( preorder @ B ) )
=> ! [F3: A > B,A4: set @ A,X: A,Y: A] :
( ( strict_mono_on @ A @ B @ F3 @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_4268_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_4269_set__remove1__eq,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( set2 @ A @ ( remove1 @ A @ X @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_remove1_eq
thf(fact_4270_nth__enumerate__eq,axiom,
! [A: $tType,M: nat,Xs: list @ A,N: nat] :
( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ M )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M ) @ ( nth @ A @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_4271_semiring__char__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiri4206861660011772517g_char @ A )
= ( ^ [Uu4: itself @ A] :
( gcd_Gcd @ nat
@ ( collect @ nat
@ ^ [N2: nat] :
( ( semiring_1_of_nat @ A @ N2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% semiring_char_def
thf(fact_4272_set__remove1__subset,axiom,
! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( remove1 @ A @ X @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_remove1_subset
thf(fact_4273_sub__num__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [L: num] :
( ( neg_numeral_sub @ A @ one2 @ ( bit0 @ L ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bitM @ L ) ) ) ) ) ).
% sub_num_simps(2)
thf(fact_4274_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B3: A,A3: A] :
( ( nO_MATCH @ B @ A @ X @ B3 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_4275_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B3: A,A3: A] :
( ( nO_MATCH @ B @ A @ X @ B3 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B3 @ A3 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_4276_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F2: B > A,A5: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( compow @ ( A > A ) @ N2 @ ( times_times @ A @ A5 ) @ ( F2 @ ( nth @ B @ Xs3 @ N2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_4277_Suc__funpow,axiom,
! [N: nat] :
( ( compow @ ( nat > nat ) @ N @ suc )
= ( plus_plus @ nat @ N ) ) ).
% Suc_funpow
thf(fact_4278_funpow__0,axiom,
! [A: $tType,F3: A > A,X: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F3 @ X )
= X ) ).
% funpow_0
thf(fact_4279_dbl__dec__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).
% dbl_dec_simps(5)
thf(fact_4280_sub__num__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ one2 )
= ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).
% sub_num_simps(4)
thf(fact_4281_funpow__swap1,axiom,
! [A: $tType,F3: A > A,N: nat,X: A] :
( ( F3 @ ( compow @ ( A > A ) @ N @ F3 @ X ) )
= ( compow @ ( A > A ) @ N @ F3 @ ( F3 @ X ) ) ) ).
% funpow_swap1
thf(fact_4282_funpow__mult,axiom,
! [A: $tType,N: nat,M: nat,F3: A > A] :
( ( compow @ ( A > A ) @ N @ ( compow @ ( A > A ) @ M @ F3 ) )
= ( compow @ ( A > A ) @ ( times_times @ nat @ M @ N ) @ F3 ) ) ).
% funpow_mult
thf(fact_4283_eval__nat__numeral_I2_J,axiom,
! [N: num] :
( ( numeral_numeral @ nat @ ( bit0 @ N ) )
= ( suc @ ( numeral_numeral @ nat @ ( bitM @ N ) ) ) ) ).
% eval_nat_numeral(2)
thf(fact_4284_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_4285_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_4286_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N2: nat] : ( compow @ ( A > A ) @ N2 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_4287_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,A3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A3 )
= ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A3 ) ) ) ).
% numeral_add_unfold_funpow
thf(fact_4288_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_4289_numeral__BitM,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bitM @ N ) )
= ( minus_minus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_BitM
thf(fact_4290_relpowp__bot,axiom,
! [A: $tType,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( compow @ ( A > A > $o ) @ N @ ( bot_bot @ ( A > A > $o ) ) )
= ( bot_bot @ ( A > A > $o ) ) ) ) ).
% relpowp_bot
thf(fact_4291_relpowp__fun__conv,axiom,
! [A: $tType] :
( ( compow @ ( A > A > $o ) )
= ( ^ [N2: nat,P3: A > A > $o,X2: A,Y3: A] :
? [F2: nat > A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= X2 )
& ( ( F2 @ N2 )
= Y3 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
=> ( P3 @ ( F2 @ I4 ) @ ( F2 @ ( suc @ I4 ) ) ) ) ) ) ) ).
% relpowp_fun_conv
thf(fact_4292_Nat_Ofunpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% Nat.funpow_code_def
thf(fact_4293_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_4294_relpowp_Osimps_I1_J,axiom,
! [A: $tType,R: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R )
= ( ^ [Y4: A,Z2: A] : Y4 = Z2 ) ) ).
% relpowp.simps(1)
thf(fact_4295_relpowp__0__E,axiom,
! [A: $tType,P: A > A > $o,X: A,Y: A] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X @ Y )
=> ( X = Y ) ) ).
% relpowp_0_E
thf(fact_4296_relpowp__0__I,axiom,
! [A: $tType,P: A > A > $o,X: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X @ X ) ).
% relpowp_0_I
thf(fact_4297_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_4298_relpowp__E,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Z: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X @ Z )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z ) )
=> ~ ! [Y2: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( compow @ ( A > A > $o ) @ M4 @ P @ X @ Y2 )
=> ~ ( P @ Y2 @ Z ) ) ) ) ) ).
% relpowp_E
thf(fact_4299_relpowp__E2,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Z: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X @ Z )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z ) )
=> ~ ! [Y2: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( P @ X @ Y2 )
=> ~ ( compow @ ( A > A > $o ) @ M4 @ P @ Y2 @ Z ) ) ) ) ) ).
% relpowp_E2
thf(fact_4300_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_4301_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_4302_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 )
@ ^ [X2: A,Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_4303_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X2: nat,Y3: nat] : ( ord_less_eq @ nat @ Y3 @ X2 )
@ ^ [X2: nat,Y3: nat] : ( ord_less @ nat @ Y3 @ X2 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_4304_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [A: $tType,F3: A > A > A,Z: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B3: A] :
( ( semila1105856199041335345_order @ A @ F3 @ Z @ Less_eq @ Less )
=> ( ( ( F3 @ A3 @ B3 )
= Z )
= ( ( A3 = Z )
& ( B3 = Z ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
thf(fact_4305_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [A: $tType,F3: A > A > A,Z: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B3: A] :
( ( semila1105856199041335345_order @ A @ F3 @ Z @ Less_eq @ Less )
=> ( ( Z
= ( F3 @ A3 @ B3 ) )
= ( ( A3 = Z )
& ( B3 = Z ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
thf(fact_4306_empty__in__Fpow,axiom,
! [A: $tType,A4: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( finite_Fpow @ A @ A4 ) ) ).
% empty_in_Fpow
thf(fact_4307_Fpow__not__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_Fpow @ A @ A4 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Fpow_not_empty
thf(fact_4308_Fpow__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A4 ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% Fpow_mono
thf(fact_4309_Fpow__subset__Pow,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A4 ) @ ( pow2 @ A @ A4 ) ) ).
% Fpow_subset_Pow
thf(fact_4310_Fpow__def,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A6: set @ A] :
( collect @ ( set @ A )
@ ^ [X8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ X8 @ A6 )
& ( finite_finite2 @ A @ X8 ) ) ) ) ) ).
% Fpow_def
thf(fact_4311_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_4312_length__mul__elem,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ! [X3: list @ A] :
( ( member @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ X3 )
= N ) )
=> ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs ) )
= ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) @ N ) ) ) ).
% length_mul_elem
thf(fact_4313_eq__numeral__iff__iszero_I8_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( one_one @ A )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_4314_eq__numeral__iff__iszero_I7_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ one2 ) ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_4315_iszero__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: num] :
( ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% iszero_neg_numeral
thf(fact_4316_iszero__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_iszero @ A )
= ( ^ [Z6: A] :
( Z6
= ( zero_zero @ A ) ) ) ) ) ).
% iszero_def
thf(fact_4317_iszero__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ring_1_iszero @ A @ ( zero_zero @ A ) ) ) ).
% iszero_0
thf(fact_4318_not__iszero__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W2: num] :
~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W2 ) ) ) ).
% not_iszero_numeral
thf(fact_4319_not__iszero__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% not_iszero_1
thf(fact_4320_eq__iff__iszero__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [X2: A,Y3: A] : ( ring_1_iszero @ A @ ( minus_minus @ A @ X2 @ Y3 ) ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_4321_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_4322_less__integer_Orep__eq,axiom,
( ( ord_less @ code_integer )
= ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_integer.rep_eq
thf(fact_4323_integer__less__eq__iff,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% integer_less_eq_iff
thf(fact_4324_less__eq__integer_Orep__eq,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_eq_integer.rep_eq
thf(fact_4325_eq__numeral__iff__iszero_I10_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( zero_zero @ A )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_4326_eq__numeral__iff__iszero_I9_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( numeral_numeral @ A @ X )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X ) ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_4327_not__iszero__Numeral1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ one2 ) ) ) ).
% not_iszero_Numeral1
thf(fact_4328_not__iszero__neg__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_iszero_neg_1
thf(fact_4329_eq__numeral__iff__iszero_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y: num] :
( ( ( numeral_numeral @ A @ X )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_4330_eq__numeral__iff__iszero_I11_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X ) ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_4331_eq__numeral__iff__iszero_I12_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_4332_not__iszero__neg__Numeral1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_4333_eq__numeral__iff__iszero_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y: num] :
( ( ( numeral_numeral @ A @ X )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_4334_eq__numeral__iff__iszero_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_4335_eq__numeral__iff__iszero_I4_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ Y @ X ) ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_4336_distinct__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A ) @ Xs )
=> ( ! [Ys4: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys4 ) )
=> ( ! [Ys4: list @ A,Zs2: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys4 != Zs2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys4 ) @ ( set2 @ A @ Zs2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_4337_eq__numeral__iff__iszero_I6_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ one2 @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_4338_eq__numeral__iff__iszero_I5_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( numeral_numeral @ A @ X )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ one2 ) ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_4339_set__n__lists,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) )
= ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys3 )
= N )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_4340_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,P6: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( P6 @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P6 @ ( insert2 @ B @ I @ I5 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P6 @ I5 ) ) )
& ( ~ ( member @ B @ I @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P6 @ ( insert2 @ B @ I @ I5 ) )
= ( times_times @ A @ ( P6 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P6 @ I5 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_4341_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( remove1 @ A @ X @ ( linord4507533701916653071of_set @ A @ A4 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_4342_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_4343_set__empty2,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_4344_set__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( set2 @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_4345_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_4346_empty__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_4347_replicate__empty,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( replicate @ A @ N @ X )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_4348_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_4349_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ A4 )
= ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_4350_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( set2 @ A @ ( linord4507533701916653071of_set @ A @ A4 ) )
= A4 ) ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_4351_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P6: B > A] :
( ( groups1962203154675924110t_prod @ B @ A @ P6 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty'
thf(fact_4352_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F3: B > A,A3: A] :
( ( groups4207007520872428315er_sum @ B @ A @ F3 @ A3 @ ( nil @ B ) )
= ( zero_zero @ A ) ) ) ).
% horner_sum_simps(1)
thf(fact_4353_prod_Oeq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,P6: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P6 @ I5 )
= ( groups7121269368397514597t_prod @ B @ A @ P6 @ I5 ) ) ) ) ).
% prod.eq_sum
thf(fact_4354_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_4355_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( linord4507533701916653071of_set @ A @ A4 )
= ( nil @ A ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_4356_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ( linord4507533701916653071of_set @ A @ A4 )
= ( linord4507533701916653071of_set @ A @ B2 ) )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( A4 = B2 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_4357_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_4358_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_4359_replicate__0,axiom,
! [A: $tType,X: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_4360_list_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_list @ A @ X @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size_gen(1)
thf(fact_4361_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_4362_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,G2: B > A,H: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I4: B] : ( times_times @ A @ ( G2 @ I4 ) @ ( H @ I4 ) )
@ I5 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G2 @ I5 ) @ ( groups1962203154675924110t_prod @ B @ A @ H @ I5 ) ) ) ) ) ).
% prod.distrib_triv'
thf(fact_4363_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T4: set @ B,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G2 @ S )
= ( groups1962203154675924110t_prod @ B @ A @ G2 @ T4 ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_4364_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T4: set @ B,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G2 @ T4 )
= ( groups1962203154675924110t_prod @ B @ A @ G2 @ S ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_4365_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T4: set @ B,H: B > A,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H @ I2 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G2 @ S )
= ( groups1962203154675924110t_prod @ B @ A @ H @ T4 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_4366_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T4: set @ B,G2: B > A,H: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G2 @ T4 )
= ( groups1962203154675924110t_prod @ B @ A @ H @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_4367_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,G2: B > A,H: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( G2 @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( H @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I4: B] : ( times_times @ A @ ( G2 @ I4 ) @ ( H @ I4 ) )
@ I5 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G2 @ I5 ) @ ( groups1962203154675924110t_prod @ B @ A @ H @ I5 ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_4368_prod_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ B @ A )
= ( ^ [P5: B > A,I7: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I7 )
& ( ( P5 @ X2 )
!= ( one_one @ A ) ) ) ) )
@ ( groups7121269368397514597t_prod @ B @ A @ P5
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I7 )
& ( ( P5 @ X2 )
!= ( one_one @ A ) ) ) ) )
@ ( one_one @ A ) ) ) ) ) ).
% prod.G_def
thf(fact_4369_Pow__set_I1_J,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( set2 @ A @ ( nil @ A ) ) )
= ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_set(1)
thf(fact_4370_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert2 @ A @ X @ A4 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_4371_rat__floor__lemma,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ ( divide_divide @ int @ A3 @ B3 ) ) @ ( fract @ A3 @ B3 ) )
& ( ord_less @ rat @ ( fract @ A3 @ B3 ) @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( one_one @ int ) ) ) ) ) ).
% rat_floor_lemma
thf(fact_4372_image__minus__const__atLeastLessThan__nat,axiom,
! [C3: nat,Y: nat,X: nat] :
( ( ( ord_less @ nat @ C3 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C3 ) @ ( minus_minus @ nat @ Y @ C3 ) ) ) )
& ( ~ ( ord_less @ nat @ C3 @ Y )
=> ( ( ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_4373_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K3: code_integer] :
( if @ num @ ( ord_less_eq @ code_integer @ K3 @ ( one_one @ code_integer ) ) @ one2
@ ( product_case_prod @ code_integer @ code_integer @ num
@ ^ [L2: code_integer,J3: code_integer] :
( if @ num
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) )
@ ( plus_plus @ num @ ( plus_plus @ num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one2 ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_4374_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A,X: B,A4: set @ B] :
( ( B3
= ( F3 @ X ) )
=> ( ( member @ B @ X @ A4 )
=> ( member @ A @ B3 @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ).
% image_eqI
thf(fact_4375_image__ident,axiom,
! [A: $tType,Y6: set @ A] :
( ( image2 @ A @ A
@ ^ [X2: A] : X2
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_4376_image__is__empty,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B] :
( ( ( image2 @ B @ A @ F3 @ A4 )
= ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_4377_empty__is__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image2 @ B @ A @ F3 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_4378_image__empty,axiom,
! [B: $tType,A: $tType,F3: B > A] :
( ( image2 @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_4379_finite__imageI,axiom,
! [B: $tType,A: $tType,F4: set @ A,H: A > B] :
( ( finite_finite2 @ A @ F4 )
=> ( finite_finite2 @ B @ ( image2 @ A @ B @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_4380_insert__image,axiom,
! [B: $tType,A: $tType,X: A,A4: set @ A,F3: A > B] :
( ( member @ A @ X @ A4 )
=> ( ( insert2 @ B @ ( F3 @ X ) @ ( image2 @ A @ B @ F3 @ A4 ) )
= ( image2 @ A @ B @ F3 @ A4 ) ) ) ).
% insert_image
thf(fact_4381_image__insert,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: B,B2: set @ B] :
( ( image2 @ B @ A @ F3 @ ( insert2 @ B @ A3 @ B2 ) )
= ( insert2 @ A @ ( F3 @ A3 ) @ ( image2 @ B @ A @ F3 @ B2 ) ) ) ).
% image_insert
thf(fact_4382_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S )
= S ) ) ).
% image_add_0
thf(fact_4383_ccSUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( bot_bot @ A )
@ A4 ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_bot
thf(fact_4384_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( bot_bot @ A )
@ A4 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_4385_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: B > A,A4: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B2 @ A4 ) )
= ( bot_bot @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ( B2 @ X2 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_4386_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: B > A,A4: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B2 @ A4 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ( B2 @ X2 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_4387_ccSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,F3: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A4 ) )
= F3 ) ) ) ).
% ccSUP_const
thf(fact_4388_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F3: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A4 ) )
= F3 ) ) ) ).
% SUP_const
thf(fact_4389_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,C3: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : C3
@ A4 ) )
= C3 ) ) ) ).
% cSUP_const
thf(fact_4390_ccINF__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,F3: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A4 ) )
= F3 ) ) ) ).
% ccINF_const
thf(fact_4391_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,C3: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : C3
@ A4 ) )
= C3 ) ) ) ).
% cINF_const
thf(fact_4392_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F3: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A4 ) )
= F3 ) ) ) ).
% INF_const
thf(fact_4393_if__image__distrib,axiom,
! [A: $tType,B: $tType,P: B > $o,F3: B > A,G2: B > A,S: set @ B] :
( ( image2 @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ S )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ S @ ( collect @ B @ P ) ) )
@ ( image2 @ B @ A @ G2
@ ( inf_inf @ ( set @ B ) @ S
@ ( collect @ B
@ ^ [X2: B] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_4394_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A4: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
=> ? [Y3: B] :
( ( member @ B @ Y3 @ A4 )
& ( ord_less @ A @ ( F3 @ Y3 ) @ X2 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_4395_ccSUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F3: B > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_empty
thf(fact_4396_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert2 @ A @ X @ A4 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ ( linord4507533701916653071of_set @ A @ A4 ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_4397_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D2 @ A3 ) @ ( times_times @ A @ D2 @ B3 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_4398_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image2 @ A @ A
@ ^ [C6: A] : ( divide_divide @ A @ C6 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D2 ) @ ( divide_divide @ A @ B3 @ D2 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_4399_less__rat,axiom,
! [B3: int,D2: int,A3: int,C3: int] :
( ( B3
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( ord_less @ rat @ ( fract @ A3 @ B3 ) @ ( fract @ C3 @ D2 ) )
= ( ord_less @ int @ ( times_times @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ B3 @ D2 ) ) @ ( times_times @ int @ ( times_times @ int @ C3 @ B3 ) @ ( times_times @ int @ B3 @ D2 ) ) ) ) ) ) ).
% less_rat
thf(fact_4400_le__rat,axiom,
! [B3: int,D2: int,A3: int,C3: int] :
( ( B3
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( ord_less_eq @ rat @ ( fract @ A3 @ B3 ) @ ( fract @ C3 @ D2 ) )
= ( ord_less_eq @ int @ ( times_times @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ B3 @ D2 ) ) @ ( times_times @ int @ ( times_times @ int @ C3 @ B3 ) @ ( times_times @ int @ B3 @ D2 ) ) ) ) ) ) ).
% le_rat
thf(fact_4401_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F3: B > A,A4: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A4 ) @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F3 ) @ ( pow2 @ B @ A4 ) ) @ ( pow2 @ A @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_4402_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F3: B > A,A4: set @ B,B2: set @ A] :
( ( ( image2 @ B @ A @ F3 @ A4 )
= B2 )
=> ( ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F3 ) @ ( pow2 @ B @ A4 ) )
= ( pow2 @ A @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_4403_image__Un,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B,B2: set @ B] :
( ( image2 @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A4 ) @ ( image2 @ B @ A @ F3 @ B2 ) ) ) ).
% image_Un
thf(fact_4404_subset__image__iff,axiom,
! [A: $tType,B: $tType,B2: set @ A,F3: B > A,A4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F3 @ A4 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A4 )
& ( B2
= ( image2 @ B @ A @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_4405_image__subset__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A4 ) @ B2 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( member @ A @ ( F3 @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_4406_subset__imageE,axiom,
! [A: $tType,B: $tType,B2: set @ A,F3: B > A,A4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ~ ! [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A4 )
=> ( B2
!= ( image2 @ B @ A @ F3 @ C7 ) ) ) ) ).
% subset_imageE
thf(fact_4407_image__subsetI,axiom,
! [A: $tType,B: $tType,A4: set @ A,F3: A > B,B2: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ B @ ( F3 @ X3 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ B2 ) ) ).
% image_subsetI
thf(fact_4408_image__mono,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ ( image2 @ A @ B @ F3 @ B2 ) ) ) ).
% image_mono
thf(fact_4409_all__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P: ( set @ A ) > $o] :
( ( ! [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A4 )
=> ( P @ ( image2 @ B @ A @ F3 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_4410_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: A > B,B2: set @ B] :
( ! [X3: A] :
( ( P @ X3 )
=> ( member @ B @ ( F3 @ X3 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( collect @ A @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_4411_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A4: set @ A,F3: A > B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ ( image2 @ A @ B @ F3 @ A4 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A4 )
& ( ( F3 @ A5 )
= ( F3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_4412_imageE,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A,A4: set @ B] :
( ( member @ A @ B3 @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ~ ! [X3: B] :
( ( B3
= ( F3 @ X3 ) )
=> ~ ( member @ B @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_4413_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,G2: C > B,A4: set @ C] :
( ( image2 @ B @ A @ F3 @ ( image2 @ C @ B @ G2 @ A4 ) )
= ( image2 @ C @ A
@ ^ [X2: C] : ( F3 @ ( G2 @ X2 ) )
@ A4 ) ) ).
% image_image
thf(fact_4414_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( image2 @ B @ A @ F3 @ A4 ) )
& ( P @ X2 ) ) )
= ( image2 @ B @ A @ F3
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( P @ ( F3 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_4415_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A4: set @ A,B3: B,F3: A > B] :
( ( member @ A @ X @ A4 )
=> ( ( B3
= ( F3 @ X ) )
=> ( member @ B @ B3 @ ( image2 @ A @ B @ F3 @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_4416_ball__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( P @ X3 ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( P @ ( F3 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_4417_image__cong,axiom,
! [B: $tType,A: $tType,M5: set @ A,N6: set @ A,F3: A > B,G2: A > B] :
( ( M5 = N6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N6 )
=> ( ( F3 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image2 @ A @ B @ F3 @ M5 )
= ( image2 @ A @ B @ G2 @ N6 ) ) ) ) ).
% image_cong
thf(fact_4418_bex__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image2 @ B @ A @ F3 @ A4 ) )
& ( P @ X5 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( P @ ( F3 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_4419_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F3: B > A,A4: set @ B] :
( ( member @ A @ Z @ ( image2 @ B @ A @ F3 @ A4 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( Z
= ( F3 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_4420_imageI,axiom,
! [B: $tType,A: $tType,X: A,A4: set @ A,F3: A > B] :
( ( member @ A @ X @ A4 )
=> ( member @ B @ ( F3 @ X ) @ ( image2 @ A @ B @ F3 @ A4 ) ) ) ).
% imageI
thf(fact_4421_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( insert2 @ A @ ( F3 @ X2 ) @ ( bot_bot @ ( set @ A ) ) )
@ A4 ) )
= ( image2 @ B @ A @ F3 @ A4 ) ) ).
% UNION_singleton_eq_range
thf(fact_4422_image__Fpow__mono,axiom,
! [B: $tType,A: $tType,F3: B > A,A4: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A4 ) @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F3 ) @ ( finite_Fpow @ B @ A4 ) ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_4423_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B2: set @ C,F3: B > A,G2: C > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B2 )
& ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B2 )
=> ? [X5: B] :
( ( member @ B @ X5 @ A4 )
& ( ord_less_eq @ A @ ( G2 @ J2 ) @ ( F3 @ X5 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G2 @ B2 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_4424_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B2: set @ C,G2: C > A,F3: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B2 )
& ( ord_less_eq @ A @ ( G2 @ X5 ) @ ( F3 @ I2 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B2 )
=> ? [X5: B] :
( ( member @ B @ X5 @ A4 )
& ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G2 @ J2 ) ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G2 @ B2 ) ) ) ) ) ) ).
% INF_eq
thf(fact_4425_zero__notin__Suc__image,axiom,
! [A4: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ A4 ) ) ).
% zero_notin_Suc_image
thf(fact_4426_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P: ( set @ A ) > $o] :
( ( ! [B6: set @ A] :
( ( ( finite_finite2 @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F3 @ A4 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set @ B] :
( ( ( finite_finite2 @ B @ B6 )
& ( ord_less_eq @ ( set @ B ) @ B6 @ A4 ) )
=> ( P @ ( image2 @ B @ A @ F3 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_4427_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P: ( set @ A ) > $o] :
( ( ? [B6: set @ A] :
( ( finite_finite2 @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F3 @ A4 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set @ B] :
( ( finite_finite2 @ B @ B6 )
& ( ord_less_eq @ ( set @ B ) @ B6 @ A4 )
& ( P @ ( image2 @ B @ A @ F3 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_4428_finite__subset__image,axiom,
! [A: $tType,B: $tType,B2: set @ A,F3: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ? [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A4 )
& ( finite_finite2 @ B @ C7 )
& ( B2
= ( image2 @ B @ A @ F3 @ C7 ) ) ) ) ) ).
% finite_subset_image
thf(fact_4429_finite__surj,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( image2 @ A @ B @ F3 @ A4 ) )
=> ( finite_finite2 @ B @ B2 ) ) ) ).
% finite_surj
thf(fact_4430_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F3: B > A,X: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( ( F3 @ I2 )
= X ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ I5 ) )
= X ) ) ) ) ).
% SUP_eq_const
thf(fact_4431_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F3: B > A,X: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( ( F3 @ I2 )
= X ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I5 ) )
= X ) ) ) ) ).
% INF_eq_const
thf(fact_4432_finite__image__absD,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ ( image2 @ A @ A @ ( abs_abs @ A ) @ S ) )
=> ( finite_finite2 @ A @ S ) ) ) ).
% finite_image_absD
thf(fact_4433_image__Int__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B,B2: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) @ ( inf_inf @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A4 ) @ ( image2 @ B @ A @ F3 @ B2 ) ) ) ).
% image_Int_subset
thf(fact_4434_sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,B2: set @ A] :
( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ B2 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ A @ A @ ( sup_sup @ A @ A3 ) @ B2 ) ) ) ) ).
% sup_Inf
thf(fact_4435_image__diff__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B,B2: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A4 ) @ ( image2 @ B @ A @ F3 @ B2 ) ) @ ( image2 @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_4436_Rat__induct__pos,axiom,
! [P: rat > $o,Q5: rat] :
( ! [A7: int,B7: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ( P @ ( fract @ A7 @ B7 ) ) )
=> ( P @ Q5 ) ) ).
% Rat_induct_pos
thf(fact_4437_set__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( set2 @ B @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( insert2 @ B @ X @ ( set2 @ B @ Xs ) ) ) ) ).
% set_insort_key
thf(fact_4438_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F3: B > A,X: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ X ) )
=> ( ! [Y2: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ Y2 ) )
=> ( ord_less_eq @ A @ X @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
= X ) ) ) ) ).
% SUP_eqI
thf(fact_4439_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B2: set @ C,F3: B > A,G2: C > A] :
( ! [N3: B] :
( ( member @ B @ N3 @ A4 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B2 )
& ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G2 @ X5 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G2 @ B2 ) ) ) ) ) ).
% SUP_mono
thf(fact_4440_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F3: B > A,U: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ U ) ) ) ).
% SUP_least
thf(fact_4441_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G2: B > A,A4: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) ) ) ) ).
% SUP_mono'
thf(fact_4442_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,F3: B > A] :
( ( member @ B @ I @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ I ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ).
% SUP_upper
thf(fact_4443_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A4: set @ B,U: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ U )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ X2 ) @ U ) ) ) ) ) ).
% SUP_le_iff
thf(fact_4444_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,U: A,F3: B > A] :
( ( member @ B @ I @ A4 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ I ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_4445_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: A,F3: B > A,A4: set @ B] :
( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ A3 @ ( F3 @ X2 ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_4446_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A4: set @ B,Y: A,I: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ Y )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ ( F3 @ I ) @ Y ) ) ) ) ).
% SUP_lessD
thf(fact_4447_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,U: A,F3: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ U @ ( F3 @ I2 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ).
% INF_greatest
thf(fact_4448_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,F3: B > A,A4: set @ B] :
( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ord_less_eq @ A @ U @ ( F3 @ X2 ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_4449_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,F3: B > A,U: A] :
( ( member @ B @ I @ A4 )
=> ( ( ord_less_eq @ A @ ( F3 @ I ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ U ) ) ) ) ).
% INF_lower2
thf(fact_4450_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G2: B > A,A4: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) ) ) ) ).
% INF_mono'
thf(fact_4451_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,F3: B > A] :
( ( member @ B @ I @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( F3 @ I ) ) ) ) ).
% INF_lower
thf(fact_4452_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: set @ B,A4: set @ C,F3: C > A,G2: B > A] :
( ! [M4: B] :
( ( member @ B @ M4 @ B2 )
=> ? [X5: C] :
( ( member @ C @ X5 @ A4 )
& ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G2 @ M4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F3 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ B2 ) ) ) ) ) ).
% INF_mono
thf(fact_4453_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,X: A,F3: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ X @ ( F3 @ I2 ) ) )
=> ( ! [Y2: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less_eq @ A @ Y2 @ ( F3 @ I3 ) ) )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
= X ) ) ) ) ).
% INF_eqI
thf(fact_4454_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A4: set @ B,A3: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ A3 )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ ( F3 @ X2 ) @ A3 ) ) ) ) ) ).
% INF_less_iff
thf(fact_4455_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A,F3: B > A,A4: set @ B,I: B] :
( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ Y @ ( F3 @ I ) ) ) ) ) ).
% less_INF_D
thf(fact_4456_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] :
? [N2: nat,F2: nat > A] :
( A6
= ( image2 @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N2 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_4457_nat__seg__image__imp__finite,axiom,
! [A: $tType,A4: set @ A,F3: nat > A,N: nat] :
( ( A4
= ( image2 @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N ) ) ) )
=> ( finite_finite2 @ A @ A4 ) ) ).
% nat_seg_image_imp_finite
thf(fact_4458_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A4: set @ A,C3: B] :
( ( member @ A @ X @ A4 )
=> ( ( image2 @ A @ B
@ ^ [X2: A] : C3
@ A4 )
= ( insert2 @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_4459_image__constant__conv,axiom,
! [B: $tType,A: $tType,A4: set @ B,C3: A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X2: B] : C3
@ A4 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X2: B] : C3
@ A4 )
= ( insert2 @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_4460_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,H: B > A,G2: B > C] :
( ( finite_finite2 @ B @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ H @ S )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y3: C] :
( groups7311177749621191930dd_sum @ B @ A @ H
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S )
& ( ( G2 @ X2 )
= Y3 ) ) ) )
@ ( image2 @ B @ C @ G2 @ S ) ) ) ) ) ).
% sum.image_gen
thf(fact_4461_SUP__absorb,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [K: B,I5: set @ B,A4: B > A] :
( ( member @ B @ K @ I5 )
=> ( ( sup_sup @ A @ ( A4 @ K ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ A4 @ I5 ) ) )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ A4 @ I5 ) ) ) ) ) ).
% SUP_absorb
thf(fact_4462_complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A4: set @ B,G2: B > A] :
( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A5: B] : ( sup_sup @ A @ ( F3 @ A5 ) @ ( G2 @ A5 ) )
@ A4 ) ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_4463_INF__sup__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F3: B > A,A4: set @ B,G2: C > A,B2: set @ C] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G2 @ B2 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A5: B] :
( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [B5: C] : ( sup_sup @ A @ ( F3 @ A5 ) @ ( G2 @ B5 ) )
@ B2 ) )
@ A4 ) ) ) ) ).
% INF_sup_distrib2
thf(fact_4464_sup__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F3: B > A,B2: set @ B] :
( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ B2 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( sup_sup @ A @ A3 @ ( F3 @ B5 ) )
@ B2 ) ) ) ) ).
% sup_INF
thf(fact_4465_Inf__sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B2: set @ A,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B2 ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image2 @ A @ A
@ ^ [B5: A] : ( sup_sup @ A @ B5 @ A3 )
@ B2 ) ) ) ) ).
% Inf_sup
thf(fact_4466_INF__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F3: B > A,B2: set @ B,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ B2 ) ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( sup_sup @ A @ ( F3 @ B5 ) @ A3 )
@ B2 ) ) ) ) ).
% INF_sup
thf(fact_4467_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,H: B > A,G2: B > C] :
( ( finite_finite2 @ B @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A @ H @ S )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y3: C] :
( groups7121269368397514597t_prod @ B @ A @ H
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S )
& ( ( G2 @ X2 )
= Y3 ) ) ) )
@ ( image2 @ B @ C @ G2 @ S ) ) ) ) ) ).
% prod.image_gen
thf(fact_4468_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A4: set @ A,F3: A > B,X: A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ A4 )
=> ( ( F3 @ Y2 )
= ( F3 @ X ) ) )
=> ( ( the_elem @ B @ ( image2 @ A @ B @ F3 @ A4 ) )
= ( F3 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_4469_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,F3: B > A,A4: set @ B] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) )
= ( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X )
=> ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ Y3 @ ( F3 @ X2 ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_4470_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A4: set @ B,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ X )
= ( ! [Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ ( F3 @ X2 ) @ Y3 ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_4471_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,C3: A,F3: B > A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( ord_less_eq @ A @ C3 @ ( F3 @ I2 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ I5 ) )
= C3 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I5 )
=> ( ( F3 @ X2 )
= C3 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_4472_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F3: B > A,M5: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ M5 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ M5 ) ) ) ) ).
% cSUP_least
thf(fact_4473_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,M: A,F3: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ M @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_4474_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F3: B > A,C3: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ C3 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I5 ) )
= C3 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I5 )
=> ( ( F3 @ X2 )
= C3 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4475_card__image__le,axiom,
! [B: $tType,A: $tType,A4: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A4 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image2 @ A @ B @ F3 @ A4 ) ) @ ( finite_card @ A @ A4 ) ) ) ).
% card_image_le
thf(fact_4476_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B2: set @ B,F3: B > A,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A4 @ B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ B2 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_4477_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: set @ B,A4: set @ B,F3: B > A,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B2 @ A4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B2 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ B2 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_4478_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,C3: A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C3
@ A4 ) )
= ( bot_bot @ A ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C3
@ A4 ) )
= C3 ) ) ) ) ).
% SUP_constant
thf(fact_4479_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% SUP_empty
thf(fact_4480_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T4: set @ C,G2: B > C,H: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G2 @ S ) @ T4 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y3: C] :
( groups7311177749621191930dd_sum @ B @ A @ H
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S )
& ( ( G2 @ X2 )
= Y3 ) ) ) )
@ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ S ) ) ) ) ) ) ).
% sum.group
thf(fact_4481_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,X: A,F3: B > A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( inf_inf @ A @ X @ ( F3 @ I4 ) )
@ I5 ) )
= ( inf_inf @ A @ X @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I5 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_4482_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F3: B > A,X: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( inf_inf @ A @ ( F3 @ I4 ) @ X )
@ I5 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I5 ) ) @ X ) ) ) ) ).
% INF_inf_const2
thf(fact_4483_SUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A3: B,A4: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( sup_sup @ A @ ( F3 @ A3 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ).
% SUP_insert
thf(fact_4484_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T4: set @ C,G2: B > C,H: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G2 @ S ) @ T4 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y3: C] :
( groups7121269368397514597t_prod @ B @ A @ H
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S )
& ( ( G2 @ X2 )
= Y3 ) ) ) )
@ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ S ) ) ) ) ) ) ).
% prod.group
thf(fact_4485_INF__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A3: B,A4: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( inf_inf @ A @ ( F3 @ A3 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ).
% INF_insert
thf(fact_4486_SUP__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M5: B > A,A4: set @ B,B2: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M5 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M5 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M5 @ B2 ) ) ) ) ) ).
% SUP_union
thf(fact_4487_INF__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M5: B > A,A4: set @ B,B2: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M5 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M5 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M5 @ B2 ) ) ) ) ) ).
% INF_union
thf(fact_4488_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F3: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_4489_surj__card__le,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( image2 @ A @ B @ F3 @ A4 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B2 ) @ ( finite_card @ A @ A4 ) ) ) ) ).
% surj_card_le
thf(fact_4490_scaleR__image__atLeastAtMost,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,X: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( image2 @ A @ A @ ( real_V8093663219630862766scaleR @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ Y ) ) ) ) ) ).
% scaleR_image_atLeastAtMost
thf(fact_4491_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H: A > A,N6: set @ A] :
( ! [X3: A,Y2: A] :
( ( H @ ( inf_inf @ A @ X3 @ Y2 ) )
= ( inf_inf @ A @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic7752659483105999362nf_fin @ A @ N6 ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image2 @ A @ A @ H @ N6 ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_4492_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H: A > A,N6: set @ A] :
( ! [X3: A,Y2: A] :
( ( H @ ( sup_sup @ A @ X3 @ Y2 ) )
= ( sup_sup @ A @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic5882676163264333800up_fin @ A @ N6 ) )
= ( lattic5882676163264333800up_fin @ A @ ( image2 @ A @ A @ H @ N6 ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_4493_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_4494_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_4495_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ N ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_4496_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost @ nat @ ( suc @ N ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_4497_Fract__less__zero__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less @ rat @ ( fract @ A3 @ B3 ) @ ( zero_zero @ rat ) )
= ( ord_less @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% Fract_less_zero_iff
thf(fact_4498_zero__less__Fract__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ ( fract @ A3 @ B3 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% zero_less_Fract_iff
thf(fact_4499_Fract__less__one__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less @ rat @ ( fract @ A3 @ B3 ) @ ( one_one @ rat ) )
= ( ord_less @ int @ A3 @ B3 ) ) ) ).
% Fract_less_one_iff
thf(fact_4500_one__less__Fract__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less @ rat @ ( one_one @ rat ) @ ( fract @ A3 @ B3 ) )
= ( ord_less @ int @ B3 @ A3 ) ) ) ).
% one_less_Fract_iff
thf(fact_4501_Fract__le__zero__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ rat @ ( fract @ A3 @ B3 ) @ ( zero_zero @ rat ) )
= ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% Fract_le_zero_iff
thf(fact_4502_zero__le__Fract__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ ( fract @ A3 @ B3 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% zero_le_Fract_iff
thf(fact_4503_Fract__le__one__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ rat @ ( fract @ A3 @ B3 ) @ ( one_one @ rat ) )
= ( ord_less_eq @ int @ A3 @ B3 ) ) ) ).
% Fract_le_one_iff
thf(fact_4504_one__le__Fract__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ rat @ ( one_one @ rat ) @ ( fract @ A3 @ B3 ) )
= ( ord_less_eq @ int @ B3 @ A3 ) ) ) ).
% one_le_Fract_iff
thf(fact_4505_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,X: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ X ) @ ( times_times @ A @ C3 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ Y ) @ ( times_times @ A @ C3 @ X ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_4506_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ A4 )
= ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_4507_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,C3: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C3 ) @ ( times_times @ A @ Y @ C3 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C3 ) @ ( times_times @ A @ X @ C3 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_4508_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B3 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_4509_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B3 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B3 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_4510_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B3 @ M ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B3 @ M ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_4511_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B3 @ M ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B3 @ M ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_4512_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S: set @ A,R: set @ B,G2: A > B,F3: B > C] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ R )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G2 @ S ) @ R )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X2: A] : ( F3 @ ( G2 @ X2 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y3: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S )
& ( ( G2 @ X2 )
= Y3 ) ) ) ) )
@ ( F3 @ Y3 ) )
@ R ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_4513_INF__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: A,B2: A] :
( ( inf_inf @ A @ A4
@ ( complete_Inf_Inf @ A
@ ( image2 @ nat @ A
@ ^ [X2: nat] : B2
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( inf_inf @ A @ A4 @ B2 ) ) ) ).
% INF_nat_binary
thf(fact_4514_SUP__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: A,B2: A] :
( ( sup_sup @ A @ A4
@ ( complete_Sup_Sup @ A
@ ( image2 @ nat @ A
@ ^ [X2: nat] : B2
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( sup_sup @ A @ A4 @ B2 ) ) ) ).
% SUP_nat_binary
thf(fact_4515_positive__rat,axiom,
! [A3: int,B3: int] :
( ( positive @ ( fract @ A3 @ B3 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ A3 @ B3 ) ) ) ).
% positive_rat
thf(fact_4516_nth__image,axiom,
! [A: $tType,L: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( image2 @ nat @ A @ ( nth @ A @ Xs ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ L ) )
= ( set2 @ A @ ( take @ A @ L @ Xs ) ) ) ) ).
% nth_image
thf(fact_4517_finite__UN,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A4 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( finite_finite2 @ B @ ( B2 @ X2 ) ) ) ) ) ) ).
% finite_UN
thf(fact_4518_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs3: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_4519_take__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( take @ A @ N @ Xs )
= ( nil @ A ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil
thf(fact_4520_take__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( nil @ A )
= ( take @ A @ N @ Xs ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil2
thf(fact_4521_take__all,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( take @ A @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_4522_take__all__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( take @ A @ N @ Xs )
= Xs )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_4523_nth__take,axiom,
! [A: $tType,I: nat,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( take @ A @ N @ Xs ) @ I )
= ( nth @ A @ Xs @ I ) ) ) ).
% nth_take
thf(fact_4524_take__update__cancel,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A,Y: A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( take @ A @ N @ ( list_update @ A @ Xs @ M @ Y ) )
= ( take @ A @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_4525_UN__constant,axiom,
! [B: $tType,A: $tType,A4: set @ B,C3: set @ A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C3
@ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C3
@ A4 ) )
= C3 ) ) ) ).
% UN_constant
thf(fact_4526_finite__UN__I,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [A7: A] :
( ( member @ A @ A7 @ A4 )
=> ( finite_finite2 @ B @ ( B2 @ A7 ) ) )
=> ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A4 ) ) ) ) ) ).
% finite_UN_I
thf(fact_4527_UN__Un,axiom,
! [A: $tType,B: $tType,M5: B > ( set @ A ),A4: set @ B,B2: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M5 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M5 @ A4 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M5 @ B2 ) ) ) ) ).
% UN_Un
thf(fact_4528_finite__INT,axiom,
! [B: $tType,A: $tType,I5: set @ A,A4: A > ( set @ B )] :
( ? [X5: A] :
( ( member @ A @ X5 @ I5 )
& ( finite_finite2 @ B @ ( A4 @ X5 ) ) )
=> ( finite_finite2 @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I5 ) ) ) ) ).
% finite_INT
thf(fact_4529_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C2: set @ B,A3: A,B2: B > ( set @ A )] :
( ( ( C2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( insert2 @ A @ A3 @ ( B2 @ X2 ) )
@ C2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( insert2 @ A @ A3 @ ( B2 @ X2 ) )
@ C2 ) )
= ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_4530_UN__singleton,axiom,
! [A: $tType,A4: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X2: A] : ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
@ A4 ) )
= A4 ) ).
% UN_singleton
thf(fact_4531_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C2: set @ C,A4: C > ( set @ D ),B2: set @ D] :
( ( ( C2
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X2: C] : ( sup_sup @ ( set @ D ) @ ( A4 @ X2 ) @ B2 )
@ C2 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X2: C] : ( sup_sup @ ( set @ D ) @ ( A4 @ X2 ) @ B2 )
@ C2 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A4 @ C2 ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_4532_UN__simps_I3_J,axiom,
! [E4: $tType,F: $tType,C2: set @ F,A4: set @ E4,B2: F > ( set @ E4 )] :
( ( ( C2
= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E4 )
@ ( image2 @ F @ ( set @ E4 )
@ ^ [X2: F] : ( sup_sup @ ( set @ E4 ) @ A4 @ ( B2 @ X2 ) )
@ C2 ) )
= ( bot_bot @ ( set @ E4 ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E4 )
@ ( image2 @ F @ ( set @ E4 )
@ ^ [X2: F] : ( sup_sup @ ( set @ E4 ) @ A4 @ ( B2 @ X2 ) )
@ C2 ) )
= ( sup_sup @ ( set @ E4 ) @ A4 @ ( complete_Sup_Sup @ ( set @ E4 ) @ ( image2 @ F @ ( set @ E4 ) @ B2 @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_4533_UN__insert,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A3: B,A4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( B2 @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A4 ) ) ) ) ).
% UN_insert
thf(fact_4534_INT__insert,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A3: B,A4: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( inf_inf @ ( set @ A ) @ ( B2 @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A4 ) ) ) ) ).
% INT_insert
thf(fact_4535_Sup__SUP__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( A > $o ) )
= ( ^ [S7: set @ ( A > $o ),X2: A] : ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S7 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_4536_SUP__Sup__eq,axiom,
! [A: $tType,S: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I4: set @ A,X2: A] : ( member @ A @ X2 @ I4 )
@ S ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_4537_SUP__UN__eq,axiom,
! [B: $tType,A: $tType,R2: B > ( set @ A ),S: set @ B] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I4: B,X2: A] : ( member @ A @ X2 @ ( R2 @ I4 ) )
@ S ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R2 @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_4538_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I4: set @ ( product_prod @ A @ B ),X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ I4 )
@ S ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_4539_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I4: C,X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( R2 @ I4 ) )
@ S ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_4540_Inf__INT__eq,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( A > $o ) )
= ( ^ [S7: set @ ( A > $o ),X2: A] : ( member @ A @ X2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S7 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_4541_INF__Int__eq,axiom,
! [A: $tType,S: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I4: set @ A,X2: A] : ( member @ A @ X2 @ I4 )
@ S ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Inf_Inf @ ( set @ A ) @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_4542_INF__INT__eq,axiom,
! [B: $tType,A: $tType,R2: B > ( set @ A ),S: set @ B] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I4: B,X2: A] : ( member @ A @ X2 @ ( R2 @ I4 ) )
@ S ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R2 @ S ) ) ) ) ) ).
% INF_INT_eq
thf(fact_4543_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I4: C,X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( R2 @ I4 ) )
@ S ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_4544_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I4: set @ ( product_prod @ A @ B ),X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ I4 )
@ S ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% INF_Int_eq2
thf(fact_4545_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S7: set @ ( A > B > $o ),X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S7 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_4546_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S7: set @ ( A > B > $o ),X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S7 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_4547_UN__Pow__subset,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A4: set @ B] :
( ord_less_eq @ ( set @ ( set @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ B @ ( set @ ( set @ A ) )
@ ^ [X2: B] : ( pow2 @ A @ ( B2 @ X2 ) )
@ A4 ) )
@ ( pow2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A4 ) ) ) ) ).
% UN_Pow_subset
thf(fact_4548_take__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( take @ A @ ( zero_zero @ nat ) @ Xs )
= ( nil @ A ) ) ).
% take_0
thf(fact_4549_set__take__subset,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_take_subset
thf(fact_4550_less__rat__def,axiom,
( ( ord_less @ rat )
= ( ^ [X2: rat,Y3: rat] : ( positive @ ( minus_minus @ rat @ Y3 @ X2 ) ) ) ) ).
% less_rat_def
thf(fact_4551_INF__filter__not__bot,axiom,
! [I6: $tType,A: $tType,B2: set @ I6,F4: I6 > ( filter @ A )] :
( ! [X9: set @ I6] :
( ( ord_less_eq @ ( set @ I6 ) @ X9 @ B2 )
=> ( ( finite_finite2 @ I6 @ X9 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I6 @ ( filter @ A ) @ F4 @ X9 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I6 @ ( filter @ A ) @ F4 @ B2 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_4552_finite__int__iff__bounded__le,axiom,
( ( finite_finite2 @ int )
= ( ^ [S7: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image2 @ int @ int @ ( abs_abs @ int ) @ S7 ) @ ( set_ord_atMost @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded_le
thf(fact_4553_finite__int__iff__bounded,axiom,
( ( finite_finite2 @ int )
= ( ^ [S7: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image2 @ int @ int @ ( abs_abs @ int ) @ S7 ) @ ( set_ord_lessThan @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded
thf(fact_4554_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A4: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A4 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ( B2 @ X2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_4555_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A4: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A4 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ( B2 @ X2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_4556_UN__empty,axiom,
! [B: $tType,A: $tType,B2: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_4557_UN__empty2,axiom,
! [B: $tType,A: $tType,A4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( bot_bot @ ( set @ A ) )
@ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_4558_UN__subset__iff,axiom,
! [A: $tType,B: $tType,A4: B > ( set @ A ),I5: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I5 ) ) @ B2 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I5 )
=> ( ord_less_eq @ ( set @ A ) @ ( A4 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_4559_UN__upper,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: A > ( set @ B )] :
( ( member @ A @ A3 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( B2 @ A3 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A4 ) ) ) ) ).
% UN_upper
thf(fact_4560_UN__least,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: A > ( set @ B ),C2: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( B2 @ X3 ) @ C2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A4 ) ) @ C2 ) ) ).
% UN_least
thf(fact_4561_UN__mono,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ A,F3: A > ( set @ B ),G2: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ A4 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ G2 @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_4562_UN__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A4: set @ A,A3: B,B2: A > ( set @ B )] :
( ( member @ A @ U @ A4 )
=> ( ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( insert2 @ B @ A3 @ ( B2 @ X2 ) )
@ A4 ) )
= ( insert2 @ B @ A3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_4563_Un__Union__image,axiom,
! [A: $tType,B: $tType,A4: B > ( set @ A ),B2: B > ( set @ A ),C2: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( sup_sup @ ( set @ A ) @ ( A4 @ X2 ) @ ( B2 @ X2 ) )
@ C2 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ C2 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ C2 ) ) ) ) ).
% Un_Union_image
thf(fact_4564_UN__Un__distrib,axiom,
! [A: $tType,B: $tType,A4: B > ( set @ A ),B2: B > ( set @ A ),I5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I4: B] : ( sup_sup @ ( set @ A ) @ ( A4 @ I4 ) @ ( B2 @ I4 ) )
@ I5 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I5 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ I5 ) ) ) ) ).
% UN_Un_distrib
thf(fact_4565_UN__absorb,axiom,
! [B: $tType,A: $tType,K: A,I5: set @ A,A4: A > ( set @ B )] :
( ( member @ A @ K @ I5 )
=> ( ( sup_sup @ ( set @ B ) @ ( A4 @ K ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I5 ) ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I5 ) ) ) ) ).
% UN_absorb
thf(fact_4566_INT__lower,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: A > ( set @ B )] :
( ( member @ A @ A3 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A4 ) ) @ ( B2 @ A3 ) ) ) ).
% INT_lower
thf(fact_4567_INT__greatest,axiom,
! [B: $tType,A: $tType,A4: set @ A,C2: set @ B,B2: A > ( set @ B )] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ C2 @ ( B2 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ C2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A4 ) ) ) ) ).
% INT_greatest
thf(fact_4568_INT__anti__mono,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ A,F3: A > ( set @ B ),G2: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ B2 ) ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ G2 @ A4 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_4569_INT__subset__iff,axiom,
! [A: $tType,B: $tType,B2: set @ A,A4: B > ( set @ A ),I5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I5 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I5 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ ( A4 @ X2 ) ) ) ) ) ).
% INT_subset_iff
thf(fact_4570_INT__extend__simps_I5_J,axiom,
! [I6: $tType,J4: $tType,A3: I6,B2: J4 > ( set @ I6 ),C2: set @ J4] :
( ( insert2 @ I6 @ A3 @ ( complete_Inf_Inf @ ( set @ I6 ) @ ( image2 @ J4 @ ( set @ I6 ) @ B2 @ C2 ) ) )
= ( complete_Inf_Inf @ ( set @ I6 )
@ ( image2 @ J4 @ ( set @ I6 )
@ ^ [X2: J4] : ( insert2 @ I6 @ A3 @ ( B2 @ X2 ) )
@ C2 ) ) ) ).
% INT_extend_simps(5)
thf(fact_4571_INT__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A4: set @ A,A3: B,B2: A > ( set @ B )] :
( ( member @ A @ U @ A4 )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( insert2 @ B @ A3 @ ( B2 @ X2 ) )
@ A4 ) )
= ( insert2 @ B @ A3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A4 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_4572_INT__extend__simps_I7_J,axiom,
! [M11: $tType,N10: $tType,A4: set @ M11,B2: N10 > ( set @ M11 ),C2: set @ N10] :
( ( sup_sup @ ( set @ M11 ) @ A4 @ ( complete_Inf_Inf @ ( set @ M11 ) @ ( image2 @ N10 @ ( set @ M11 ) @ B2 @ C2 ) ) )
= ( complete_Inf_Inf @ ( set @ M11 )
@ ( image2 @ N10 @ ( set @ M11 )
@ ^ [X2: N10] : ( sup_sup @ ( set @ M11 ) @ A4 @ ( B2 @ X2 ) )
@ C2 ) ) ) ).
% INT_extend_simps(7)
thf(fact_4573_INT__extend__simps_I6_J,axiom,
! [L4: $tType,K8: $tType,A4: K8 > ( set @ L4 ),C2: set @ K8,B2: set @ L4] :
( ( sup_sup @ ( set @ L4 ) @ ( complete_Inf_Inf @ ( set @ L4 ) @ ( image2 @ K8 @ ( set @ L4 ) @ A4 @ C2 ) ) @ B2 )
= ( complete_Inf_Inf @ ( set @ L4 )
@ ( image2 @ K8 @ ( set @ L4 )
@ ^ [X2: K8] : ( sup_sup @ ( set @ L4 ) @ ( A4 @ X2 ) @ B2 )
@ C2 ) ) ) ).
% INT_extend_simps(6)
thf(fact_4574_Un__INT__distrib,axiom,
! [A: $tType,B: $tType,B2: set @ A,A4: B > ( set @ A ),I5: set @ B] :
( ( sup_sup @ ( set @ A ) @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I4: B] : ( sup_sup @ ( set @ A ) @ B2 @ ( A4 @ I4 ) )
@ I5 ) ) ) ).
% Un_INT_distrib
thf(fact_4575_Un__INT__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A4: B > ( set @ A ),I5: set @ B,B2: C > ( set @ A ),J5: set @ C] :
( ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B2 @ J5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I4: B] :
( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [J3: C] : ( sup_sup @ ( set @ A ) @ ( A4 @ I4 ) @ ( B2 @ J3 ) )
@ J5 ) )
@ I5 ) ) ) ).
% Un_INT_distrib2
thf(fact_4576_Un__Inter,axiom,
! [A: $tType,A4: set @ A,B2: set @ ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( complete_Inf_Inf @ ( set @ A ) @ B2 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 ) @ B2 ) ) ) ).
% Un_Inter
thf(fact_4577_set__take__subset__set__take,axiom,
! [A: $tType,M: nat,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ M @ Xs ) ) @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_4578_in__image__insert__iff,axiom,
! [A: $tType,B2: set @ ( set @ A ),X: A,A4: set @ A] :
( ! [C7: set @ A] :
( ( member @ ( set @ A ) @ C7 @ B2 )
=> ~ ( member @ A @ X @ C7 ) )
=> ( ( member @ ( set @ A ) @ A4 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X ) @ B2 ) )
= ( ( member @ A @ X @ A4 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_4579_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C2: set @ B,A3: A,B2: B > ( set @ A )] :
( ( ( C2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ C2 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ C2 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( insert2 @ A @ A3 @ ( B2 @ X2 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_4580_UN__extend__simps_I3_J,axiom,
! [E4: $tType,F: $tType,C2: set @ F,A4: set @ E4,B2: F > ( set @ E4 )] :
( ( ( C2
= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E4 ) @ A4 @ ( complete_Sup_Sup @ ( set @ E4 ) @ ( image2 @ F @ ( set @ E4 ) @ B2 @ C2 ) ) )
= A4 ) )
& ( ( C2
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E4 ) @ A4 @ ( complete_Sup_Sup @ ( set @ E4 ) @ ( image2 @ F @ ( set @ E4 ) @ B2 @ C2 ) ) )
= ( complete_Sup_Sup @ ( set @ E4 )
@ ( image2 @ F @ ( set @ E4 )
@ ^ [X2: F] : ( sup_sup @ ( set @ E4 ) @ A4 @ ( B2 @ X2 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_4581_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C2: set @ C,A4: C > ( set @ D ),B2: set @ D] :
( ( ( C2
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A4 @ C2 ) ) @ B2 )
= B2 ) )
& ( ( C2
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A4 @ C2 ) ) @ B2 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X2: C] : ( sup_sup @ ( set @ D ) @ ( A4 @ X2 ) @ B2 )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_4582_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C2: set @ A,A4: A > ( set @ B ),B2: set @ B] :
( ( ( C2
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ C2 ) ) @ B2 )
= B2 ) )
& ( ( C2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ C2 ) ) @ B2 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( inf_inf @ ( set @ B ) @ ( A4 @ X2 ) @ B2 )
@ C2 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_4583_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C2: set @ D,A4: set @ C,B2: D > ( set @ C )] :
( ( ( C2
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A4 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B2 @ C2 ) ) )
= A4 ) )
& ( ( C2
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A4 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B2 @ C2 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X2: D] : ( inf_inf @ ( set @ C ) @ A4 @ ( B2 @ X2 ) )
@ C2 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_4584_Pow__insert,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( pow2 @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A4 ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ A3 ) @ ( pow2 @ A @ A4 ) ) ) ) ).
% Pow_insert
thf(fact_4585_INT__Un,axiom,
! [A: $tType,B: $tType,M5: B > ( set @ A ),A4: set @ B,B2: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M5 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M5 @ A4 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M5 @ B2 ) ) ) ) ).
% INT_Un
thf(fact_4586_nth__take__lemma,axiom,
! [A: $tType,K: nat,Xs: list @ A,Ys2: list @ A] :
( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys2 @ I2 ) ) )
=> ( ( take @ A @ K @ Xs )
= ( take @ A @ K @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_4587_INT__extend__simps_I4_J,axiom,
! [G5: $tType,H4: $tType,C2: set @ H4,A4: set @ G5,B2: H4 > ( set @ G5 )] :
( ( ( C2
= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( minus_minus @ ( set @ G5 ) @ A4 @ ( complete_Sup_Sup @ ( set @ G5 ) @ ( image2 @ H4 @ ( set @ G5 ) @ B2 @ C2 ) ) )
= A4 ) )
& ( ( C2
!= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( minus_minus @ ( set @ G5 ) @ A4 @ ( complete_Sup_Sup @ ( set @ G5 ) @ ( image2 @ H4 @ ( set @ G5 ) @ B2 @ C2 ) ) )
= ( complete_Inf_Inf @ ( set @ G5 )
@ ( image2 @ H4 @ ( set @ G5 )
@ ^ [X2: H4] : ( minus_minus @ ( set @ G5 ) @ A4 @ ( B2 @ X2 ) )
@ C2 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_4588_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U )
=> ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U )
= ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_4589_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,A4: B > ( set @ C ),G2: C > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( finite_finite2 @ C @ ( A4 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ! [Xa2: B] :
( ( member @ B @ Xa2 @ I5 )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf @ ( set @ C ) @ ( A4 @ X3 ) @ ( A4 @ Xa2 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G2 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I5 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( groups7311177749621191930dd_sum @ C @ A @ G2 @ ( A4 @ X2 ) )
@ I5 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_4590_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,A4: B > ( set @ C ),G2: C > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( finite_finite2 @ C @ ( A4 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ! [Xa2: B] :
( ( member @ B @ Xa2 @ I5 )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf @ ( set @ C ) @ ( A4 @ X3 ) @ ( A4 @ Xa2 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G2 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I5 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( groups7121269368397514597t_prod @ C @ A @ G2 @ ( A4 @ X2 ) )
@ I5 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_4591_card__UN__le,axiom,
! [B: $tType,A: $tType,I5: set @ A,A4: A > ( set @ B )] :
( ( finite_finite2 @ A @ I5 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I5 ) ) )
@ ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] : ( finite_card @ B @ ( A4 @ I4 ) )
@ I5 ) ) ) ).
% card_UN_le
thf(fact_4592_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I5: set @ A,A4: A > ( set @ B )] :
( ( finite_finite2 @ A @ I5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ I5 )
=> ( finite_finite2 @ B @ ( A4 @ X3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ I5 )
=> ! [Xa2: A] :
( ( member @ A @ Xa2 @ I5 )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf @ ( set @ B ) @ ( A4 @ X3 ) @ ( A4 @ Xa2 ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I5 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] : ( finite_card @ B @ ( A4 @ I4 ) )
@ I5 ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_4593_UN__le__eq__Un0,axiom,
! [A: $tType,M5: nat > ( set @ A ),N: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M5 @ ( set_ord_atMost @ nat @ N ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M5 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) @ ( M5 @ ( zero_zero @ nat ) ) ) ) ).
% UN_le_eq_Un0
thf(fact_4594_Union__image__insert,axiom,
! [A: $tType,B: $tType,F3: B > ( set @ A ),A3: B,B2: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ ( insert2 @ B @ A3 @ B2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( F3 @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ B2 ) ) ) ) ).
% Union_image_insert
thf(fact_4595_Union__image__empty,axiom,
! [B: $tType,A: $tType,A4: set @ A,F3: B > ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= A4 ) ).
% Union_image_empty
thf(fact_4596_UN__image__subset,axiom,
! [C: $tType,A: $tType,B: $tType,F3: B > ( set @ A ),G2: C > ( set @ B ),X: C,X4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ ( G2 @ X ) ) ) @ X4 )
= ( ord_less_eq @ ( set @ B ) @ ( G2 @ X )
@ ( collect @ B
@ ^ [X2: B] : ( ord_less_eq @ ( set @ A ) @ ( F3 @ X2 ) @ X4 ) ) ) ) ).
% UN_image_subset
thf(fact_4597_Pow__fold,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( pow2 @ A @ A4 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X2: A,A6: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A6 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X2 ) @ A6 ) )
@ ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A4 ) ) ) ).
% Pow_fold
thf(fact_4598_fold__empty,axiom,
! [B: $tType,A: $tType,F3: B > A > A,Z: A] :
( ( finite_fold @ B @ A @ F3 @ Z @ ( bot_bot @ ( set @ B ) ) )
= Z ) ).
% fold_empty
thf(fact_4599_fold__infinite,axiom,
! [A: $tType,B: $tType,A4: set @ A,F3: A > B > B,Z: B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ A4 )
= Z ) ) ).
% fold_infinite
thf(fact_4600_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I5: set @ A,F4: A > ( filter @ B )] :
( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I5 )
=> ? [X5: A] :
( ( member @ A @ X5 @ I5 )
& ( ord_less_eq @ ( filter @ B ) @ ( F4 @ X5 ) @ ( inf_inf @ ( filter @ B ) @ ( F4 @ I2 ) @ ( F4 @ J2 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F4 @ I5 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ I5 )
& ( ( F4 @ X2 )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_4601_fold__closed__eq,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,F3: A > B > B,G2: A > B > B,Z: B] :
( ! [A7: A,B7: B] :
( ( member @ A @ A7 @ A4 )
=> ( ( member @ B @ B7 @ B2 )
=> ( ( F3 @ A7 @ B7 )
= ( G2 @ A7 @ B7 ) ) ) )
=> ( ! [A7: A,B7: B] :
( ( member @ A @ A7 @ A4 )
=> ( ( member @ B @ B7 @ B2 )
=> ( member @ B @ ( G2 @ A7 @ B7 ) @ B2 ) ) )
=> ( ( member @ B @ Z @ B2 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ A4 )
= ( finite_fold @ A @ B @ G2 @ Z @ A4 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_4602_Inf__filter__not__bot,axiom,
! [A: $tType,B2: set @ ( filter @ A )] :
( ! [X9: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X9 @ B2 )
=> ( ( finite_finite2 @ ( filter @ A ) @ X9 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ X9 )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ B2 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% Inf_filter_not_bot
thf(fact_4603_union__fold__insert,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= ( finite_fold @ A @ ( set @ A ) @ ( insert2 @ A ) @ B2 @ A4 ) ) ) ).
% union_fold_insert
thf(fact_4604_sup__Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B2: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A4 ) @ B2 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ B2 @ A4 ) ) ) ) ).
% sup_Sup_fold_sup
thf(fact_4605_inf__Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B2: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A4 ) @ B2 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ B2 @ A4 ) ) ) ) ).
% inf_Inf_fold_inf
thf(fact_4606_minus__fold__remove,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ B2 @ A4 )
= ( finite_fold @ A @ ( set @ A ) @ ( remove @ A ) @ B2 @ A4 ) ) ) ).
% minus_fold_remove
thf(fact_4607_Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( complete_Sup_Sup @ A @ A4 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) @ A4 ) ) ) ) ).
% Sup_fold_sup
thf(fact_4608_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X @ A4 ) ) ) ) ).
% Inf_fin.eq_fold
thf(fact_4609_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X @ A4 ) ) ) ) ).
% Sup_fin.eq_fold
thf(fact_4610_image__fold__insert,axiom,
! [B: $tType,A: $tType,A4: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( image2 @ A @ B @ F3 @ A4 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K3: A] : ( insert2 @ B @ ( F3 @ K3 ) )
@ ( bot_bot @ ( set @ B ) )
@ A4 ) ) ) ).
% image_fold_insert
thf(fact_4611_conj__subset__def,axiom,
! [A: $tType,A4: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A4
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_4612_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F3: nat > ( set @ A ),S: set @ A] :
( ! [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( F3 @ I2 ) @ S )
=> ( ( finite_finite2 @ A @ S )
=> ( ? [N8: nat] :
( ! [N3: nat] :
( ( ord_less_eq @ nat @ N3 @ N8 )
=> ! [M4: nat] :
( ( ord_less_eq @ nat @ M4 @ N8 )
=> ( ( ord_less @ nat @ M4 @ N3 )
=> ( ord_less @ ( set @ A ) @ ( F3 @ M4 ) @ ( F3 @ N3 ) ) ) ) )
& ! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ( F3 @ N8 )
= ( F3 @ N3 ) ) ) )
=> ( ( F3 @ ( finite_card @ A @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ F3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_4613_Set__filter__fold,axiom,
! [A: $tType,A4: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( filter3 @ A @ P @ A4 )
= ( finite_fold @ A @ ( set @ A )
@ ^ [X2: A,A14: set @ A] : ( if @ ( set @ A ) @ ( P @ X2 ) @ ( insert2 @ A @ X2 @ A14 ) @ A14 )
@ ( bot_bot @ ( set @ A ) )
@ A4 ) ) ) ).
% Set_filter_fold
thf(fact_4614_suminf__eq__SUP__real,axiom,
! [X4: nat > real] :
( ( summable @ real @ X4 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( X4 @ I2 ) )
=> ( ( suminf @ real @ X4 )
= ( complete_Sup_Sup @ real
@ ( image2 @ nat @ real
@ ^ [I4: nat] : ( groups7311177749621191930dd_sum @ nat @ real @ X4 @ ( set_ord_lessThan @ nat @ I4 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% suminf_eq_SUP_real
thf(fact_4615_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_4616_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_4617_member__filter,axiom,
! [A: $tType,X: A,P: A > $o,A4: set @ A] :
( ( member @ A @ X @ ( filter3 @ A @ P @ A4 ) )
= ( ( member @ A @ X @ A4 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_4618_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_4619_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_4620_inf__top__left,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% inf_top_left
thf(fact_4621_inf__top__right,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% inf_top_right
thf(fact_4622_inf__eq__top__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( top_top @ A ) )
= ( ( X
= ( top_top @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% inf_eq_top_iff
thf(fact_4623_top__eq__inf__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A,Y: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ X @ Y ) )
= ( ( X
= ( top_top @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% top_eq_inf_iff
thf(fact_4624_inf__top_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A,B3: A] :
( ( ( inf_inf @ A @ A3 @ B3 )
= ( top_top @ A ) )
= ( ( A3
= ( top_top @ A ) )
& ( B3
= ( top_top @ A ) ) ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_4625_inf__top_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ A3 )
= A3 ) ) ).
% inf_top.left_neutral
thf(fact_4626_inf__top_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A,B3: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ A3 @ B3 ) )
= ( ( A3
= ( top_top @ A ) )
& ( B3
= ( top_top @ A ) ) ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_4627_inf__top_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ A3 @ ( top_top @ A ) )
= A3 ) ) ).
% inf_top.right_neutral
thf(fact_4628_boolean__algebra_Odisj__one__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_one_left
thf(fact_4629_boolean__algebra_Odisj__one__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_one_right
thf(fact_4630_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_4631_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_4632_Int__UNIV,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( top_top @ ( set @ A ) ) )
= ( ( A4
= ( top_top @ ( set @ A ) ) )
& ( B2
= ( top_top @ ( set @ A ) ) ) ) ) ).
% Int_UNIV
thf(fact_4633_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_4634_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_4635_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_4636_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_4637_finite__Collect__not,axiom,
! [A: $tType,P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_4638_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( ( complete_Sup_Sup @ A @ A4 )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A4 )
& ( ord_less @ A @ X2 @ Y3 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_4639_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.compl_zero
thf(fact_4640_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.compl_one
thf(fact_4641_boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( uminus_uminus @ A @ X ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_cancel_right
thf(fact_4642_boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ X )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_cancel_left
thf(fact_4643_sup__compl__top__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ Y ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left2
thf(fact_4644_sup__compl__top__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( sup_sup @ A @ X @ Y ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left1
thf(fact_4645_Inf__UNIV,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Inf_UNIV
thf(fact_4646_ccInf__empty,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% ccInf_empty
thf(fact_4647_Inf__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Inf_empty
thf(fact_4648_Diff__UNIV,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_4649_surj__fn,axiom,
! [A: $tType,F3: A > A,N: nat] :
( ( ( image2 @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_fn
thf(fact_4650_finite__compl,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_compl
thf(fact_4651_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A4: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ? [Y3: B] :
( ( member @ B @ Y3 @ A4 )
& ( ord_less @ A @ X2 @ ( F3 @ Y3 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_4652_range__constant,axiom,
! [B: $tType,A: $tType,X: A] :
( ( image2 @ B @ A
@ ^ [Uu3: B] : X
@ ( top_top @ ( set @ B ) ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_4653_ccINF__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F3: B > A] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% ccINF_empty
thf(fact_4654_INT__constant,axiom,
! [B: $tType,A: $tType,A4: set @ B,C3: set @ A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C3
@ A4 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C3
@ A4 ) )
= C3 ) ) ) ).
% INT_constant
thf(fact_4655_Inf__atMostLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( top_top @ A ) @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X ) )
= ( bot_bot @ A ) ) ) ) ).
% Inf_atMostLessThan
thf(fact_4656_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C2: set @ A,A4: A > ( set @ B ),B2: set @ B] :
( ( ( C2
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( inf_inf @ ( set @ B ) @ ( A4 @ X2 ) @ B2 )
@ C2 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( inf_inf @ ( set @ B ) @ ( A4 @ X2 ) @ B2 )
@ C2 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ C2 ) ) @ B2 ) ) ) ) ).
% INT_simps(1)
thf(fact_4657_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C2: set @ D,A4: set @ C,B2: D > ( set @ C )] :
( ( ( C2
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X2: D] : ( inf_inf @ ( set @ C ) @ A4 @ ( B2 @ X2 ) )
@ C2 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X2: D] : ( inf_inf @ ( set @ C ) @ A4 @ ( B2 @ X2 ) )
@ C2 ) )
= ( inf_inf @ ( set @ C ) @ A4 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B2 @ C2 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_4658_INT__simps_I3_J,axiom,
! [E4: $tType,F: $tType,C2: set @ E4,A4: E4 > ( set @ F ),B2: set @ F] :
( ( ( C2
= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image2 @ E4 @ ( set @ F )
@ ^ [X2: E4] : ( minus_minus @ ( set @ F ) @ ( A4 @ X2 ) @ B2 )
@ C2 ) )
= ( top_top @ ( set @ F ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image2 @ E4 @ ( set @ F )
@ ^ [X2: E4] : ( minus_minus @ ( set @ F ) @ ( A4 @ X2 ) @ B2 )
@ C2 ) )
= ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image2 @ E4 @ ( set @ F ) @ A4 @ C2 ) ) @ B2 ) ) ) ) ).
% INT_simps(3)
thf(fact_4659_INT__simps_I4_J,axiom,
! [G5: $tType,H4: $tType,C2: set @ H4,A4: set @ G5,B2: H4 > ( set @ G5 )] :
( ( ( C2
= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G5 )
@ ( image2 @ H4 @ ( set @ G5 )
@ ^ [X2: H4] : ( minus_minus @ ( set @ G5 ) @ A4 @ ( B2 @ X2 ) )
@ C2 ) )
= ( top_top @ ( set @ G5 ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G5 )
@ ( image2 @ H4 @ ( set @ G5 )
@ ^ [X2: H4] : ( minus_minus @ ( set @ G5 ) @ A4 @ ( B2 @ X2 ) )
@ C2 ) )
= ( minus_minus @ ( set @ G5 ) @ A4 @ ( complete_Sup_Sup @ ( set @ G5 ) @ ( image2 @ H4 @ ( set @ G5 ) @ B2 @ C2 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_4660_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A,X: B] :
( ( B3
= ( F3 @ X ) )
=> ( member @ A @ B3 @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_4661_rangeI,axiom,
! [A: $tType,B: $tType,F3: B > A,X: B] : ( member @ A @ ( F3 @ X ) @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_4662_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,G2: B > C] :
( ( image2 @ B @ A
@ ^ [X2: B] : ( F3 @ ( G2 @ X2 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image2 @ C @ A @ F3 @ ( image2 @ B @ C @ G2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_4663_rangeE,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A] :
( ( member @ A @ B3 @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X3: B] :
( B3
!= ( F3 @ X3 ) ) ) ).
% rangeE
thf(fact_4664_less__filter__def,axiom,
! [A: $tType] :
( ( ord_less @ ( filter @ A ) )
= ( ^ [F8: filter @ A,F9: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F8 @ F9 )
& ~ ( ord_less_eq @ ( filter @ A ) @ F9 @ F8 ) ) ) ) ).
% less_filter_def
thf(fact_4665_atLeastAtMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( bounded_lattice @ A )
=> ! [X: A,Y: A] :
( ( ( set_or1337092689740270186AtMost @ A @ X @ Y )
= ( top_top @ ( set @ A ) ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_4666_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_4667_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_4668_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_4669_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_4670_subset__UNIV,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_4671_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A3 ) ) ).
% top.extremum_strict
thf(fact_4672_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( A3
!= ( top_top @ A ) )
= ( ord_less @ A @ A3 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_4673_infinite__UNIV__char__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% infinite_UNIV_char_0
thf(fact_4674_ex__new__if__finite,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ A4 )
=> ? [A7: A] :
~ ( member @ A @ A7 @ A4 ) ) ) ).
% ex_new_if_finite
thf(fact_4675_finite__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_UNIV
thf(fact_4676_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_4677_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_4678_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_4679_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_4680_UNIV__eq__I,axiom,
! [A: $tType,A4: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A4 )
=> ( ( top_top @ ( set @ A ) )
= A4 ) ) ).
% UNIV_eq_I
thf(fact_4681_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_4682_Set_Ofilter__def,axiom,
! [A: $tType] :
( ( filter3 @ A )
= ( ^ [P3: A > $o,A6: set @ A] :
( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A6 )
& ( P3 @ A5 ) ) ) ) ) ).
% Set.filter_def
thf(fact_4683_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X2: A] : $true ) ) ).
% UNIV_def
thf(fact_4684_nat__not__finite,axiom,
~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ).
% nat_not_finite
thf(fact_4685_infinite__UNIV__nat,axiom,
~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ).
% infinite_UNIV_nat
thf(fact_4686_insert__UNIV,axiom,
! [A: $tType,X: A] :
( ( insert2 @ A @ X @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_4687_Un__UNIV__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B2 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_4688_Un__UNIV__right,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_4689_Int__UNIV__right,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) )
= A4 ) ).
% Int_UNIV_right
thf(fact_4690_Int__UNIV__left,axiom,
! [A: $tType,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_4691_boolean__algebra_Oconj__one__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% boolean_algebra.conj_one_right
thf(fact_4692_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_4693_sup__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( sup_sup @ A @ X @ B3 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left2
thf(fact_4694_sup__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ A3 ) @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ B3 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left1
thf(fact_4695_Inf__sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B2: set @ A,A3: A] :
( ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B2 ) @ A3 )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( ( sup_sup @ A @ X2 @ A3 )
= ( top_top @ A ) ) ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_4696_finite__filter,axiom,
! [A: $tType,S: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ S )
=> ( finite_finite2 @ A @ ( filter3 @ A @ P @ S ) ) ) ).
% finite_filter
thf(fact_4697_range__subsetD,axiom,
! [B: $tType,A: $tType,F3: B > A,B2: set @ A,I: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) @ B2 )
=> ( member @ A @ ( F3 @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_4698_perfect__space__class_OUNIV__not__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X: A] :
( ( top_top @ ( set @ A ) )
!= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% perfect_space_class.UNIV_not_singleton
thf(fact_4699_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_4700_not__UNIV__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L @ H ) ) ) ).
% not_UNIV_le_Icc
thf(fact_4701_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( ( finite_card @ A @ A4 )
= ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( A4
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_4702_finite__range__Some,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_range_Some
thf(fact_4703_not__UNIV__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atMost @ A @ H ) ) ) ).
% not_UNIV_le_Iic
thf(fact_4704_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_4705_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_4706_Compl__partition2,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ A4 )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition2
thf(fact_4707_Compl__partition,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition
thf(fact_4708_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_4709_finite__range__imageI,axiom,
! [C: $tType,A: $tType,B: $tType,G2: B > A,F3: A > C] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ G2 @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ C
@ ( image2 @ B @ C
@ ^ [X2: B] : ( F3 @ ( G2 @ X2 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_imageI
thf(fact_4710_sup__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( top_top @ A ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% sup_shunt
thf(fact_4711_boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [A3: A,X: A,Y: A] :
( ( ( inf_inf @ A @ A3 @ X )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A3 @ X )
= ( top_top @ A ) )
=> ( ( ( inf_inf @ A @ A3 @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A3 @ Y )
= ( top_top @ A ) )
=> ( X = Y ) ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_4712_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: A,X: B] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( F3 @ X )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_4713_INF__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% INF_empty
thf(fact_4714_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,C3: A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C3
@ A4 ) )
= ( top_top @ A ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C3
@ A4 ) )
= C3 ) ) ) ) ).
% INF_constant
thf(fact_4715_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_4716_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_4717_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( image2 @ B @ A @ F3 @ ( uminus_uminus @ ( set @ B ) @ A4 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_4718_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu3: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_4719_INT__empty,axiom,
! [B: $tType,A: $tType,B2: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_4720_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ X @ Y )
= ( top_top @ A ) )
=> ( ( uminus_uminus @ A @ X )
= Y ) ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_4721_Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( complete_Inf_Inf @ A @ A4 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ A4 ) ) ) ) ).
% Inf_fold_inf
thf(fact_4722_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_4723_range__enumerate,axiom,
! [S: set @ nat] :
( ~ ( finite_finite2 @ nat @ S )
=> ( ( image2 @ nat @ nat @ ( infini527867602293511546merate @ nat @ S ) @ ( top_top @ ( set @ nat ) ) )
= S ) ) ).
% range_enumerate
thf(fact_4724_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_4725_inter__Set__filter,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( filter3 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 )
@ B2 ) ) ) ).
% inter_Set_filter
thf(fact_4726_UNIV__nat__eq,axiom,
( ( top_top @ ( set @ nat ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UNIV_nat_eq
thf(fact_4727_INT__extend__simps_I3_J,axiom,
! [F: $tType,E4: $tType,C2: set @ E4,A4: E4 > ( set @ F ),B2: set @ F] :
( ( ( C2
= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image2 @ E4 @ ( set @ F ) @ A4 @ C2 ) ) @ B2 )
= ( minus_minus @ ( set @ F ) @ ( top_top @ ( set @ F ) ) @ B2 ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image2 @ E4 @ ( set @ F ) @ A4 @ C2 ) ) @ B2 )
= ( complete_Inf_Inf @ ( set @ F )
@ ( image2 @ E4 @ ( set @ F )
@ ^ [X2: E4] : ( minus_minus @ ( set @ F ) @ ( A4 @ X2 ) @ B2 )
@ C2 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_4728_UN__UN__finite__eq,axiom,
! [A: $tType,A4: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [N2: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_4729_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F3: B > A] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_4730_UN__finite__subset,axiom,
! [A: $tType,A4: nat > ( set @ A ),C2: set @ A] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ C2 ) ) ).
% UN_finite_subset
thf(fact_4731_UN__finite2__eq,axiom,
! [A: $tType,A4: nat > ( set @ A ),B2: nat > ( set @ A ),K: nat] :
( ! [N3: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_4732_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image2 @ nat @ nat
@ ^ [M2: nat] : ( modulo_modulo @ nat @ M2 @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_4733_fold__union__pair,axiom,
! [B: $tType,A: $tType,B2: set @ A,X: B,A4: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B2 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B2 ) )
@ A4 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) )
@ A4
@ B2 ) ) ) ).
% fold_union_pair
thf(fact_4734_UN__finite2__subset,axiom,
! [A: $tType,A4: nat > ( set @ A ),B2: nat > ( set @ A ),K: nat] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_4735_cclfp__def,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( order_532582986084564980_cclfp @ A )
= ( ^ [F2: A > A] :
( complete_Sup_Sup @ A
@ ( image2 @ nat @ A
@ ^ [I4: nat] : ( compow @ ( A > A ) @ I4 @ F2 @ ( bot_bot @ A ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% cclfp_def
thf(fact_4736_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A4: B > ( set @ A ),I: B,B2: set @ A,J5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( fun_upd @ B @ ( set @ A ) @ A4 @ I @ B2 ) @ J5 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ B ) @ J5 @ ( insert2 @ B @ I @ ( bot_bot @ ( set @ B ) ) ) ) ) ) @ ( if @ ( set @ A ) @ ( member @ B @ I @ J5 ) @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% UNION_fun_upd
thf(fact_4737_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,A4: set @ A,B2: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( finite_fold @ A @ B @ F3 @ ( finite_fold @ A @ B @ F3 @ Z @ A4 ) @ B2 ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
thf(fact_4738_card__UNIV__unit,axiom,
( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
= ( one_one @ nat ) ) ).
% card_UNIV_unit
thf(fact_4739_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_4740_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_4741_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_4742_comp__fun__commute__on_Ofun__left__comm,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,X: A,Y: A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( member @ A @ X @ S )
=> ( ( member @ A @ Y @ S )
=> ( ( F3 @ Y @ ( F3 @ X @ Z ) )
= ( F3 @ X @ ( F3 @ Y @ Z ) ) ) ) ) ) ).
% comp_fun_commute_on.fun_left_comm
thf(fact_4743_finite__update__induct,axiom,
! [B: $tType,A: $tType,F3: A > B,C3: B,P: ( A > B ) > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( F3 @ A5 )
!= C3 ) ) )
=> ( ( P
@ ^ [A5: A] : C3 )
=> ( ! [A7: A,B7: B,F6: A > B] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [C6: A] :
( ( F6 @ C6 )
!= C3 ) ) )
=> ( ( ( F6 @ A7 )
= C3 )
=> ( ( B7 != C3 )
=> ( ( P @ F6 )
=> ( P @ ( fun_upd @ A @ B @ F6 @ A7 @ B7 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_update_induct
thf(fact_4744_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,G2: A > nat] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( finite4664212375090638736ute_on @ A @ B @ S
@ ^ [X2: A] : ( compow @ ( B > B ) @ ( G2 @ X2 ) @ ( F3 @ X2 ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on_funpow
thf(fact_4745_Inter__UNIV,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Inter_UNIV
thf(fact_4746_SUP__UNIV__bool__expand,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: $o > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ $o @ A @ A4 @ ( top_top @ ( set @ $o ) ) ) )
= ( sup_sup @ A @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_4747_Un__eq__UN,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ $o @ ( set @ A )
@ ^ [B5: $o] : ( if @ ( set @ A ) @ B5 @ A6 @ B6 )
@ ( top_top @ ( set @ $o ) ) ) ) ) ) ).
% Un_eq_UN
thf(fact_4748_UN__bool__eq,axiom,
! [A: $tType,A4: $o > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ $o @ ( set @ A ) @ A4 @ ( top_top @ ( set @ $o ) ) ) )
= ( sup_sup @ ( set @ A ) @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% UN_bool_eq
thf(fact_4749_Finite__Set_Ofold__cong,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,G2: A > B > B,A4: set @ A,S3: B,T2: B,B2: set @ A] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( finite4664212375090638736ute_on @ A @ B @ S @ G2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( F3 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( S3 = T2 )
=> ( ( A4 = B2 )
=> ( ( finite_fold @ A @ B @ F3 @ S3 @ A4 )
= ( finite_fold @ A @ B @ G2 @ T2 @ B2 ) ) ) ) ) ) ) ) ) ).
% Finite_Set.fold_cong
thf(fact_4750_fun__upd__image,axiom,
! [A: $tType,B: $tType,X: B,A4: set @ B,F3: B > A,Y: A] :
( ( ( member @ B @ X @ A4 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F3 @ X @ Y ) @ A4 )
= ( insert2 @ A @ Y @ ( image2 @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ X @ A4 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F3 @ X @ Y ) @ A4 )
= ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ).
% fun_upd_image
thf(fact_4751_comp__fun__commute__on_Ofold__insert,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,A4: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert2 @ A @ X @ A4 ) )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ A4 ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert
thf(fact_4752_comp__fun__commute__on_Ofold__insert2,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,A4: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert2 @ A @ X @ A4 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X @ Z ) @ A4 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert2
thf(fact_4753_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,A4: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ A4 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X @ Z ) @ A4 ) ) ) ) ) ).
% comp_fun_commute_on.fold_fun_left_comm
thf(fact_4754_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,A4: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert2 @ A @ X @ A4 ) )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert_remove
thf(fact_4755_comp__fun__commute__on_Ofold__rec,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,A4: set @ A,X: A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ A4 )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_rec
thf(fact_4756_root__def,axiom,
( root
= ( ^ [N2: nat,X2: real] :
( if @ real
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ real )
@ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) )
@ ^ [Y3: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y3 ) @ ( power_power @ real @ ( abs_abs @ real @ Y3 ) @ N2 ) )
@ X2 ) ) ) ) ).
% root_def
thf(fact_4757_Id__on__fold,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( id_on @ A @ A4 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X2: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A4 ) ) ) ).
% Id_on_fold
thf(fact_4758_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A6: set @ A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X2: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A6 ) ) ) ) ).
% Id_on_def
thf(fact_4759_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_4760_top2I,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : ( top_top @ ( A > B > $o ) @ X @ Y ) ).
% top2I
thf(fact_4761_Id__onI,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id_on @ A @ A4 ) ) ) ).
% Id_onI
thf(fact_4762_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_4763_UNIV__bool,axiom,
( ( top_top @ ( set @ $o ) )
= ( insert2 @ $o @ $false @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% UNIV_bool
thf(fact_4764_Id__on__iff,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id_on @ A @ A4 ) )
= ( ( X = Y )
& ( member @ A @ X @ A4 ) ) ) ).
% Id_on_iff
thf(fact_4765_Id__on__eqI,axiom,
! [A: $tType,A3: A,B3: A,A4: set @ A] :
( ( A3 = B3 )
=> ( ( member @ A @ A3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( id_on @ A @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_4766_Id__onE,axiom,
! [A: $tType,C3: product_prod @ A @ A,A4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A4 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( C3
!= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_4767_Id__on__def_H,axiom,
! [A: $tType,A4: A > $o] :
( ( id_on @ A @ ( collect @ A @ A4 ) )
= ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y3: A] :
( ( X2 = Y3 )
& ( A4 @ X2 ) ) ) ) ) ).
% Id_on_def'
thf(fact_4768_DERIV__real__root__generic,axiom,
! [N: nat,X: real,D3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D3
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D3
= ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( D3
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ D3 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_4769_DERIV__even__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_4770_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real
@ ^ [X10: real] :
( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X10 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) )
@ ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( power_power @ real @ X @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_arctan_series
thf(fact_4771_at__within__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: A] :
( ( topolo174197925503356063within @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ) ).
% at_within_empty
thf(fact_4772_deriv__nonneg__imp__mono,axiom,
! [A3: real,B3: real,G2: real > real,G6: real > real] :
( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) )
=> ( has_field_derivative @ real @ G2 @ ( G6 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G6 @ X3 ) ) )
=> ( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ord_less_eq @ real @ ( G2 @ A3 ) @ ( G2 @ B3 ) ) ) ) ) ).
% deriv_nonneg_imp_mono
thf(fact_4773_DERIV__nonneg__imp__nondecreasing,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y5 ) ) ) )
=> ( ord_less_eq @ real @ ( F3 @ A3 ) @ ( F3 @ B3 ) ) ) ) ).
% DERIV_nonneg_imp_nondecreasing
thf(fact_4774_DERIV__nonpos__imp__nonincreasing,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ Y5 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less_eq @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) ) ) ) ).
% DERIV_nonpos_imp_nonincreasing
thf(fact_4775_DERIV__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F10: A,X: A,S3: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S3 )
=> ( has_field_derivative @ A @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% DERIV_subset
thf(fact_4776_has__field__derivative__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Y: A,X: A,S3: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F3 @ Y @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S3 )
=> ( has_field_derivative @ A @ F3 @ Y @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% has_field_derivative_subset
thf(fact_4777_at__within__union,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X: A,S: set @ A,T4: set @ A] :
( ( topolo174197925503356063within @ A @ X @ ( sup_sup @ ( set @ A ) @ S @ T4 ) )
= ( sup_sup @ ( filter @ A ) @ ( topolo174197925503356063within @ A @ X @ S ) @ ( topolo174197925503356063within @ A @ X @ T4 ) ) ) ) ).
% at_within_union
thf(fact_4778_DERIV__const,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [K: A,F4: filter @ A] :
( has_field_derivative @ A
@ ^ [X2: A] : K
@ ( zero_zero @ A )
@ F4 ) ) ).
% DERIV_const
thf(fact_4779_has__real__derivative__pos__inc__left,axiom,
! [F3: real > real,L: real,X: real,S: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H5 ) @ S )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ ( minus_minus @ real @ X @ H5 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_4780_has__real__derivative__neg__dec__left,axiom,
! [F3: real > real,L: real,X: real,S: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H5 ) @ S )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( minus_minus @ real @ X @ H5 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_4781_has__real__derivative__pos__inc__right,axiom,
! [F3: real > real,L: real,X: real,S: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( plus_plus @ real @ X @ H5 ) @ S )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( plus_plus @ real @ X @ H5 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_4782_has__real__derivative__neg__dec__right,axiom,
! [F3: real > real,L: real,X: real,S: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( plus_plus @ real @ X @ H5 ) @ S )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ ( plus_plus @ real @ X @ H5 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_4783_DERIV__neg__imp__decreasing,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_4784_DERIV__pos__imp__increasing,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) ) )
=> ( ord_less @ real @ ( F3 @ A3 ) @ ( F3 @ B3 ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_4785_DERIV__pos__inc__right,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( plus_plus @ real @ X @ H5 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_4786_DERIV__neg__dec__right,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ ( plus_plus @ real @ X @ H5 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_4787_DERIV__pos__inc__left,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ ( minus_minus @ real @ X @ H5 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_4788_DERIV__neg__dec__left,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( minus_minus @ real @ X @ H5 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_4789_DERIV__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S3: set @ A,G2: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( has_field_derivative @ A @ G2 @ E5 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ( G2 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D3 @ ( G2 @ X ) ) @ ( times_times @ A @ ( F3 @ X ) @ E5 ) ) @ ( times_times @ A @ ( G2 @ X ) @ ( G2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ) ).
% DERIV_divide
thf(fact_4790_DERIV__inverse_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S3: set @ A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( inverse_inverse @ A @ ( F3 @ X2 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F3 @ X ) ) @ D3 ) @ ( inverse_inverse @ A @ ( F3 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ).
% DERIV_inverse'
thf(fact_4791_at__le,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,T2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ T2 )
=> ( ord_less_eq @ ( filter @ A ) @ ( topolo174197925503356063within @ A @ X @ S3 ) @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ).
% at_le
thf(fact_4792_MVT2,axiom,
! [A3: real,B3: real,F3: real > real,F10: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B3 )
=> ( has_field_derivative @ real @ F3 @ ( F10 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A3 @ Z3 )
& ( ord_less @ real @ Z3 @ B3 )
& ( ( minus_minus @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) )
= ( times_times @ real @ ( minus_minus @ real @ B3 @ A3 ) @ ( F10 @ Z3 ) ) ) ) ) ) ).
% MVT2
thf(fact_4793_DERIV__local__const,axiom,
! [F3: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y2 ) ) @ D2 )
=> ( ( F3 @ X )
= ( F3 @ Y2 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_const
thf(fact_4794_DERIV__ln,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( inverse_inverse @ real @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln
thf(fact_4795_DERIV__inverse,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,S3: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% DERIV_inverse
thf(fact_4796_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S3: set @ A,N: nat] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( power_power @ A @ ( F3 @ X2 ) @ N )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( times_times @ A @ D3 @ ( power_power @ A @ ( F3 @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% DERIV_power
thf(fact_4797_DERIV__local__min,axiom,
! [F3: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y2 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_min
thf(fact_4798_DERIV__local__max,axiom,
! [F3: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y2 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F3 @ Y2 ) @ ( F3 @ X ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_max
thf(fact_4799_DERIV__ln__divide,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln_divide
thf(fact_4800_DERIV__pow,axiom,
! [N: nat,X: real,S3: set @ real] :
( has_field_derivative @ real
@ ^ [X2: real] : ( power_power @ real @ X2 @ N )
@ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ X @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ S3 ) ) ).
% DERIV_pow
thf(fact_4801_at__within__Icc__at,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,X: A,B3: A] :
( ( ord_less @ A @ A3 @ X )
=> ( ( ord_less @ A @ X @ B3 )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% at_within_Icc_at
thf(fact_4802_at__within__Icc__at__left,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( topolo174197925503356063within @ A @ B3 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( topolo174197925503356063within @ A @ B3 @ ( set_ord_lessThan @ A @ B3 ) ) ) ) ) ).
% at_within_Icc_at_left
thf(fact_4803_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_4804_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D2: A,X: A,S3: set @ A,G2: A > A,E2: A] :
( ( has_field_derivative @ A @ F3 @ D2 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( has_field_derivative @ A @ G2 @ E2 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ( G2 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y3: A] : ( divide_divide @ A @ ( F3 @ Y3 ) @ ( G2 @ Y3 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D2 @ ( G2 @ X ) ) @ ( times_times @ A @ E2 @ ( F3 @ X ) ) ) @ ( power_power @ A @ ( G2 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ) ).
% DERIV_quotient
thf(fact_4805_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D2: A,X: A,S3: set @ A] :
( ( has_field_derivative @ A @ F3 @ D2 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( inverse_inverse @ A @ ( F3 @ X2 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ D2 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F3 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_4806_termdiffs__sums__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K4: real,C3: nat > A,F3: A > A,F10: A,Z: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K4 )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) )
@ ( F3 @ Z3 ) ) )
=> ( ( has_field_derivative @ A @ F3 @ F10 @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K4 )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) )
@ F10 ) ) ) ) ) ).
% termdiffs_sums_strong
thf(fact_4807_has__real__derivative__powr,axiom,
! [Z: real,R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Z )
=> ( has_field_derivative @ real
@ ^ [Z6: real] : ( powr @ real @ Z6 @ R2 )
@ ( times_times @ real @ R2 @ ( powr @ real @ Z @ ( minus_minus @ real @ R2 @ ( one_one @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) ) ) ).
% has_real_derivative_powr
thf(fact_4808_termdiffs__strong_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K4: real,C3: nat > A,Z: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K4 )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K4 )
=> ( has_field_derivative @ A
@ ^ [Z6: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ Z6 @ N2 ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong'
thf(fact_4809_termdiffs__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K4: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ K4 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K4 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ X2 @ N2 ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong
thf(fact_4810_termdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K4: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ K4 @ N2 ) ) )
=> ( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ K4 @ N2 ) ) )
=> ( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C3 ) @ N2 ) @ ( power_power @ A @ K4 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K4 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ X2 @ N2 ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% termdiffs
thf(fact_4811_DERIV__log,axiom,
! [X: real,B3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( log @ B3 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( times_times @ real @ ( ln_ln @ real @ B3 ) @ X ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_log
thf(fact_4812_DERIV__fun__powr,axiom,
! [G2: real > real,M: real,X: real,R2: real] :
( ( has_field_derivative @ real @ G2 @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( powr @ real @ ( G2 @ X2 ) @ R2 )
@ ( times_times @ real @ ( times_times @ real @ R2 @ ( powr @ real @ ( G2 @ X ) @ ( minus_minus @ real @ R2 @ ( semiring_1_of_nat @ real @ ( one_one @ nat ) ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_fun_powr
thf(fact_4813_DERIV__powr,axiom,
! [G2: real > real,M: real,X: real,F3: real > real,R2: real] :
( ( has_field_derivative @ real @ G2 @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X ) )
=> ( ( has_field_derivative @ real @ F3 @ R2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( powr @ real @ ( G2 @ X2 ) @ ( F3 @ X2 ) )
@ ( times_times @ real @ ( powr @ real @ ( G2 @ X ) @ ( F3 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ R2 @ ( ln_ln @ real @ ( G2 @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ M @ ( F3 @ X ) ) @ ( G2 @ X ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_powr
thf(fact_4814_DERIV__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( tan @ A ) @ ( inverse_inverse @ A @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_tan
thf(fact_4815_DERIV__real__sqrt,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ sqrt @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_real_sqrt
thf(fact_4816_DERIV__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( cot @ A ) @ ( uminus_uminus @ A @ ( inverse_inverse @ A @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_cot
thf(fact_4817_has__field__derivative__tanh,axiom,
! [A15: $tType] :
( ( ( real_Vector_banach @ A15 )
& ( real_V3459762299906320749_field @ A15 ) )
=> ! [G2: A15 > A15,X: A15,Db: A15,S3: set @ A15] :
( ( ( cosh @ A15 @ ( G2 @ X ) )
!= ( zero_zero @ A15 ) )
=> ( ( has_field_derivative @ A15 @ G2 @ Db @ ( topolo174197925503356063within @ A15 @ X @ S3 ) )
=> ( has_field_derivative @ A15
@ ^ [X2: A15] : ( tanh @ A15 @ ( G2 @ X2 ) )
@ ( times_times @ A15 @ ( minus_minus @ A15 @ ( one_one @ A15 ) @ ( power_power @ A15 @ ( tanh @ A15 @ ( G2 @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A15 @ X @ S3 ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_4818_DERIV__real__sqrt__generic,axiom,
! [X: real,D3: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D3
= ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D3
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( has_field_derivative @ real @ sqrt @ D3 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_4819_arcosh__real__has__field__derivative,axiom,
! [X: real,A4: set @ real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( has_field_derivative @ real @ ( arcosh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A4 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_4820_artanh__real__has__field__derivative,axiom,
! [X: real,A4: set @ real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ ( artanh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A4 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_4821_DERIV__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_real_root
thf(fact_4822_DERIV__arccos,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arccos @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arccos
thf(fact_4823_DERIV__arcsin,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arcsin @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arcsin
thf(fact_4824_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F3: real > real,X: real,N: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
& ! [M4: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T6: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_4825_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F3: real > real,X: real,N: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T6: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_4826_DERIV__odd__real__root,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_4827_Maclaurin__minus,axiom,
! [H: real,N: nat,Diff: nat > real > real,F3: real > real] :
( ( ord_less @ real @ H @ ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T6: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ H @ T6 )
& ( ord_less_eq @ real @ T6 @ ( zero_zero @ real ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo174197925503356063within @ real @ T6 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T6: real] :
( ( ord_less @ real @ H @ T6 )
& ( ord_less @ real @ T6 @ ( zero_zero @ real ) )
& ( ( F3 @ H )
= ( 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 @ H @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_4828_Maclaurin2,axiom,
! [H: real,Diff: nat > real > real,F3: real > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T6: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less_eq @ real @ T6 @ H ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo174197925503356063within @ real @ T6 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less_eq @ real @ T6 @ H )
& ( ( F3 @ H )
= ( 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 @ H @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_4829_Maclaurin,axiom,
! [H: real,N: nat,Diff: nat > real > real,F3: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T6: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less_eq @ real @ T6 @ H ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo174197925503356063within @ real @ T6 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less @ real @ T6 @ H )
& ( ( F3 @ H )
= ( 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 @ H @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_4830_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F3: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ! [M4: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T6 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_4831_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F3: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T6: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo174197925503356063within @ real @ T6 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T6: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_4832_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A3: real,B3: real,C3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T6: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ A3 @ T6 )
& ( ord_less_eq @ real @ T6 @ B3 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo174197925503356063within @ real @ T6 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less @ real @ A3 @ C3 )
=> ( ( ord_less_eq @ real @ C3 @ B3 )
=> ? [T6: real] :
( ( ord_less @ real @ A3 @ T6 )
& ( ord_less @ real @ T6 @ C3 )
& ( ( F3 @ A3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ C3 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A3 @ C3 ) @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ A3 @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_4833_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A3: real,B3: real,C3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T6: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ A3 @ T6 )
& ( ord_less_eq @ real @ T6 @ B3 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo174197925503356063within @ real @ T6 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ C3 )
=> ( ( ord_less @ real @ C3 @ B3 )
=> ? [T6: real] :
( ( ord_less @ real @ C3 @ T6 )
& ( ord_less @ real @ T6 @ B3 )
& ( ( F3 @ B3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ C3 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B3 @ C3 ) @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ B3 @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_4834_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A3: real,B3: real,C3: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T6: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ A3 @ T6 )
& ( ord_less_eq @ real @ T6 @ B3 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo174197925503356063within @ real @ T6 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ C3 )
=> ( ( ord_less_eq @ real @ C3 @ B3 )
=> ( ( ord_less_eq @ real @ A3 @ X )
=> ( ( ord_less_eq @ real @ X @ B3 )
=> ( ( X != C3 )
=> ? [T6: real] :
( ( ( ord_less @ real @ X @ C3 )
=> ( ( ord_less @ real @ X @ T6 )
& ( ord_less @ real @ T6 @ C3 ) ) )
& ( ~ ( ord_less @ real @ X @ C3 )
=> ( ( ord_less @ real @ C3 @ T6 )
& ( ord_less @ real @ T6 @ X ) ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ C3 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C3 ) @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_4835_Maclaurin__lemma2,axiom,
! [N: nat,H: real,Diff: nat > real > real,K: nat,B2: real] :
( ! [M4: nat,T6: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less_eq @ real @ T6 @ H ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo174197925503356063within @ real @ T6 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( N
= ( suc @ K ) )
=> ! [M3: nat,T8: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ H ) )
=> ( has_field_derivative @ real
@ ^ [U2: real] :
( minus_minus @ real @ ( Diff @ M3 @ U2 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M3 @ P5 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ real @ U2 @ P5 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ M3 ) ) )
@ ( times_times @ real @ B2 @ ( divide_divide @ real @ ( power_power @ real @ U2 @ ( minus_minus @ nat @ N @ M3 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ M3 ) ) ) ) ) )
@ ( minus_minus @ real @ ( Diff @ ( suc @ M3 ) @ T8 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M3 ) @ P5 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ real @ T8 @ P5 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) )
@ ( times_times @ real @ B2 @ ( divide_divide @ real @ ( power_power @ real @ T8 @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_4836_DERIV__power__series_H,axiom,
! [R: real,F3: nat > real,X0: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( summable @ real
@ ^ [N2: nat] : ( times_times @ real @ ( times_times @ real @ ( F3 @ N2 ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( has_field_derivative @ real
@ ^ [X2: real] :
( suminf @ real
@ ^ [N2: nat] : ( times_times @ real @ ( F3 @ N2 ) @ ( power_power @ real @ X2 @ ( suc @ N2 ) ) ) )
@ ( suminf @ real
@ ^ [N2: nat] : ( times_times @ real @ ( times_times @ real @ ( F3 @ N2 ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) @ ( power_power @ real @ X0 @ N2 ) ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_4837_has__derivative__arcsin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: A > real,X: A,G6: A > real,S3: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G2 @ X ) )
=> ( ( ord_less @ real @ ( G2 @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G2 @ G6 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arcsin @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G6 @ X2 ) @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G2 @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ) ).
% has_derivative_arcsin
thf(fact_4838_has__derivative__arccos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: A > real,X: A,G6: A > real,S3: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G2 @ X ) )
=> ( ( ord_less @ real @ ( G2 @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G2 @ G6 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arccos @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G6 @ X2 ) @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G2 @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ) ).
% has_derivative_arccos
thf(fact_4839_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_4840_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_4841_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_4842_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_4843_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% infinite_Ioo_iff
thf(fact_4844_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanLessThan
thf(fact_4845_cSup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_greaterThanLessThan
thf(fact_4846_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanLessThan
thf(fact_4847_cInf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_greaterThanLessThan
thf(fact_4848_has__derivative__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [C3: B,F4: filter @ A] :
( has_derivative @ A @ B
@ ^ [X2: A] : C3
@ ^ [X2: A] : ( zero_zero @ B )
@ F4 ) ) ).
% has_derivative_const
thf(fact_4849_has__derivative__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F10: A > B,X: A,S3: set @ A,T2: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S3 )
=> ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% has_derivative_subset
thf(fact_4850_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) ) ) ) ).
% infinite_Ioo
thf(fact_4851_has__derivative__zero__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: A > B,X: A] :
( ( has_derivative @ A @ B
@ ^ [X2: A] : ( zero_zero @ B )
@ F4
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( F4
= ( ^ [H2: A] : ( zero_zero @ B ) ) ) ) ) ).
% has_derivative_zero_unique
thf(fact_4852_has__derivative__in__compose2,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [T2: set @ A,G2: A > B,G6: A > A > B,F3: C > A,S3: set @ C,X: C,F10: C > A] :
( ! [X3: A] :
( ( member @ A @ X3 @ T2 )
=> ( has_derivative @ A @ B @ G2 @ ( G6 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ F3 @ S3 ) @ T2 )
=> ( ( member @ C @ X @ S3 )
=> ( ( has_derivative @ C @ A @ F3 @ F10 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( has_derivative @ C @ B
@ ^ [X2: C] : ( G2 @ ( F3 @ X2 ) )
@ ^ [Y3: C] : ( G6 @ ( F3 @ X ) @ ( F10 @ Y3 ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ) ) ).
% has_derivative_in_compose2
thf(fact_4853_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) @ ( set_or5935395276787703475ssThan @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_4854_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_4855_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_4856_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_4857_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_4858_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_4859_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_4860_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_4861_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_4862_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_4863_has__derivative__divide_H,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,F10: C > A,X: C,S: set @ C,G2: C > A,G6: C > A] :
( ( has_derivative @ C @ A @ F3 @ F10 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( ( has_derivative @ C @ A @ G2 @ G6 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( ( ( G2 @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ^ [H2: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F10 @ H2 ) @ ( G2 @ X ) ) @ ( times_times @ A @ ( F3 @ X ) @ ( G6 @ H2 ) ) ) @ ( times_times @ A @ ( G2 @ X ) @ ( G2 @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_4864_has__derivative__inverse,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: C > A,X: C,F10: C > A,S: set @ C] :
( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F3 @ F10 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( inverse_inverse @ A @ ( F3 @ X2 ) )
@ ^ [H2: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F3 @ X ) ) @ ( F10 @ H2 ) ) @ ( inverse_inverse @ A @ ( F3 @ X ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ).
% has_derivative_inverse
thf(fact_4865_has__derivative__inverse_H,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,S: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A @ ( inverse_inverse @ A )
@ ^ [H2: A] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ X ) @ H2 ) @ ( inverse_inverse @ A @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_inverse'
thf(fact_4866_DERIV__isconst3,axiom,
! [A3: real,B3: real,X: real,Y: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( member @ real @ X @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( ( member @ real @ Y @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( has_field_derivative @ real @ F3 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( F3 @ X )
= ( F3 @ Y ) ) ) ) ) ) ).
% DERIV_isconst3
thf(fact_4867_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_4868_ivl__disj__un__singleton_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_4869_has__derivative__ln,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: A > real,X: A,G6: A > real,S3: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X ) )
=> ( ( has_derivative @ A @ real @ G2 @ G6 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( ln_ln @ real @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G6 @ X2 ) @ ( inverse_inverse @ real @ ( G2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ).
% has_derivative_ln
thf(fact_4870_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: C > A,F10: C > A,X: C,S: set @ C,G2: C > A,G6: C > A] :
( ( has_derivative @ C @ A @ F3 @ F10 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( ( has_derivative @ C @ A @ G2 @ G6 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( ( ( G2 @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ^ [H2: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F3 @ X ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G2 @ X ) ) @ ( G6 @ H2 ) ) @ ( inverse_inverse @ A @ ( G2 @ X ) ) ) ) @ ( divide_divide @ A @ ( F10 @ H2 ) @ ( G2 @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_4871_has__derivative__prod,axiom,
! [B: $tType,I6: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [I5: set @ I6,F3: I6 > A > B,F10: I6 > A > B,X: A,S: set @ A] :
( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( has_derivative @ A @ B @ ( F3 @ I2 ) @ ( F10 @ I2 ) @ ( topolo174197925503356063within @ A @ X @ S ) ) )
=> ( has_derivative @ A @ B
@ ^ [X2: A] :
( groups7121269368397514597t_prod @ I6 @ B
@ ^ [I4: I6] : ( F3 @ I4 @ X2 )
@ I5 )
@ ^ [Y3: A] :
( groups7311177749621191930dd_sum @ I6 @ B
@ ^ [I4: I6] :
( times_times @ B @ ( F10 @ I4 @ Y3 )
@ ( groups7121269368397514597t_prod @ I6 @ B
@ ^ [J3: I6] : ( F3 @ J3 @ X )
@ ( minus_minus @ ( set @ I6 ) @ I5 @ ( insert2 @ I6 @ I4 @ ( bot_bot @ ( set @ I6 ) ) ) ) ) )
@ I5 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_prod
thf(fact_4872_has__derivative__powr,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: A > real,G6: A > real,X: A,X4: set @ A,F3: A > real,F10: A > real] :
( ( has_derivative @ A @ real @ G2 @ G6 @ ( topolo174197925503356063within @ A @ X @ X4 ) )
=> ( ( has_derivative @ A @ real @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ X4 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X ) )
=> ( ( member @ A @ X @ X4 )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( G2 @ X2 ) @ ( F3 @ X2 ) )
@ ^ [H2: A] : ( times_times @ real @ ( powr @ real @ ( G2 @ X ) @ ( F3 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( F10 @ H2 ) @ ( ln_ln @ real @ ( G2 @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G6 @ H2 ) @ ( F3 @ X ) ) @ ( G2 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ X4 ) ) ) ) ) ) ) ).
% has_derivative_powr
thf(fact_4873_has__derivative__real__sqrt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: A > real,X: A,G6: A > real,S3: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X ) )
=> ( ( has_derivative @ A @ real @ G2 @ G6 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( sqrt @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G6 @ X2 ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G2 @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ).
% has_derivative_real_sqrt
thf(fact_4874_DERIV__series_H,axiom,
! [F3: real > nat > real,F10: real > nat > real,X0: real,A3: real,B3: real,L5: nat > real] :
( ! [N3: nat] :
( has_field_derivative @ real
@ ^ [X2: real] : ( F3 @ X2 @ N3 )
@ ( F10 @ X0 @ N3 )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( summable @ real @ ( F3 @ X3 ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( ( summable @ real @ ( F10 @ X0 ) )
=> ( ( summable @ real @ L5 )
=> ( ! [N3: nat,X3: real,Y2: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( ( member @ real @ Y2 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( F3 @ X3 @ N3 ) @ ( F3 @ Y2 @ N3 ) ) ) @ ( times_times @ real @ ( L5 @ N3 ) @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y2 ) ) ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( suminf @ real @ ( F3 @ X2 ) )
@ ( suminf @ real @ ( F10 @ X0 ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_4875_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K4: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C3 ) @ N2 ) @ ( power_power @ A @ K4 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K4 ) )
=> ( filterlim @ A @ A
@ ^ [H2: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X @ H2 ) @ N2 ) @ ( power_power @ A @ X @ N2 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_aux
thf(fact_4876_isCont__powser,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K4: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ K4 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K4 ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ X2 @ N2 ) ) ) ) ) ) ) ).
% isCont_powser
thf(fact_4877_isCont__powser_H,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ Aa )
& ( real_V3459762299906320749_field @ Aa ) )
=> ! [A3: A,F3: A > Aa,C3: nat > Aa,K4: Aa] :
( ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( summable @ Aa
@ ^ [N2: nat] : ( times_times @ Aa @ ( C3 @ N2 ) @ ( power_power @ Aa @ K4 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ Aa @ ( F3 @ A3 ) ) @ ( real_V7770717601297561774m_norm @ Aa @ K4 ) )
=> ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ Aa
@ ^ [N2: nat] : ( times_times @ Aa @ ( C3 @ N2 ) @ ( power_power @ Aa @ ( F3 @ X2 ) @ N2 ) ) ) ) ) ) ) ) ).
% isCont_powser'
thf(fact_4878_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_4879_tendsto__mult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,F3: B > A,L: A,F4: filter @ B] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C3 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C3 @ L ) )
@ F4 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% tendsto_mult_left_iff
thf(fact_4880_tendsto__mult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,F3: B > A,L: A,F4: filter @ B] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F3 @ X2 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C3 ) )
@ F4 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% tendsto_mult_right_iff
thf(fact_4881_power__tendsto__0__iff,axiom,
! [A: $tType,N: nat,F3: A > real,F4: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real
@ ^ [X2: A] : ( power_power @ real @ ( F3 @ X2 ) @ N )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% power_tendsto_0_iff
thf(fact_4882_isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F3 )
= ( filterlim @ A @ B
@ ^ [H2: A] : ( F3 @ ( plus_plus @ A @ X @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ X ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% isCont_iff
thf(fact_4883_LIM__not__zero,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( zero @ Aa )
& ( topological_t2_space @ Aa ) )
=> ! [K: Aa,A3: A] :
( ( K
!= ( zero_zero @ Aa ) )
=> ~ ( filterlim @ A @ Aa
@ ^ [X2: A] : K
@ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_not_zero
thf(fact_4884_filterlim__sup,axiom,
! [B: $tType,A: $tType,F3: A > B,F4: filter @ B,F1: filter @ A,F22: filter @ A] :
( ( filterlim @ A @ B @ F3 @ F4 @ F1 )
=> ( ( filterlim @ A @ B @ F3 @ F4 @ F22 )
=> ( filterlim @ A @ B @ F3 @ F4 @ ( sup_sup @ ( filter @ A ) @ F1 @ F22 ) ) ) ) ).
% filterlim_sup
thf(fact_4885_tendsto__arcosh,axiom,
! [B: $tType,F3: B > real,A3: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( arcosh @ real @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A3 ) )
@ F4 ) ) ) ).
% tendsto_arcosh
thf(fact_4886_filterlim__top,axiom,
! [B: $tType,A: $tType,F3: A > B,F4: filter @ A] : ( filterlim @ A @ B @ F3 @ ( top_top @ ( filter @ B ) ) @ F4 ) ).
% filterlim_top
thf(fact_4887_filterlim__mono,axiom,
! [B: $tType,A: $tType,F3: A > B,F22: filter @ B,F1: filter @ A,F23: filter @ B,F12: filter @ A] :
( ( filterlim @ A @ B @ F3 @ F22 @ F1 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F22 @ F23 )
=> ( ( ord_less_eq @ ( filter @ A ) @ F12 @ F1 )
=> ( filterlim @ A @ B @ F3 @ F23 @ F12 ) ) ) ) ).
% filterlim_mono
thf(fact_4888_tendsto__mono,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F4: filter @ B,F11: filter @ B,F3: B > A,L: A] :
( ( ord_less_eq @ ( filter @ B ) @ F4 @ F11 )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F11 )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% tendsto_mono
thf(fact_4889_filterlim__inf,axiom,
! [B: $tType,A: $tType,F3: A > B,F22: filter @ B,F32: filter @ B,F1: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( inf_inf @ ( filter @ B ) @ F22 @ F32 ) @ F1 )
= ( ( filterlim @ A @ B @ F3 @ F22 @ F1 )
& ( filterlim @ A @ B @ F3 @ F32 @ F1 ) ) ) ).
% filterlim_inf
thf(fact_4890_tendsto__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: A > A,A3: A,F4: filter @ A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( ( cos @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( tan @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tan @ A @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_tan
thf(fact_4891_filterlim__ident,axiom,
! [A: $tType,F4: filter @ A] :
( filterlim @ A @ A
@ ^ [X2: A] : X2
@ F4
@ F4 ) ).
% filterlim_ident
thf(fact_4892_filterlim__compose,axiom,
! [B: $tType,A: $tType,C: $tType,G2: A > B,F32: filter @ B,F22: filter @ A,F3: C > A,F1: filter @ C] :
( ( filterlim @ A @ B @ G2 @ F32 @ F22 )
=> ( ( filterlim @ C @ A @ F3 @ F22 @ F1 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( G2 @ ( F3 @ X2 ) )
@ F32
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_4893_tendsto__norm__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_norm_zero
thf(fact_4894_tendsto__norm__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 ) ) ) ).
% tendsto_norm_zero_iff
thf(fact_4895_tendsto__norm__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 ) ) ) ).
% tendsto_norm_zero_cancel
thf(fact_4896_tendsto__divide__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,F4: filter @ B,C3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F3 @ X2 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_divide_zero
thf(fact_4897_tendsto__divide,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,A3: A,F4: filter @ B,G2: B > A,B3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ B3 ) @ F4 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A3 @ B3 ) )
@ F4 ) ) ) ) ) ).
% tendsto_divide
thf(fact_4898_tendsto__sgn,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,L: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( L
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( sgn_sgn @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L ) )
@ F4 ) ) ) ) ).
% tendsto_sgn
thf(fact_4899_tendsto__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > A,A3: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( inverse_inverse @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_inverse
thf(fact_4900_tendsto__mult__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F4: filter @ D,G2: D > A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( ( filterlim @ D @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D @ A
@ ^ [X2: D] : ( times_times @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_mult_zero
thf(fact_4901_tendsto__mult__left__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F4: filter @ D,C3: A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D @ A
@ ^ [X2: D] : ( times_times @ A @ ( F3 @ X2 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_mult_left_zero
thf(fact_4902_tendsto__mult__right__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F4: filter @ D,C3: A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D @ A
@ ^ [X2: D] : ( times_times @ A @ C3 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_mult_right_zero
thf(fact_4903_tendsto__add__zero,axiom,
! [B: $tType,D: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F3: D > B,F4: filter @ D,G2: D > B] :
( ( filterlim @ D @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( filterlim @ D @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ D @ B
@ ^ [X2: D] : ( plus_plus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% tendsto_add_zero
thf(fact_4904_tendsto__null__sum,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I5: set @ B,F3: A > B > C,F4: filter @ A] :
( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( F3 @ X2 @ I2 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) )
=> ( filterlim @ A @ C
@ ^ [I4: A] : ( groups7311177749621191930dd_sum @ B @ C @ ( F3 @ I4 ) @ I5 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) ) ) ).
% tendsto_null_sum
thf(fact_4905_Lim__transform__eq,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,G2: B > A,F4: filter @ B,A3: A] :
( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
= ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 ) ) ) ) ).
% Lim_transform_eq
thf(fact_4906_LIM__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F3 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ).
% LIM_zero_cancel
thf(fact_4907_Lim__transform2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,A3: A,F4: filter @ B,G2: B > A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 ) ) ) ) ).
% Lim_transform2
thf(fact_4908_Lim__transform,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: B > A,A3: A,F4: filter @ B,F3: B > A] :
( ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 ) ) ) ) ).
% Lim_transform
thf(fact_4909_LIM__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F3 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 )
= ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ).
% LIM_zero_iff
thf(fact_4910_LIM__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F3 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ).
% LIM_zero
thf(fact_4911_tendsto__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,A3: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( ( cosh @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( tanh @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tanh @ A @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_tanh
thf(fact_4912_tendsto__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: A > A,A3: A,F4: filter @ A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( ( sin @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( cot @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cot @ A @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_cot
thf(fact_4913_real__LIM__sandwich__zero,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: A > real,A3: A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ! [X3: A] :
( ( X3 != A3 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G2 @ X3 ) ) )
=> ( ! [X3: A] :
( ( X3 != A3 )
=> ( ord_less_eq @ real @ ( G2 @ X3 ) @ ( F3 @ X3 ) ) )
=> ( filterlim @ A @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% real_LIM_sandwich_zero
thf(fact_4914_isCont__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A3: A,F3: A > B,G2: B > C,L: C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( filterlim @ B @ C @ G2 @ ( topolo7230453075368039082e_nhds @ C @ L ) @ ( topolo174197925503356063within @ B @ ( F3 @ A3 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A3 ) ) @ D4 ) )
=> ( ( F3 @ X3 )
!= ( F3 @ A3 ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% isCont_LIM_compose2
thf(fact_4915_tendsto__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: A > B,L: filter @ B,X: A,S: set @ A,T4: set @ A] :
( ( filterlim @ A @ B @ F3 @ L @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T4 @ S )
=> ( filterlim @ A @ B @ F3 @ L @ ( topolo174197925503356063within @ A @ X @ T4 ) ) ) ) ) ).
% tendsto_within_subset
thf(fact_4916_LIM__isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,A3: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [H2: A] : ( F3 @ ( plus_plus @ A @ A3 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A3 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_isCont_iff
thf(fact_4917_LIM__offset__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,L5: B,A3: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H2: A] : ( F3 @ ( plus_plus @ A @ A3 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero
thf(fact_4918_LIM__offset__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,A3: A,L5: B] :
( ( filterlim @ A @ B
@ ^ [H2: A] : ( F3 @ ( plus_plus @ A @ A3 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero_cancel
thf(fact_4919_tendsto__null__power,axiom,
! [B: $tType,A: $tType] :
( ( real_V2822296259951069270ebra_1 @ B )
=> ! [F3: A > B,F4: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F3 @ X2 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% tendsto_null_power
thf(fact_4920_tendsto__log,axiom,
! [A: $tType,F3: A > real,A3: real,F4: filter @ A,G2: A > real,B3: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( filterlim @ A @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( log @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( log @ A3 @ B3 ) )
@ F4 ) ) ) ) ) ) ).
% tendsto_log
thf(fact_4921_tendsto__artanh,axiom,
! [A: $tType,F3: A > real,A3: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ A3 )
=> ( ( ord_less @ real @ A3 @ ( one_one @ real ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( artanh @ real @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( artanh @ real @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_artanh
thf(fact_4922_LIM__imp__LIM,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,L: B,A3: A,G2: A > C,M: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ! [X3: A] :
( ( X3 != A3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( minus_minus @ C @ ( G2 @ X3 ) @ M ) ) @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F3 @ X3 ) @ L ) ) ) )
=> ( filterlim @ A @ C @ G2 @ ( topolo7230453075368039082e_nhds @ C @ M ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_imp_LIM
thf(fact_4923_LIM__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A3: A,F3: A > D,L5: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A3 )
=> ( ( filterlim @ A @ D @ F3 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ D
@ ^ [H2: A] : ( F3 @ ( plus_plus @ A @ A3 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ D @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_offset_zero_iff
thf(fact_4924_filterlim__INF_H,axiom,
! [C: $tType,B: $tType,A: $tType,X: A,A4: set @ A,F3: B > C,F4: filter @ C,G3: A > ( filter @ B )] :
( ( member @ A @ X @ A4 )
=> ( ( filterlim @ B @ C @ F3 @ F4 @ ( G3 @ X ) )
=> ( filterlim @ B @ C @ F3 @ F4 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ G3 @ A4 ) ) ) ) ) ).
% filterlim_INF'
thf(fact_4925_filterlim__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,G3: C > ( filter @ B ),B2: set @ C,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ G3 @ B2 ) ) @ F4 )
= ( ! [X2: C] :
( ( member @ C @ X2 @ B2 )
=> ( filterlim @ A @ B @ F3 @ ( G3 @ X2 ) @ F4 ) ) ) ) ).
% filterlim_INF
thf(fact_4926_IVT,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F3: A > B,A3: A,Y: B,B3: A] :
( ( ord_less_eq @ B @ ( F3 @ A3 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F3 @ B3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ! [X3: A] :
( ( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less_eq @ A @ X3 @ B3 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F3 ) )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less_eq @ A @ X3 @ B3 )
& ( ( F3 @ X3 )
= Y ) ) ) ) ) ) ) ).
% IVT
thf(fact_4927_IVT2,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F3: A > B,B3: A,Y: B,A3: A] :
( ( ord_less_eq @ B @ ( F3 @ B3 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F3 @ A3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ! [X3: A] :
( ( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less_eq @ A @ X3 @ B3 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F3 ) )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less_eq @ A @ X3 @ B3 )
& ( ( F3 @ X3 )
= Y ) ) ) ) ) ) ) ).
% IVT2
thf(fact_4928_LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,L5: B,A3: A,R2: real] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [S4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S4 )
& ! [X5: A] :
( ( ( X5 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X5 @ A3 ) ) @ S4 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F3 @ X5 ) @ L5 ) ) @ R2 ) ) ) ) ) ) ).
% LIM_D
thf(fact_4929_LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F3: A > B,L5: B] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [S9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S9 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A3 ) ) @ S9 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F3 @ X3 ) @ L5 ) ) @ R3 ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_I
thf(fact_4930_LIM__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,L5: B,A3: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S8 )
& ! [X2: A] :
( ( ( X2 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ A3 ) ) @ S8 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F3 @ X2 ) @ L5 ) ) @ R5 ) ) ) ) ) ) ) ).
% LIM_eq
thf(fact_4931_LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [R: real,A3: A,F3: A > B,G2: A > B,L: B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( ! [X3: A] :
( ( X3 != A3 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A3 ) ) @ R )
=> ( ( F3 @ X3 )
= ( G2 @ X3 ) ) ) )
=> ( ( filterlim @ A @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_equal2
thf(fact_4932_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > A,A3: A,D3: A] :
( ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ A3 @ H2 ) ) @ ( F3 @ A3 ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D3 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ X2 ) @ ( F3 @ A3 ) ) @ ( minus_minus @ A @ X2 @ A3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D3 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_4933_isCont__Lb__Ub,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [L6: real,M9: real] :
( ! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B3 ) )
=> ( ( ord_less_eq @ real @ L6 @ ( F3 @ X5 ) )
& ( ord_less_eq @ real @ ( F3 @ X5 ) @ M9 ) ) )
& ! [Y5: real] :
( ( ( ord_less_eq @ real @ L6 @ Y5 )
& ( ord_less_eq @ real @ Y5 @ M9 ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 )
& ( ( F3 @ X3 )
= Y5 ) ) ) ) ) ) ).
% isCont_Lb_Ub
thf(fact_4934_LIM__fun__gt__zero,axiom,
! [F3: real > real,L: real,C3: real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X5: real] :
( ( ( X5 != C3 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C3 @ X5 ) ) @ R3 ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X5 ) ) ) ) ) ) ).
% LIM_fun_gt_zero
thf(fact_4935_LIM__fun__not__zero,axiom,
! [F3: real > real,L: real,C3: real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( L
!= ( zero_zero @ real ) )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X5: real] :
( ( ( X5 != C3 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C3 @ X5 ) ) @ R3 ) )
=> ( ( F3 @ X5 )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_not_zero
thf(fact_4936_LIM__fun__less__zero,axiom,
! [F3: real > real,L: real,C3: real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X5: real] :
( ( ( X5 != C3 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C3 @ X5 ) ) @ R3 ) )
=> ( ord_less @ real @ ( F3 @ X5 ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_less_zero
thf(fact_4937_LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,B3: B,A3: A,G2: B > C,C3: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ B3 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G2 @ ( topolo7230453075368039082e_nhds @ C @ C3 ) @ ( topolo174197925503356063within @ B @ B3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A3 ) ) @ D4 ) )
=> ( ( F3 @ X3 )
!= B3 ) ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C3 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose2
thf(fact_4938_continuous__at__within__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,S3: set @ A,F3: A > B,G2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S3 ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S3 ) @ G2 )
=> ( ( ( G2 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S3 )
@ ^ [X2: A] : ( divide_divide @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_4939_continuous__at__within__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A3: A,S3: set @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S3 ) @ F3 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S3 )
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_4940_continuous__at__within__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,S3: set @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S3 ) @ F3 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S3 )
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_at_within_sgn
thf(fact_4941_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H2 ) ) @ ( F3 @ X ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D3 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_4942_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H2 ) ) @ ( F3 @ X ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D3 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_4943_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_4944_lemma__termdiff4,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [K: real,F3: A > B,K4: 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 @ ( F3 @ H6 ) ) @ ( times_times @ real @ K4 @ ( real_V7770717601297561774m_norm @ A @ H6 ) ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% lemma_termdiff4
thf(fact_4945_isCont__eq__Lb,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B3: real,F3: real > A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M9: A] :
( ! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B3 ) )
=> ( ord_less_eq @ A @ M9 @ ( F3 @ X5 ) ) )
& ? [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 )
& ( ( F3 @ X3 )
= M9 ) ) ) ) ) ) ).
% isCont_eq_Lb
thf(fact_4946_isCont__eq__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B3: real,F3: real > A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M9: A] :
( ! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B3 ) )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ M9 ) )
& ? [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 )
& ( ( F3 @ X3 )
= M9 ) ) ) ) ) ) ).
% isCont_eq_Ub
thf(fact_4947_isCont__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B3: real,F3: real > A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M9: A] :
! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B3 ) )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ M9 ) ) ) ) ) ).
% isCont_bounded
thf(fact_4948_isCont__inverse__function2,axiom,
! [A3: real,X: real,B3: real,G2: real > real,F3: real > real] :
( ( ord_less @ real @ A3 @ X )
=> ( ( ord_less @ real @ X @ B3 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A3 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B3 )
=> ( ( G2 @ ( F3 @ Z3 ) )
= Z3 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A3 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B3 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F3 @ X ) @ ( top_top @ ( set @ real ) ) ) @ G2 ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_4949_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A] :
( ( has_derivative @ A @ A @ F3 @ ( times_times @ A @ D3 ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H2 ) ) @ ( F3 @ X ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D3 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_4950_isCont__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,F3: A > B,G2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G2 )
=> ( ( ( G2 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( divide_divide @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% isCont_divide
thf(fact_4951_isCont__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F3 @ X2 ) ) ) ) ) ) ).
% isCont_sgn
thf(fact_4952_filterlim__at__to__0,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: A > B,F4: filter @ B,A3: A] :
( ( filterlim @ A @ B @ F3 @ F4 @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [X2: A] : ( F3 @ ( plus_plus @ A @ X2 @ A3 ) )
@ F4
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_4953_continuous__within__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S3: set @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S3 ) @ F3 )
=> ( ( ( cos @ A @ ( F3 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S3 )
@ ^ [X2: A] : ( tan @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_within_tan
thf(fact_4954_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ ( suc @ I ) ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_4955_continuous__within__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S3: set @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S3 ) @ F3 )
=> ( ( ( sin @ A @ ( F3 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S3 )
@ ^ [X2: A] : ( cot @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_within_cot
thf(fact_4956_continuous__at__within__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: C,A4: set @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A4 ) @ F3 )
=> ( ( ( cosh @ A @ ( F3 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A4 )
@ ^ [X2: C] : ( tanh @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_at_within_tanh
thf(fact_4957_isCont__has__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B3: real,F3: real > A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M9: A] :
( ! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B3 ) )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ M9 ) )
& ! [N8: A] :
( ( ord_less @ A @ N8 @ M9 )
=> ? [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 )
& ( ord_less @ A @ N8 @ ( F3 @ X3 ) ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_4958_isCont__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( tan @ A ) ) ) ) ).
% isCont_tan
thf(fact_4959_isCont__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( cot @ A ) ) ) ) ).
% isCont_cot
thf(fact_4960_isCont__tanh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cosh @ A @ X )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( tanh @ A ) ) ) ) ).
% isCont_tanh
thf(fact_4961_powser__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S3: real,A3: nat > A,F3: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S3 )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S3 )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A3 @ N2 ) @ ( power_power @ A @ X3 @ N2 ) )
@ ( F3 @ X3 ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( A3 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0
thf(fact_4962_powser__limit__0__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S3: real,A3: nat > A,F3: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S3 )
=> ( ! [X3: A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S3 )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A3 @ N2 ) @ ( power_power @ A @ X3 @ N2 ) )
@ ( F3 @ X3 ) ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( A3 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0_strong
thf(fact_4963_lemma__termdiff5,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_Vector_banach @ B ) )
=> ! [K: real,F3: nat > real,G2: A > nat > B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ( summable @ real @ F3 )
=> ( ! [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 @ ( G2 @ H6 @ N3 ) ) @ ( times_times @ real @ ( F3 @ N3 ) @ ( real_V7770717601297561774m_norm @ A @ H6 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H2: A] : ( suminf @ B @ ( G2 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% lemma_termdiff5
thf(fact_4964_isCont__tan_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( cos @ A @ ( F3 @ A3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( tan @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% isCont_tan'
thf(fact_4965_isCont__arcosh,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( arcosh @ real ) ) ) ).
% isCont_arcosh
thf(fact_4966_continuous__at__within__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A3: A,S3: set @ A,F3: A > real,G2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ S3 ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ S3 ) @ G2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ A3 ) )
=> ( ( ( F3 @ A3 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ A3 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ S3 )
@ ^ [X2: A] : ( log @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_at_within_log
thf(fact_4967_isCont__cot_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( sin @ A @ ( F3 @ A3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( cot @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% isCont_cot'
thf(fact_4968_DERIV__inverse__function,axiom,
! [F3: real > real,D3: real,G2: real > real,X: real,A3: real,B3: real] :
( ( has_field_derivative @ real @ F3 @ D3 @ ( topolo174197925503356063within @ real @ ( G2 @ X ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( D3
!= ( zero_zero @ real ) )
=> ( ( ord_less @ real @ A3 @ X )
=> ( ( ord_less @ real @ X @ B3 )
=> ( ! [Y2: real] :
( ( ord_less @ real @ A3 @ Y2 )
=> ( ( ord_less @ real @ Y2 @ B3 )
=> ( ( F3 @ ( G2 @ Y2 ) )
= Y2 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ G2 )
=> ( has_field_derivative @ real @ G2 @ ( inverse_inverse @ real @ D3 ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_inverse_function
thf(fact_4969_isCont__arccos,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ arccos ) ) ) ).
% isCont_arccos
thf(fact_4970_isCont__arcsin,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ arcsin ) ) ) ).
% isCont_arcsin
thf(fact_4971_LIM__less__bound,axiom,
! [B3: real,X: real,F3: real > real] :
( ( ord_less @ real @ B3 @ X )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ B3 @ X ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ F3 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X ) ) ) ) ) ).
% LIM_less_bound
thf(fact_4972_isCont__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A3: A,F3: A > real,G2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ A3 ) )
=> ( ( ( F3 @ A3 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ A3 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( log @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ) ) ).
% isCont_log
thf(fact_4973_isCont__artanh,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( artanh @ real ) ) ) ) ).
% isCont_artanh
thf(fact_4974_isCont__inverse__function,axiom,
! [D2: real,X: real,G2: real > real,F3: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z3 @ X ) ) @ D2 )
=> ( ( G2 @ ( F3 @ Z3 ) )
= Z3 ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z3 @ X ) ) @ D2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F3 @ X ) @ ( top_top @ ( set @ real ) ) ) @ G2 ) ) ) ) ).
% isCont_inverse_function
thf(fact_4975_GMVT_H,axiom,
! [A3: real,B3: real,F3: real > real,G2: real > real,G6: real > real,F10: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A3 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B3 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A3 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B3 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ G2 ) ) )
=> ( ! [Z3: real] :
( ( ord_less @ real @ A3 @ Z3 )
=> ( ( ord_less @ real @ Z3 @ B3 )
=> ( has_field_derivative @ real @ G2 @ ( G6 @ Z3 ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ! [Z3: real] :
( ( ord_less @ real @ A3 @ Z3 )
=> ( ( ord_less @ real @ Z3 @ B3 )
=> ( has_field_derivative @ real @ F3 @ ( F10 @ Z3 ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [C5: real] :
( ( ord_less @ real @ A3 @ C5 )
& ( ord_less @ real @ C5 @ B3 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) ) @ ( G6 @ C5 ) )
= ( times_times @ real @ ( minus_minus @ real @ ( G2 @ B3 ) @ ( G2 @ A3 ) ) @ ( F10 @ C5 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_4976_summable__Leibniz_I3_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( ( ord_less @ real @ ( A3 @ ( zero_zero @ nat ) ) @ ( zero_zero @ real ) )
=> ! [N4: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_4977_summable__Leibniz_I2_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( A3 @ ( zero_zero @ nat ) ) )
=> ! [N4: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_4978_summable__Leibniz_H_I5_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_4979_trivial__limit__sequentially,axiom,
( ( at_top @ nat )
!= ( bot_bot @ ( filter @ nat ) ) ) ).
% trivial_limit_sequentially
thf(fact_4980_tendsto__zero__mult__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,A3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A3 @ N2 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_right_iff
thf(fact_4981_tendsto__zero__mult__left__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,A3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ C3 @ ( A3 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_left_iff
thf(fact_4982_tendsto__zero__divide__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,A3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( A3 @ N2 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_4983_filterlim__sequentially__Suc,axiom,
! [A: $tType,F3: nat > A,F4: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X2: nat] : ( F3 @ ( suc @ X2 ) )
@ F4
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F3 @ F4 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_4984_filterlim__Suc,axiom,
filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ).
% filterlim_Suc
thf(fact_4985_trivial__limit__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_top @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_top_linorder
thf(fact_4986_approx__from__above__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [U4: nat > A] :
( ! [N4: nat] : ( ord_less @ A @ X @ ( U4 @ N4 ) )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_above_dense_linorder
thf(fact_4987_approx__from__below__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ? [U4: nat > A] :
( ! [N4: nat] : ( ord_less @ A @ ( U4 @ N4 ) @ X )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_below_dense_linorder
thf(fact_4988_LIMSEQ__le__const2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X4: nat > A,X: A,A3: A] :
( ( filterlim @ nat @ A @ X4 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ A @ ( X4 @ N3 ) @ A3 ) )
=> ( ord_less_eq @ A @ X @ A3 ) ) ) ) ).
% LIMSEQ_le_const2
thf(fact_4989_LIMSEQ__le__const,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X4: nat > A,X: A,A3: A] :
( ( filterlim @ nat @ A @ X4 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ A @ A3 @ ( X4 @ N3 ) ) )
=> ( ord_less_eq @ A @ A3 @ X ) ) ) ) ).
% LIMSEQ_le_const
thf(fact_4990_Lim__bounded2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: nat > A,L: A,N6: nat,C2: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( ord_less_eq @ A @ C2 @ ( F3 @ N3 ) ) )
=> ( ord_less_eq @ A @ C2 @ L ) ) ) ) ).
% Lim_bounded2
thf(fact_4991_Lim__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: nat > A,L: A,M5: nat,C2: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
=> ( ord_less_eq @ A @ ( F3 @ N3 ) @ C2 ) )
=> ( ord_less_eq @ A @ L @ C2 ) ) ) ) ).
% Lim_bounded
thf(fact_4992_LIMSEQ__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X4: nat > A,X: A,Y6: nat > A,Y: A] :
( ( filterlim @ nat @ A @ X4 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y6 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ A @ ( X4 @ N3 ) @ ( Y6 @ N3 ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ).
% LIMSEQ_le
thf(fact_4993_lim__mono,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [N6: nat,X4: nat > A,Y6: nat > A,X: A,Y: A] :
( ! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( ord_less_eq @ A @ ( X4 @ N3 ) @ ( Y6 @ N3 ) ) )
=> ( ( filterlim @ nat @ A @ X4 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y6 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ).
% lim_mono
thf(fact_4994_Sup__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B3: nat > A,S3: set @ A,A3: A] :
( ! [N3: nat] : ( member @ A @ ( B3 @ N3 ) @ S3 )
=> ( ( filterlim @ nat @ A @ B3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ A3 @ ( complete_Sup_Sup @ A @ S3 ) ) ) ) ) ).
% Sup_lim
thf(fact_4995_Inf__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B3: nat > A,S3: set @ A,A3: A] :
( ! [N3: nat] : ( member @ A @ ( B3 @ N3 ) @ S3 )
=> ( ( filterlim @ nat @ A @ B3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ S3 ) @ A3 ) ) ) ) ).
% Inf_lim
thf(fact_4996_Inf__as__limit,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ? [U4: nat > A] :
( ! [N4: nat] : ( member @ A @ ( U4 @ N4 ) @ A4 )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ ( complete_Inf_Inf @ A @ A4 ) ) @ ( at_top @ nat ) ) ) ) ) ).
% Inf_as_limit
thf(fact_4997_summable__LIMSEQ__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ_zero
thf(fact_4998_mult__nat__right__at__top,axiom,
! [C3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( filterlim @ nat @ nat
@ ^ [X2: nat] : ( times_times @ nat @ X2 @ C3 )
@ ( at_top @ nat )
@ ( at_top @ nat ) ) ) ).
% mult_nat_right_at_top
thf(fact_4999_mult__nat__left__at__top,axiom,
! [C3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( filterlim @ nat @ nat @ ( times_times @ nat @ C3 ) @ ( at_top @ nat ) @ ( at_top @ nat ) ) ) ).
% mult_nat_left_at_top
thf(fact_5000_monoseq__convergent,axiom,
! [X4: nat > real,B2: real] :
( ( topological_monoseq @ real @ X4 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( X4 @ I2 ) ) @ B2 )
=> ~ ! [L6: real] :
~ ( filterlim @ nat @ real @ X4 @ ( topolo7230453075368039082e_nhds @ real @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% monoseq_convergent
thf(fact_5001_monoseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: nat > A,X: A] :
( ( topological_monoseq @ A @ A3 )
=> ( ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ! [N4: nat] : ( ord_less_eq @ A @ ( A3 @ N4 ) @ X )
& ! [M3: nat,N4: nat] :
( ( ord_less_eq @ nat @ M3 @ N4 )
=> ( ord_less_eq @ A @ ( A3 @ M3 ) @ ( A3 @ N4 ) ) ) )
| ( ! [N4: nat] : ( ord_less_eq @ A @ X @ ( A3 @ N4 ) )
& ! [M3: nat,N4: nat] :
( ( ord_less_eq @ nat @ M3 @ N4 )
=> ( ord_less_eq @ A @ ( A3 @ N4 ) @ ( A3 @ M3 ) ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_5002_lim__const__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A] :
( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_const_over_n
thf(fact_5003_lim__inverse__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_inverse_n
thf(fact_5004_LIMSEQ__linear,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X4: nat > A,X: A,L: nat] :
( ( filterlim @ nat @ A @ X4 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( X4 @ ( times_times @ nat @ N2 @ L ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_linear
thf(fact_5005_nested__sequence__unique,axiom,
! [F3: nat > real,G2: nat > real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( G2 @ ( suc @ N3 ) ) @ ( G2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( F3 @ N3 ) @ ( G2 @ N3 ) )
=> ( ( filterlim @ nat @ real
@ ^ [N2: nat] : ( minus_minus @ real @ ( F3 @ N2 ) @ ( G2 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) )
=> ? [L7: real] :
( ! [N4: nat] : ( ord_less_eq @ real @ ( F3 @ N4 ) @ L7 )
& ( filterlim @ nat @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L7 ) @ ( at_top @ nat ) )
& ! [N4: nat] : ( ord_less_eq @ real @ L7 @ ( G2 @ N4 ) )
& ( filterlim @ nat @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ L7 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_5006_LIMSEQ__inverse__zero,axiom,
! [X4: nat > real] :
( ! [R3: real] :
? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less @ real @ R3 @ ( X4 @ N3 ) ) )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( inverse_inverse @ real @ ( X4 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_zero
thf(fact_5007_LIMSEQ__root__const,axiom,
! [C3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( root @ N2 @ C3 )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_root_const
thf(fact_5008_increasing__LIMSEQ,axiom,
! [F3: nat > real,L: real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( F3 @ N3 ) @ L )
=> ( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [N4: nat] : ( ord_less_eq @ real @ L @ ( plus_plus @ real @ ( F3 @ N4 ) @ E ) ) )
=> ( filterlim @ nat @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( at_top @ nat ) ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_5009_lim__1__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_1_over_n
thf(fact_5010_LIMSEQ__realpow__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ X ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_5011_telescope__sums_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C3: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
@ ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ C3 ) ) ) ) ).
% telescope_sums'
thf(fact_5012_telescope__sums,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C3: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
@ ( minus_minus @ A @ C3 @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% telescope_sums
thf(fact_5013_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( divide_divide @ real @ A3 @ ( power_power @ real @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_5014_LIMSEQ__abs__realpow__zero2,axiom,
! [C3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ C3 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ C3 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ).
% LIMSEQ_abs_realpow_zero2
thf(fact_5015_LIMSEQ__abs__realpow__zero,axiom,
! [C3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ C3 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ ( abs_abs @ real @ C3 ) ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ).
% LIMSEQ_abs_realpow_zero
thf(fact_5016_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( inverse_inverse @ real @ ( power_power @ real @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_5017_sums__def_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F2: nat > A,S8: A] :
( filterlim @ nat @ A
@ ^ [N2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S8 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def'
thf(fact_5018_root__test__convergence,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,X: real] :
( ( filterlim @ nat @ real
@ ^ [N2: nat] : ( root @ N2 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ nat ) )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% root_test_convergence
thf(fact_5019_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X4: nat > A,L5: A,R2: real] :
( ( filterlim @ nat @ A @ X4 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ No @ N4 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X4 @ N4 ) @ L5 ) ) @ R2 ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_5020_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X4: nat > A,L5: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No2: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No2 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X4 @ N3 ) @ L5 ) ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X4 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_5021_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X4: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X4 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No3 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X4 @ N2 ) @ L5 ) ) @ R5 ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_5022_LIMSEQ__power__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_power_zero
thf(fact_5023_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F3: B > nat,F4: filter @ B,X: A] :
( ( filterlim @ B @ nat @ F3 @ ( at_top @ nat ) @ F4 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y3: B] : ( power_power @ A @ X @ ( F3 @ Y3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_power_zero
thf(fact_5024_LIMSEQ__norm__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_norm_0
thf(fact_5025_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Df: A,Z: A,S3: nat > A,A3: A] :
( ( has_field_derivative @ A @ F3 @ Df @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ nat @ A @ S3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( S3 @ N3 )
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ Z @ ( S3 @ N2 ) ) ) @ ( F3 @ Z ) ) @ ( S3 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( at_top @ nat ) )
=> ( Df = A3 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_5026_powser__times__n__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_5027_lim__n__over__pown,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_5028_summable,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( summable @ real
@ ^ [N2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) @ ( A3 @ N2 ) ) ) ) ) ) ).
% summable
thf(fact_5029_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% zeroseq_arctan_series
thf(fact_5030_summable__Leibniz_H_I2_J,axiom,
! [A3: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_5031_summable__Leibniz_H_I3_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_5032_sums__alternating__upper__lower,axiom,
! [A3: nat > real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ? [L7: real] :
( ! [N4: nat] :
( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ L7 )
& ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L7 )
@ ( at_top @ nat ) )
& ! [N4: nat] :
( ord_less_eq @ real @ L7
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) ) )
& ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L7 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_5033_summable__Leibniz_H_I4_J,axiom,
! [A3: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( ord_less_eq @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_5034_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F10: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F10 )
& ( filterlim @ A @ B
@ ^ [Y3: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ X ) ) ) @ ( minus_minus @ B @ ( F3 @ Y3 ) @ ( plus_plus @ B @ ( F3 @ X ) @ ( F10 @ ( minus_minus @ A @ Y3 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_5035_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,D3: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ D3 )
& ( filterlim @ A @ real
@ ^ [H2: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ ( plus_plus @ A @ X @ H2 ) ) @ ( F3 @ X ) ) @ ( D3 @ H2 ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at
thf(fact_5036_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F10: A > B,X: A,S3: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F10 )
& ( filterlim @ A @ B
@ ^ [Y3: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ X ) ) ) @ ( minus_minus @ B @ ( F3 @ Y3 ) @ ( plus_plus @ B @ ( F3 @ X ) @ ( F10 @ ( minus_minus @ A @ Y3 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ).
% has_derivative_within
thf(fact_5037_bounded__linear__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] : ( zero_zero @ B ) ) ) ).
% bounded_linear_zero
thf(fact_5038_bounded__linear_Obounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ? [K9: real] :
! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K9 ) ) ) ) ).
% bounded_linear.bounded
thf(fact_5039_bounded__linear_Otendsto__zero,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G2: C > A,F4: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( filterlim @ C @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( F3 @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% bounded_linear.tendsto_zero
thf(fact_5040_bounded__linear_Ononneg__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ? [K9: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ K9 )
& ! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K9 ) ) ) ) ) ).
% bounded_linear.nonneg_bounded
thf(fact_5041_has__derivative__within__singleton__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G2: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ G2 @ ( topolo174197925503356063within @ A @ X @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( real_V3181309239436604168linear @ A @ B @ G2 ) ) ) ).
% has_derivative_within_singleton_iff
thf(fact_5042_filterlim__pow__at__top,axiom,
! [A: $tType,N: nat,F3: A > real,F4: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( power_power @ real @ ( F3 @ X2 ) @ N )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_pow_at_top
thf(fact_5043_bounded__linear_Opos__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ? [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
& ! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K9 ) ) ) ) ) ).
% bounded_linear.pos_bounded
thf(fact_5044_bounded__linear__intro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,K4: real] :
( ! [X3: A,Y2: A] :
( ( F3 @ ( plus_plus @ A @ X3 @ Y2 ) )
= ( plus_plus @ B @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ! [R3: real,X3: A] :
( ( F3 @ ( real_V8093663219630862766scaleR @ A @ R3 @ X3 ) )
= ( real_V8093663219630862766scaleR @ B @ R3 @ ( F3 @ X3 ) ) )
=> ( ! [X3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X3 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K4 ) )
=> ( real_V3181309239436604168linear @ A @ B @ F3 ) ) ) ) ) ).
% bounded_linear_intro
thf(fact_5045_filterlim__tendsto__pos__mult__at__top,axiom,
! [A: $tType,F3: A > real,C3: real,F4: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_top
thf(fact_5046_filterlim__at__top__mult__tendsto__pos,axiom,
! [A: $tType,F3: A > real,C3: real,F4: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( G2 @ X2 ) @ ( F3 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_at_top_mult_tendsto_pos
thf(fact_5047_tendsto__neg__powr,axiom,
! [A: $tType,S3: real,F3: A > real,F4: filter @ A] :
( ( ord_less @ real @ S3 @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F3 @ X2 ) @ S3 )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_neg_powr
thf(fact_5048_DERIV__neg__imp__decreasing__at__top,axiom,
! [B3: real,F3: real > real,Flim: real] :
( ! [X3: real] :
( ( ord_less_eq @ real @ B3 @ X3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_top @ real ) )
=> ( ord_less @ real @ Flim @ ( F3 @ B3 ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_5049_has__derivativeI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F10: A > B,X: A,F3: A > B,S3: set @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F10 )
=> ( ( filterlim @ A @ B
@ ^ [Y3: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y3 ) @ ( F3 @ X ) ) @ ( F10 @ ( minus_minus @ A @ Y3 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ).
% has_derivativeI
thf(fact_5050_has__derivative__at__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F10: A > B,X: A,S3: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F10 )
& ( filterlim @ A @ B
@ ^ [Y3: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y3 ) @ ( F3 @ X ) ) @ ( F10 @ ( minus_minus @ A @ Y3 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ).
% has_derivative_at_within
thf(fact_5051_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F10: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F10 )
& ? [E3: A > B] :
( ! [H2: A] :
( ( F3 @ ( plus_plus @ A @ X @ H2 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F3 @ X ) @ ( F10 @ H2 ) ) @ ( E3 @ H2 ) ) )
& ( filterlim @ A @ real
@ ^ [H2: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E3 @ H2 ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% has_derivative_iff_Ex
thf(fact_5052_has__derivative__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( has_derivative @ A @ B )
= ( ^ [F2: A > B,F13: A > B,F8: filter @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F13 )
& ( filterlim @ A @ B
@ ^ [Y3: A] :
( real_V8093663219630862766scaleR @ B
@ ( inverse_inverse @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( minus_minus @ A @ Y3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X2: A] : X2 ) ) ) )
@ ( minus_minus @ B
@ ( minus_minus @ B @ ( F2 @ Y3 )
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X2: A] : X2 ) ) )
@ ( F13
@ ( minus_minus @ A @ Y3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X2: A] : X2 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F8 ) ) ) ) ) ).
% has_derivative_def
thf(fact_5053_has__derivative__at__within__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X: A,S: set @ A,F3: A > B,F10: A > B] :
( ( member @ A @ X @ S )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F10 )
& ? [E3: A > B] :
( ! [H2: A] :
( ( member @ A @ ( plus_plus @ A @ X @ H2 ) @ S )
=> ( ( F3 @ ( plus_plus @ A @ X @ H2 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F3 @ X ) @ ( F10 @ H2 ) ) @ ( E3 @ H2 ) ) ) )
& ( filterlim @ A @ real
@ ^ [H2: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E3 @ H2 ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% has_derivative_at_within_iff_Ex
thf(fact_5054_has__derivativeI__sandwich,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [E2: real,F10: A > B,S3: set @ A,X: A,F3: A > B,H7: A > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( ( real_V3181309239436604168linear @ A @ B @ F10 )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ S3 )
=> ( ( Y2 != X )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y2 @ X ) @ E2 )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) ) @ ( F10 @ ( minus_minus @ A @ Y2 @ X ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X ) ) ) @ ( H7 @ Y2 ) ) ) ) )
=> ( ( filterlim @ A @ real @ H7 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ) ) ).
% has_derivativeI_sandwich
thf(fact_5055_open__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo1002775350975398744n_open @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% open_empty
thf(fact_5056_open__Un,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,T4: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( topolo1002775350975398744n_open @ A @ T4 )
=> ( topolo1002775350975398744n_open @ A @ ( sup_sup @ ( set @ A ) @ S @ T4 ) ) ) ) ) ).
% open_Un
thf(fact_5057_dist__0__norm,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( real_V557655796197034286t_dist @ A @ ( zero_zero @ A ) @ X )
= ( real_V7770717601297561774m_norm @ A @ X ) ) ) ).
% dist_0_norm
thf(fact_5058_zero__less__dist__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) )
= ( X != Y ) ) ) ).
% zero_less_dist_iff
thf(fact_5059_dist__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) )
= ( X = Y ) ) ) ).
% dist_le_zero_iff
thf(fact_5060_open__INT,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A4: set @ B,B2: B > ( set @ A )] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( topolo1002775350975398744n_open @ A @ ( B2 @ X3 ) ) )
=> ( topolo1002775350975398744n_open @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A4 ) ) ) ) ) ) ).
% open_INT
thf(fact_5061_open__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S7: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S7 )
=> ? [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
& ! [Y3: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y3 @ X2 ) @ E3 )
=> ( member @ A @ Y3 @ S7 ) ) ) ) ) ) ) ).
% open_dist
thf(fact_5062_dist__commute__lessI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y: A,X: A,E2: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X ) @ E2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E2 ) ) ) ).
% dist_commute_lessI
thf(fact_5063_open__ball,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,D2: real] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [Y3: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y3 ) @ D2 ) ) ) ) ).
% open_ball
thf(fact_5064_openI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ? [T9: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T9 )
& ( member @ A @ X3 @ T9 )
& ( ord_less_eq @ ( set @ A ) @ T9 @ S ) ) )
=> ( topolo1002775350975398744n_open @ A @ S ) ) ) ).
% openI
thf(fact_5065_open__subopen,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S7: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S7 )
=> ? [T10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T10 )
& ( member @ A @ X2 @ T10 )
& ( ord_less_eq @ ( set @ A ) @ T10 @ S7 ) ) ) ) ) ) ).
% open_subopen
thf(fact_5066_first__countable__basis,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X: A] :
? [A9: nat > ( set @ A )] :
( ! [I3: nat] :
( ( member @ A @ X @ ( A9 @ I3 ) )
& ( topolo1002775350975398744n_open @ A @ ( A9 @ I3 ) ) )
& ! [S10: set @ A] :
( ( ( topolo1002775350975398744n_open @ A @ S10 )
& ( member @ A @ X @ S10 ) )
=> ? [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( A9 @ I2 ) @ S10 ) ) ) ) ).
% first_countable_basis
thf(fact_5067_norm__conv__dist,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A )
= ( ^ [X2: A] : ( real_V557655796197034286t_dist @ A @ X2 @ ( zero_zero @ A ) ) ) ) ) ).
% norm_conv_dist
thf(fact_5068_dist__pos__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) ) ) ) ).
% dist_pos_lt
thf(fact_5069_dist__not__less__zero,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) ) ) ).
% dist_not_less_zero
thf(fact_5070_zero__le__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) ) ) ).
% zero_le_dist
thf(fact_5071_Sup__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A4: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less @ A @ X3 @ X ) )
=> ~ ( member @ A @ ( complete_Sup_Sup @ A @ A4 ) @ A4 ) ) ) ) ).
% Sup_notin_open
thf(fact_5072_Inf__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A4: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less @ A @ X @ X3 ) )
=> ~ ( member @ A @ ( complete_Inf_Inf @ A @ A4 ) @ A4 ) ) ) ) ).
% Inf_notin_open
thf(fact_5073_not__open__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X: A] :
~ ( topolo1002775350975398744n_open @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% not_open_singleton
thf(fact_5074_dist__triangle__less__add,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X15: A,Y: A,E1: real,X23: A,E22: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ Y ) @ E1 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X23 @ Y ) @ E22 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X23 ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% dist_triangle_less_add
thf(fact_5075_dist__triangle__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z: A,Y: A,E2: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z ) ) @ E2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E2 ) ) ) ).
% dist_triangle_lt
thf(fact_5076_separation__t2,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ? [U5: set @ A,V4: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U5 )
& ( topolo1002775350975398744n_open @ A @ V4 )
& ( member @ A @ X @ U5 )
& ( member @ A @ Y @ V4 )
& ( ( inf_inf @ ( set @ A ) @ U5 @ V4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% separation_t2
thf(fact_5077_hausdorff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ? [U6: set @ A,V5: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U6 )
& ( topolo1002775350975398744n_open @ A @ V5 )
& ( member @ A @ X @ U6 )
& ( member @ A @ Y @ V5 )
& ( ( inf_inf @ ( set @ A ) @ U6 @ V5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% hausdorff
thf(fact_5078_dist__triangle,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z: A,Y: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z ) ) ) ) ).
% dist_triangle
thf(fact_5079_dist__triangle2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A,Z: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z ) ) ) ) ).
% dist_triangle2
thf(fact_5080_dist__triangle3,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A,A3: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ A3 @ X ) @ ( real_V557655796197034286t_dist @ A @ A3 @ Y ) ) ) ) ).
% dist_triangle3
thf(fact_5081_dist__triangle__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z: A,Y: A,E2: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z ) ) @ E2 )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E2 ) ) ) ).
% dist_triangle_le
thf(fact_5082_at__within__open__subset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: A,S: set @ A,T4: set @ A] :
( ( member @ A @ A3 @ S )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ T4 )
=> ( ( topolo174197925503356063within @ A @ A3 @ T4 )
= ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% at_within_open_subset
thf(fact_5083_open__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S: set @ A,X: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( member @ A @ X @ S )
=> ( ( ord_less @ A @ X @ Y )
=> ? [B7: A] :
( ( ord_less @ A @ X @ B7 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ B7 ) @ S ) ) ) ) ) ) ).
% open_right
thf(fact_5084_abs__dist__diff__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [A3: A,B3: A,C3: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V557655796197034286t_dist @ A @ A3 @ B3 ) @ ( real_V557655796197034286t_dist @ A @ B3 @ C3 ) ) ) @ ( real_V557655796197034286t_dist @ A @ A3 @ C3 ) ) ) ).
% abs_dist_diff_le
thf(fact_5085_has__field__derivative__transform__within,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F10: A,A3: A,S: set @ A,D2: real,G2: A > A] :
( ( has_field_derivative @ A @ F3 @ F10 @ ( topolo174197925503356063within @ A @ A3 @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ( member @ A @ A3 @ S )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A3 ) @ D2 )
=> ( ( F3 @ X3 )
= ( G2 @ X3 ) ) ) )
=> ( has_field_derivative @ A @ G2 @ F10 @ ( topolo174197925503356063within @ A @ A3 @ S ) ) ) ) ) ) ) ).
% has_field_derivative_transform_within
thf(fact_5086_has__derivative__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F10: A > B,X: A,S3: set @ A,D2: real,G2: A > B] :
( ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ( member @ A @ X @ S3 )
=> ( ! [X16: A] :
( ( member @ A @ X16 @ S3 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X ) @ D2 )
=> ( ( F3 @ X16 )
= ( G2 @ X16 ) ) ) )
=> ( has_derivative @ A @ B @ G2 @ F10 @ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ) ) ).
% has_derivative_transform_within
thf(fact_5087_Cauchy__def,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M8: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M8 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M8 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_def
thf(fact_5088_Cauchy__altdef2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [S8: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S8 @ N2 ) @ ( S8 @ N5 ) ) @ E3 ) ) ) ) ) ) ).
% Cauchy_altdef2
thf(fact_5089_metric__CauchyD,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X4: nat > A,E2: real] :
( ( topolo3814608138187158403Cauchy @ A @ X4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M9: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M9 @ M3 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ M9 @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X4 @ M3 ) @ ( X4 @ N4 ) ) @ E2 ) ) ) ) ) ) ).
% metric_CauchyD
thf(fact_5090_metric__CauchyI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X4: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M10 @ M4 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M10 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X4 @ M4 ) @ ( X4 @ N3 ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X4 ) ) ) ).
% metric_CauchyI
thf(fact_5091_lim__explicit,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,F0: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ F0 ) @ ( at_top @ nat ) )
= ( ! [S7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S7 )
=> ( ( member @ A @ F0 @ S7 )
=> ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( member @ A @ ( F3 @ N2 ) @ S7 ) ) ) ) ) ) ) ).
% lim_explicit
thf(fact_5092_continuous__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F4: filter @ A,F3: A > B,G2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G2 )
=> ( ( ( G2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( divide_divide @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% continuous_divide
thf(fact_5093_continuous__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F4: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_inverse
thf(fact_5094_continuous__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_sgn
thf(fact_5095_metric__LIM__imp__LIM,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V7819770556892013058_space @ B )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F3: C > A,L: A,A3: C,G2: C > B,M: B] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) )
=> ( ! [X3: C] :
( ( X3 != A3 )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G2 @ X3 ) @ M ) @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X3 ) @ L ) ) )
=> ( filterlim @ C @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ M ) @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) ) ) ) ) ).
% metric_LIM_imp_LIM
thf(fact_5096_Lim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,L: B,X: A,S: set @ A,D2: real,G2: A > B] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [X16: A] :
( ( member @ A @ X16 @ S )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X16 @ X ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X ) @ D2 )
=> ( ( F3 @ X16 )
= ( G2 @ X16 ) ) ) ) )
=> ( filterlim @ A @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% Lim_transform_within
thf(fact_5097_dist__triangle__half__l,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X15: A,Y: A,E2: real,X23: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ Y ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X23 @ Y ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X23 ) @ E2 ) ) ) ) ).
% dist_triangle_half_l
thf(fact_5098_dist__triangle__half__r,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y: A,X15: A,E2: real,X23: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X15 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X23 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X23 ) @ E2 ) ) ) ) ).
% dist_triangle_half_r
thf(fact_5099_dist__triangle__third,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X15: A,X23: A,E2: real,X33: A,X42: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X23 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X23 @ X33 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X33 @ X42 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X42 ) @ E2 ) ) ) ) ) ).
% dist_triangle_third
thf(fact_5100_at__within__nhd,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X: A,S: set @ A,T4: set @ A,U3: set @ A] :
( ( member @ A @ X @ S )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ T4 @ S ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ U3 @ S ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( topolo174197925503356063within @ A @ X @ T4 )
= ( topolo174197925503356063within @ A @ X @ U3 ) ) ) ) ) ) ).
% at_within_nhd
thf(fact_5101_filterlim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [G2: A > B,G3: filter @ B,X: A,S: set @ A,F4: filter @ B,D2: real,F3: A > B] :
( ( filterlim @ A @ B @ G2 @ G3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ G3 @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [X16: A] :
( ( member @ A @ X16 @ S )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X16 @ X ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X ) @ D2 )
=> ( ( F3 @ X16 )
= ( G2 @ X16 ) ) ) ) )
=> ( filterlim @ A @ B @ F3 @ F4 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% filterlim_transform_within
thf(fact_5102_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X4: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M10 @ M4 )
=> ! [N3: nat] :
( ( ord_less @ nat @ M4 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X4 @ M4 ) @ ( X4 @ N3 ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X4 ) ) ) ).
% CauchyI'
thf(fact_5103_Cauchy__altdef,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [F2: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M8: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M8 @ M2 )
=> ! [N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ M2 ) @ ( F2 @ N2 ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_5104_at__eq__bot__iff,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: A] :
( ( ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) )
= ( topolo1002775350975398744n_open @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% at_eq_bot_iff
thf(fact_5105_metric__LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [G2: A > B,L: B,A3: A,R: real,F3: A > B] :
( ( filterlim @ A @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( ! [X3: A] :
( ( X3 != A3 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A3 ) @ R )
=> ( ( F3 @ X3 )
= ( G2 @ X3 ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_equal2
thf(fact_5106_metric__LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [A3: A,F3: A > B,L5: B] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [S9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S9 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A3 ) @ S9 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X3 ) @ L5 ) @ R3 ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% metric_LIM_I
thf(fact_5107_metric__LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F3: A > B,L5: B,A3: A,R2: real] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [S4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S4 )
& ! [X5: A] :
( ( ( X5 != A3 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A3 ) @ S4 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X5 ) @ L5 ) @ R2 ) ) ) ) ) ) ).
% metric_LIM_D
thf(fact_5108_LIM__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F3: A > B,L5: B,A3: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S8 )
& ! [X2: A] :
( ( ( X2 != A3 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A3 ) @ S8 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X2 ) @ L5 ) @ R5 ) ) ) ) ) ) ) ).
% LIM_def
thf(fact_5109_lim__sequentially,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X4: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X4 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No3 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X4 @ N2 ) @ L5 ) @ R5 ) ) ) ) ) ) ).
% lim_sequentially
thf(fact_5110_metric__LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X4: nat > A,L5: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No2: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No2 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X4 @ N3 ) @ L5 ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X4 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% metric_LIMSEQ_I
thf(fact_5111_metric__LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X4: nat > A,L5: A,R2: real] :
( ( filterlim @ nat @ A @ X4 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ No @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X4 @ N4 ) @ L5 ) @ R2 ) ) ) ) ) ).
% metric_LIMSEQ_D
thf(fact_5112_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [J3: nat] :
? [M8: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M8 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M8 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_5113_metric__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,B3: B,A3: A,G2: B > C,C3: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ B3 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G2 @ ( topolo7230453075368039082e_nhds @ C @ C3 ) @ ( topolo174197925503356063within @ B @ B3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A3 ) @ D4 ) )
=> ( ( F3 @ X3 )
!= B3 ) ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C3 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_compose2
thf(fact_5114_continuous__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F4 @ F3 )
=> ( ( ( cos @ A
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F4
@ ^ [X2: A] : ( tan @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_tan
thf(fact_5115_continuous__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F4 @ F3 )
=> ( ( ( sin @ A
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F4
@ ^ [X2: A] : ( cot @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_cot
thf(fact_5116_continuous__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F3 )
=> ( ( ( cosh @ A
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ C @ C @ F4
@ ^ [X2: C] : X2 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X2: C] : ( tanh @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_tanh
thf(fact_5117_continuous__arcosh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( ( ord_less @ real @ ( one_one @ real )
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( arcosh @ real @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_arcosh
thf(fact_5118_metric__isCont__LIM__compose2,axiom,
! [D: $tType,C: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ D ) )
=> ! [A3: A,F3: A > C,G2: C > D,L: D] :
( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( filterlim @ C @ D @ G2 @ ( topolo7230453075368039082e_nhds @ D @ L ) @ ( topolo174197925503356063within @ C @ ( F3 @ A3 ) @ ( top_top @ ( set @ C ) ) ) )
=> ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A3 ) @ D4 ) )
=> ( ( F3 @ X3 )
!= ( F3 @ A3 ) ) ) )
=> ( filterlim @ A @ D
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ D @ L )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_isCont_LIM_compose2
thf(fact_5119_tendsto__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A3: A,S: set @ A,F3: A > D,L5: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A3 )
=> ( ( member @ A @ A3 @ S )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( filterlim @ A @ D @ F3 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ S ) )
= ( filterlim @ A @ D
@ ^ [H2: A] : ( F3 @ ( plus_plus @ A @ A3 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ D @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% tendsto_offset_zero_iff
thf(fact_5120_continuous__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real,G2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F4 @ G2 )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( G2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( log @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_log
thf(fact_5121_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X4: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X4 @ ( 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 )
& ! [N2: nat] :
( ( ord_less_eq @ nat @ No3 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X4 @ N2 ) @ L5 ) @ R5 ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_5122_totally__bounded__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo6688025880775521714ounded @ A )
= ( ^ [S7: set @ A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [K3: set @ A] :
( ( finite_finite2 @ A @ K3 )
& ( ord_less_eq @ ( set @ A ) @ S7
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X2: A] :
( collect @ A
@ ^ [Y3: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y3 ) @ E3 ) )
@ K3 ) ) ) ) ) ) ) ) ).
% totally_bounded_metric
thf(fact_5123_filterlim__pow__at__bot__even,axiom,
! [N: nat,F3: real > real,F4: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F3 @ ( at_bot @ real ) @ F4 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( power_power @ real @ ( F3 @ X2 ) @ N )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_5124_lim__zero__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,L: A] :
( ( filterlim @ A @ A
@ ^ [X2: A] : ( F3 @ ( divide_divide @ A @ ( one_one @ A ) @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) ) ) ) ).
% lim_zero_infinity
thf(fact_5125_totally__bounded__empty,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( topolo6688025880775521714ounded @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% totally_bounded_empty
thf(fact_5126_at__top__le__at__infinity,axiom,
ord_less_eq @ ( filter @ real ) @ ( at_top @ real ) @ ( at_infinity @ real ) ).
% at_top_le_at_infinity
thf(fact_5127_at__bot__le__at__infinity,axiom,
ord_less_eq @ ( filter @ real ) @ ( at_bot @ real ) @ ( at_infinity @ real ) ).
% at_bot_le_at_infinity
thf(fact_5128_trivial__limit__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_bot @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_bot_linorder
thf(fact_5129_totally__bounded__subset,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [S: set @ A,T4: set @ A] :
( ( topolo6688025880775521714ounded @ A @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ T4 @ S )
=> ( topolo6688025880775521714ounded @ A @ T4 ) ) ) ) ).
% totally_bounded_subset
thf(fact_5130_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_5131_tendsto__mult__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,C3: A,F4: filter @ B,G2: B > A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F4 )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A @ G2 @ ( at_infinity @ A ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_infinity @ A )
@ F4 ) ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_5132_tendsto__divide__0,axiom,
! [A: $tType,C: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: C > A,C3: A,F4: filter @ C,G2: C > A] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F4 )
=> ( ( filterlim @ C @ A @ G2 @ ( at_infinity @ A ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_divide_0
thf(fact_5133_filterlim__power__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [F3: A > B,F4: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F3 @ X2 ) @ N )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_5134_filterlim__tendsto__pos__mult__at__bot,axiom,
! [A: $tType,F3: A > real,C3: real,F4: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ A @ real @ G2 @ ( at_bot @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_bot
thf(fact_5135_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_5136_filterlim__inverse__at__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [G2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( inverse_inverse @ B @ ( G2 @ X2 ) )
@ ( topolo174197925503356063within @ B @ ( zero_zero @ B ) @ ( top_top @ ( set @ B ) ) )
@ F4 )
= ( filterlim @ A @ B @ G2 @ ( at_infinity @ B ) @ F4 ) ) ) ).
% filterlim_inverse_at_iff
thf(fact_5137_filterlim__tendsto__neg__mult__at__bot,axiom,
! [A: $tType,F3: A > real,C3: real,F4: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F4 )
=> ( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_neg_mult_at_bot
thf(fact_5138_filterlim__divide__at__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,C3: A,F4: filter @ A,G2: A > A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F4 )
=> ( ( filterlim @ A @ A @ G2 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) @ F4 )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_infinity @ A )
@ F4 ) ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_5139_filterlim__realpow__sequentially__gt1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X ) @ ( at_infinity @ A ) @ ( at_top @ nat ) ) ) ) ).
% filterlim_realpow_sequentially_gt1
thf(fact_5140_DERIV__pos__imp__increasing__at__bot,axiom,
! [B3: real,F3: real > real,Flim: real] :
( ! [X3: real] :
( ( ord_less_eq @ real @ X3 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) )
=> ( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_bot @ real ) )
=> ( ord_less @ real @ Flim @ ( F3 @ B3 ) ) ) ) ).
% DERIV_pos_imp_increasing_at_bot
thf(fact_5141_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F3: real > real,F4: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F3 @ ( at_bot @ real ) @ F4 )
=> ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( power_power @ real @ ( F3 @ X2 ) @ N )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_5142_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C3: nat > A,K: nat,N: nat,B2: real] :
( ( ( C3 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( eventually @ A
@ ^ [Z6: A] :
( ord_less_eq @ real @ B2
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z6 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_5143_GMVT,axiom,
! [A3: real,B3: real,F3: real > real,G2: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ( ! [X3: real] :
( ( ( ord_less @ real @ A3 @ X3 )
& ( ord_less @ real @ X3 @ B3 ) )
=> ( differentiable @ real @ real @ F3 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ G2 ) )
=> ( ! [X3: real] :
( ( ( ord_less @ real @ A3 @ X3 )
& ( ord_less @ real @ X3 @ B3 ) )
=> ( differentiable @ real @ real @ G2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [G_c: real,F_c: real,C5: real] :
( ( has_field_derivative @ real @ G2 @ G_c @ ( topolo174197925503356063within @ real @ C5 @ ( top_top @ ( set @ real ) ) ) )
& ( has_field_derivative @ real @ F3 @ F_c @ ( topolo174197925503356063within @ real @ C5 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ A3 @ C5 )
& ( ord_less @ real @ C5 @ B3 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) ) @ G_c )
= ( times_times @ real @ ( minus_minus @ real @ ( G2 @ B3 ) @ ( G2 @ A3 ) ) @ F_c ) ) ) ) ) ) ) ) ).
% GMVT
thf(fact_5144_Bfun__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > A,A3: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( bfun @ B @ A
@ ^ [X2: B] : ( inverse_inverse @ A @ ( F3 @ X2 ) )
@ F4 ) ) ) ) ).
% Bfun_inverse
thf(fact_5145_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_5146_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat
@ ^ [N2: nat] : ( P @ ( plus_plus @ nat @ N2 @ K ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_5147_eventually__top,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( top_top @ ( filter @ A ) ) )
= ( ! [X8: A] : ( P @ X8 ) ) ) ).
% eventually_top
thf(fact_5148_eventually__const,axiom,
! [A: $tType,F4: filter @ A,P: $o] :
( ( F4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X2: A] : P
@ F4 )
= P ) ) ).
% eventually_const
thf(fact_5149_differentiable__cmult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Q5: B > A,C3: A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ ( Q5 @ T3 ) @ C3 )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q5 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_right_iff
thf(fact_5150_differentiable__cmult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: A,Q5: B > A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ C3 @ ( Q5 @ T3 ) )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q5 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_left_iff
thf(fact_5151_eventually__at__bot__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [C3: A] :
( eventually @ A
@ ^ [X2: A] : X2 != C3
@ ( at_bot @ A ) ) ) ).
% eventually_at_bot_not_equal
thf(fact_5152_filterlim__iff,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F2: A > B,F24: filter @ B,F14: filter @ A] :
! [P3: B > $o] :
( ( eventually @ B @ P3 @ F24 )
=> ( eventually @ A
@ ^ [X2: A] : ( P3 @ ( F2 @ X2 ) )
@ F14 ) ) ) ) ).
% filterlim_iff
thf(fact_5153_filterlim__cong,axiom,
! [A: $tType,B: $tType,F1: filter @ A,F12: filter @ A,F22: filter @ B,F23: filter @ B,F3: B > A,G2: B > A] :
( ( F1 = F12 )
=> ( ( F22 = F23 )
=> ( ( eventually @ B
@ ^ [X2: B] :
( ( F3 @ X2 )
= ( G2 @ X2 ) )
@ F22 )
=> ( ( filterlim @ B @ A @ F3 @ F1 @ F22 )
= ( filterlim @ B @ A @ G2 @ F12 @ F23 ) ) ) ) ) ).
% filterlim_cong
thf(fact_5154_eventually__compose__filterlim,axiom,
! [A: $tType,B: $tType,P: A > $o,F4: filter @ A,F3: B > A,G3: filter @ B] :
( ( eventually @ A @ P @ F4 )
=> ( ( filterlim @ B @ A @ F3 @ F4 @ G3 )
=> ( eventually @ B
@ ^ [X2: B] : ( P @ ( F3 @ X2 ) )
@ G3 ) ) ) ).
% eventually_compose_filterlim
thf(fact_5155_trivial__limit__def,axiom,
! [A: $tType,F4: filter @ A] :
( ( F4
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X2: A] : $false
@ F4 ) ) ).
% trivial_limit_def
thf(fact_5156_eventually__const__iff,axiom,
! [A: $tType,P: $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : P
@ F4 )
= ( P
| ( F4
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_5157_False__imp__not__eventually,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ! [X3: A] :
~ ( P @ X3 )
=> ( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ~ ( eventually @ A @ P @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_5158_eventually__happens_H,axiom,
! [A: $tType,F4: filter @ A,P: A > $o] :
( ( F4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A @ P @ F4 )
=> ? [X_1: A] : ( P @ X_1 ) ) ) ).
% eventually_happens'
thf(fact_5159_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_5160_eventually__bot,axiom,
! [A: $tType,P: A > $o] : ( eventually @ A @ P @ ( bot_bot @ ( filter @ A ) ) ) ).
% eventually_bot
thf(fact_5161_eventuallyI,axiom,
! [A: $tType,P: A > $o,F4: filter @ A] :
( ! [X_1: A] : ( P @ X_1 )
=> ( eventually @ A @ P @ F4 ) ) ).
% eventuallyI
thf(fact_5162_filter__eq__iff,axiom,
! [A: $tType] :
( ( ^ [Y4: filter @ A,Z2: filter @ A] : Y4 = Z2 )
= ( ^ [F8: filter @ A,F9: filter @ A] :
! [P3: A > $o] :
( ( eventually @ A @ P3 @ F8 )
= ( eventually @ A @ P3 @ F9 ) ) ) ) ).
% filter_eq_iff
thf(fact_5163_eventually__mono,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_mono
thf(fact_5164_not__eventuallyD,axiom,
! [A: $tType,P: A > $o,F4: filter @ A] :
( ~ ( eventually @ A @ P @ F4 )
=> ? [X3: A] :
~ ( P @ X3 ) ) ).
% not_eventuallyD
thf(fact_5165_always__eventually,axiom,
! [A: $tType,P: A > $o,F4: filter @ A] :
( ! [X_1: A] : ( P @ X_1 )
=> ( eventually @ A @ P @ F4 ) ) ).
% always_eventually
thf(fact_5166_not__eventually__impI,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ~ ( eventually @ A @ Q @ F4 )
=> ~ ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F4 ) ) ) ).
% not_eventually_impI
thf(fact_5167_eventually__conj__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ F4 )
= ( ( eventually @ A @ P @ F4 )
& ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_conj_iff
thf(fact_5168_eventually__rev__mp,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F4 )
=> ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_rev_mp
thf(fact_5169_eventually__subst,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [N2: A] :
( ( P @ N2 )
= ( Q @ N2 ) )
@ F4 )
=> ( ( eventually @ A @ P @ F4 )
= ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_subst
thf(fact_5170_eventually__elim2,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o,R: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A @ Q @ F4 )
=> ( ! [I2: A] :
( ( P @ I2 )
=> ( ( Q @ I2 )
=> ( R @ I2 ) ) )
=> ( eventually @ A @ R @ F4 ) ) ) ) ).
% eventually_elim2
thf(fact_5171_eventually__conj,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A @ Q @ F4 )
=> ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ F4 ) ) ) ).
% eventually_conj
thf(fact_5172_eventually__True,axiom,
! [A: $tType,F4: filter @ A] :
( eventually @ A
@ ^ [X2: A] : $true
@ F4 ) ).
% eventually_True
thf(fact_5173_eventually__mp,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F4 )
=> ( ( eventually @ A @ P @ F4 )
=> ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_mp
thf(fact_5174_eventually__frequently__const__simps_I3_J,axiom,
! [A: $tType,P: A > $o,C2: $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
| C2 )
@ F4 )
= ( ( eventually @ A @ P @ F4 )
| C2 ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_5175_eventually__frequently__const__simps_I4_J,axiom,
! [A: $tType,C2: $o,P: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( C2
| ( P @ X2 ) )
@ F4 )
= ( C2
| ( eventually @ A @ P @ F4 ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_5176_eventually__frequently__const__simps_I6_J,axiom,
! [A: $tType,C2: $o,P: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( C2
=> ( P @ X2 ) )
@ F4 )
= ( C2
=> ( eventually @ A @ P @ F4 ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_5177_eventually__inf,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,F11: filter @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F4 @ F11 ) )
= ( ? [Q7: A > $o,R6: A > $o] :
( ( eventually @ A @ Q7 @ F4 )
& ( eventually @ A @ R6 @ F11 )
& ! [X2: A] :
( ( ( Q7 @ X2 )
& ( R6 @ X2 ) )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_inf
thf(fact_5178_filter__leD,axiom,
! [A: $tType,F4: filter @ A,F11: filter @ A,P: A > $o] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ F11 )
=> ( ( eventually @ A @ P @ F11 )
=> ( eventually @ A @ P @ F4 ) ) ) ).
% filter_leD
thf(fact_5179_filter__leI,axiom,
! [A: $tType,F11: filter @ A,F4: filter @ A] :
( ! [P8: A > $o] :
( ( eventually @ A @ P8 @ F11 )
=> ( eventually @ A @ P8 @ F4 ) )
=> ( ord_less_eq @ ( filter @ A ) @ F4 @ F11 ) ) ).
% filter_leI
thf(fact_5180_le__filter__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( filter @ A ) )
= ( ^ [F8: filter @ A,F9: filter @ A] :
! [P3: A > $o] :
( ( eventually @ A @ P3 @ F9 )
=> ( eventually @ A @ P3 @ F8 ) ) ) ) ).
% le_filter_def
thf(fact_5181_BfunI,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,K4: real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) @ K4 )
@ F4 )
=> ( bfun @ A @ B @ F3 @ F4 ) ) ) ).
% BfunI
thf(fact_5182_eventually__Sup,axiom,
! [A: $tType,P: A > $o,S: set @ ( filter @ A )] :
( ( eventually @ A @ P @ ( complete_Sup_Sup @ ( filter @ A ) @ S ) )
= ( ! [X2: filter @ A] :
( ( member @ ( filter @ A ) @ X2 @ S )
=> ( eventually @ A @ P @ X2 ) ) ) ) ).
% eventually_Sup
thf(fact_5183_eventually__sup,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,F11: filter @ A] :
( ( eventually @ A @ P @ ( sup_sup @ ( filter @ A ) @ F4 @ F11 ) )
= ( ( eventually @ A @ P @ F4 )
& ( eventually @ A @ P @ F11 ) ) ) ).
% eventually_sup
thf(fact_5184_Bseq__eventually__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: nat > A,G2: nat > B] :
( ( eventually @ nat
@ ^ [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( real_V7770717601297561774m_norm @ B @ ( G2 @ N2 ) ) )
@ ( at_top @ nat ) )
=> ( ( bfun @ nat @ B @ G2 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_eventually_mono
thf(fact_5185_eventually__at__top__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C3: A] :
( eventually @ A
@ ^ [X2: A] : X2 != C3
@ ( at_top @ A ) ) ) ).
% eventually_at_top_not_equal
thf(fact_5186_eventually__False__sequentially,axiom,
~ ( eventually @ nat
@ ^ [N2: nat] : $false
@ ( at_top @ nat ) ) ).
% eventually_False_sequentially
thf(fact_5187_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N5: A] :
! [N2: A] :
( ( ord_less_eq @ A @ N5 @ N2 )
=> ( P @ N2 ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_5188_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,P: A > $o] :
( ! [X3: A] :
( ( ord_less_eq @ A @ C3 @ X3 )
=> ( P @ X3 ) )
=> ( eventually @ A @ P @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_5189_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N5: A] :
! [N2: A] :
( ( ord_less @ A @ N5 @ N2 )
=> ( P @ N2 ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_5190_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
= ( ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( P @ N2 ) ) ) ) ).
% eventually_sequentially
thf(fact_5191_eventually__sequentiallyI,axiom,
! [C3: nat,P: nat > $o] :
( ! [X3: nat] :
( ( ord_less_eq @ nat @ C3 @ X3 )
=> ( P @ X3 ) )
=> ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_5192_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N5: A] :
! [N2: A] :
( ( ord_less_eq @ A @ N2 @ N5 )
=> ( P @ N2 ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_5193_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N5: A] :
! [N2: A] :
( ( ord_less @ A @ N2 @ N5 )
=> ( P @ N2 ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_5194_eventually__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A] : ( eventually @ A @ ( ord_less_eq @ A @ C3 ) @ ( at_top @ A ) ) ) ).
% eventually_ge_at_top
thf(fact_5195_eventually__gt__at__top,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C3: A] : ( eventually @ A @ ( ord_less @ A @ C3 ) @ ( at_top @ A ) ) ) ).
% eventually_gt_at_top
thf(fact_5196_differentiable__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,X: A,S3: set @ A,T2: set @ A] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S3 )
=> ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% differentiable_within_subset
thf(fact_5197_le__sequentially,axiom,
! [F4: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F4 @ ( at_top @ nat ) )
= ( ! [N5: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N5 ) @ F4 ) ) ) ).
% le_sequentially
thf(fact_5198_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ C3 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_5199_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C3: A] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ C3 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_5200_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F3: A > B,F4: filter @ B,G3: filter @ A,F11: filter @ B,G7: filter @ A,F10: A > B] :
( ( filterlim @ A @ B @ F3 @ F4 @ G3 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F4 @ F11 )
=> ( ( ord_less_eq @ ( filter @ A ) @ G7 @ G3 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F3 @ X2 )
= ( F10 @ X2 ) )
@ G7 )
=> ( filterlim @ A @ B @ F10 @ F11 @ G7 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_5201_eventually__INF1,axiom,
! [B: $tType,A: $tType,I: A,I5: set @ A,P: B > $o,F4: A > ( filter @ B )] :
( ( member @ A @ I @ I5 )
=> ( ( eventually @ B @ P @ ( F4 @ I ) )
=> ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F4 @ I5 ) ) ) ) ) ).
% eventually_INF1
thf(fact_5202_BfunE,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( bfun @ A @ B @ F3 @ F4 )
=> ~ ! [B4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B4 )
=> ~ ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) @ B4 )
@ F4 ) ) ) ) ).
% BfunE
thf(fact_5203_Bfun__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F2: A > B,F8: filter @ A] :
? [K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) @ K5 )
@ F8 ) ) ) ) ) ).
% Bfun_def
thf(fact_5204_Bfun__metric__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F2: A > B,F8: filter @ A] :
? [Y3: B,K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X2 ) @ Y3 ) @ K5 )
@ F8 ) ) ) ) ) ).
% Bfun_metric_def
thf(fact_5205_differentiable__sum,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [S3: set @ A,F3: A > B > C,Net: filter @ B] :
( ( finite_finite2 @ A @ S3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( differentiable @ B @ C @ ( F3 @ X3 ) @ Net ) )
=> ( differentiable @ B @ C
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [A5: A] : ( F3 @ A5 @ X2 )
@ S3 )
@ Net ) ) ) ) ).
% differentiable_sum
thf(fact_5206_eventually__nhds__top,axiom,
! [A: $tType] :
( ( ( order_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B3: A,P: A > $o] :
( ( ord_less @ A @ B3 @ ( top_top @ A ) )
=> ( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ ( top_top @ A ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ ( top_top @ A ) )
& ! [Z6: A] :
( ( ord_less @ A @ B5 @ Z6 )
=> ( P @ Z6 ) ) ) ) ) ) ) ).
% eventually_nhds_top
thf(fact_5207_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P: B > $o,G2: B > A] :
( ! [X3: A,Y2: A] :
( ( Q @ X3 )
=> ( ( Q @ Y2 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F3 @ ( G2 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G2 @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( at_top @ A ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_5208_eventually__at__left__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ X )
& ! [Y3: A] :
( ( ord_less @ A @ B5 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ X )
=> ( P @ Y3 ) ) ) ) ) ) ) ).
% eventually_at_left_field
thf(fact_5209_eventually__at__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y: A,X: A,P: A > $o] :
( ( ord_less @ A @ Y @ X )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ X )
& ! [Y3: A] :
( ( ord_less @ A @ B5 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ X )
=> ( P @ Y3 ) ) ) ) ) ) ) ) ).
% eventually_at_left
thf(fact_5210_eventually__at__infinity,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_infinity @ A ) )
= ( ? [B5: real] :
! [X2: A] :
( ( ord_less_eq @ real @ B5 @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_at_infinity
thf(fact_5211_tendsto__sandwich,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,G2: B > A,Net: filter @ B,H: B > A,C3: A] :
( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( G2 @ N2 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ ( G2 @ N2 ) @ ( H @ N2 ) )
@ Net )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ Net )
=> ( ( filterlim @ B @ A @ H @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ Net )
=> ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ Net ) ) ) ) ) ) ).
% tendsto_sandwich
thf(fact_5212_order__tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,X: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
= ( ! [L2: A] :
( ( ord_less @ A @ L2 @ X )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ L2 @ ( F3 @ X2 ) )
@ F4 ) )
& ! [U2: A] :
( ( ord_less @ A @ X @ U2 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F3 @ X2 ) @ U2 )
@ F4 ) ) ) ) ) ).
% order_tendsto_iff
thf(fact_5213_order__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [Y: A,F3: B > A,F4: filter @ B] :
( ! [A7: A] :
( ( ord_less @ A @ A7 @ Y )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ A7 @ ( F3 @ X2 ) )
@ F4 ) )
=> ( ! [A7: A] :
( ( ord_less @ A @ Y @ A7 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F3 @ X2 ) @ A7 )
@ F4 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 ) ) ) ) ).
% order_tendstoI
thf(fact_5214_order__tendstoD_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,Y: A,F4: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 )
=> ( ( ord_less @ A @ A3 @ Y )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ A3 @ ( F3 @ X2 ) )
@ F4 ) ) ) ) ).
% order_tendstoD(1)
thf(fact_5215_order__tendstoD_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,Y: A,F4: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 )
=> ( ( ord_less @ A @ Y @ A3 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F3 @ X2 ) @ A3 )
@ F4 ) ) ) ) ).
% order_tendstoD(2)
thf(fact_5216_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F4 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z9 @ ( F3 @ X2 ) )
@ F4 ) ) ) ) ).
% filterlim_at_top
thf(fact_5217_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F4: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F4 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ C3 @ Z9 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z9 @ ( F3 @ X2 ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_5218_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,F4: filter @ B,G2: B > A] :
( ( filterlim @ B @ A @ F3 @ ( at_top @ A ) @ F4 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ F4 )
=> ( filterlim @ B @ A @ G2 @ ( at_top @ A ) @ F4 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_5219_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F4 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ Z9 @ ( F3 @ X2 ) )
@ F4 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_5220_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F4 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ Z9 )
@ F4 ) ) ) ) ).
% filterlim_at_bot
thf(fact_5221_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F4: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F4 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ Z9 @ C3 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ Z9 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_5222_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F4 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F3 @ X2 ) @ Z9 )
@ F4 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_5223_BseqI_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X4: nat > A,K4: real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X4 @ N3 ) ) @ K4 )
=> ( bfun @ nat @ A @ X4 @ ( at_top @ nat ) ) ) ) ).
% BseqI'
thf(fact_5224_real__tendsto__sandwich,axiom,
! [B: $tType,F3: B > real,G2: B > real,Net: filter @ B,H: B > real,C3: real] :
( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ real @ ( F3 @ N2 ) @ ( G2 @ N2 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ real @ ( G2 @ N2 ) @ ( H @ N2 ) )
@ Net )
=> ( ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ Net )
=> ( ( filterlim @ B @ real @ H @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ Net )
=> ( filterlim @ B @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ Net ) ) ) ) ) ).
% real_tendsto_sandwich
thf(fact_5225_countable__basis__at__decseq,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X: A] :
~ ! [A9: nat > ( set @ A )] :
( ! [I3: nat] : ( topolo1002775350975398744n_open @ A @ ( A9 @ I3 ) )
=> ( ! [I3: nat] : ( member @ A @ X @ ( A9 @ I3 ) )
=> ~ ! [S10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S10 )
=> ( ( member @ A @ X @ S10 )
=> ( eventually @ nat
@ ^ [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( A9 @ I4 ) @ S10 )
@ ( at_top @ nat ) ) ) ) ) ) ) ).
% countable_basis_at_decseq
thf(fact_5226_eventually__Inf__base,axiom,
! [A: $tType,B2: set @ ( filter @ A ),P: A > $o] :
( ( B2
!= ( bot_bot @ ( set @ ( filter @ A ) ) ) )
=> ( ! [F5: filter @ A] :
( ( member @ ( filter @ A ) @ F5 @ B2 )
=> ! [G4: filter @ A] :
( ( member @ ( filter @ A ) @ G4 @ B2 )
=> ? [X5: filter @ A] :
( ( member @ ( filter @ A ) @ X5 @ B2 )
& ( ord_less_eq @ ( filter @ A ) @ X5 @ ( inf_inf @ ( filter @ A ) @ F5 @ G4 ) ) ) ) )
=> ( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B2 ) )
= ( ? [X2: filter @ A] :
( ( member @ ( filter @ A ) @ X2 @ B2 )
& ( eventually @ A @ P @ X2 ) ) ) ) ) ) ).
% eventually_Inf_base
thf(fact_5227_eventually__INF__finite,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: B > $o,F4: A > ( filter @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F4 @ A4 ) ) )
= ( ? [Q7: A > B > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( eventually @ B @ ( Q7 @ X2 ) @ ( F4 @ X2 ) ) )
& ! [Y3: B] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( Q7 @ X2 @ Y3 ) )
=> ( P @ Y3 ) ) ) ) ) ) ).
% eventually_INF_finite
thf(fact_5228_eventually__at__left__real,axiom,
! [B3: real,A3: real] :
( ( ord_less @ real @ B3 @ A3 )
=> ( eventually @ real
@ ^ [X2: real] : ( member @ real @ X2 @ ( set_or5935395276787703475ssThan @ real @ B3 @ A3 ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) ) ) ).
% eventually_at_left_real
thf(fact_5229_Bseq__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( bfun @ nat @ A
@ ^ [X2: nat] : ( times_times @ A @ C3 @ ( F3 @ X2 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_cmult_iff
thf(fact_5230_differentiable__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,X: A,S3: set @ A,G2: A > B] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( differentiable @ A @ B @ G2 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ( G2 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( divide_divide @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ) ).
% differentiable_divide
thf(fact_5231_differentiable__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,X: A,S3: set @ A] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F3 @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ).
% differentiable_inverse
thf(fact_5232_eventually__at,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A3: A,S: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ S ) )
= ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( ( ( X2 != A3 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A3 ) @ D5 ) )
=> ( P @ X2 ) ) ) ) ) ) ) ).
% eventually_at
thf(fact_5233_eventually__nhds__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A3: A] :
( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ A3 ) )
= ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A3 ) @ D5 )
=> ( P @ X2 ) ) ) ) ) ) ).
% eventually_nhds_metric
thf(fact_5234_eventually__at__leftI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B3: A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ B3 @ ( set_ord_lessThan @ A @ B3 ) ) ) ) ) ) ).
% eventually_at_leftI
thf(fact_5235_eventually__at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o,A3: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( eventually @ A
@ ^ [X2: A] : ( P @ ( plus_plus @ A @ X2 @ A3 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% eventually_at_to_0
thf(fact_5236_increasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,L: A,F4: filter @ B] :
( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ ( F3 @ N2 ) @ L )
@ F4 )
=> ( ! [X3: A] :
( ( ord_less @ A @ X3 @ L )
=> ( eventually @ B
@ ^ [N2: B] : ( ord_less @ A @ X3 @ ( F3 @ N2 ) )
@ F4 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% increasing_tendsto
thf(fact_5237_decreasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [L: A,F3: B > A,F4: filter @ B] :
( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ L @ ( F3 @ N2 ) )
@ F4 )
=> ( ! [X3: A] :
( ( ord_less @ A @ L @ X3 )
=> ( eventually @ B
@ ^ [N2: B] : ( ord_less @ A @ ( F3 @ N2 ) @ X3 )
@ F4 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% decreasing_tendsto
thf(fact_5238_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F4: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F4 )
= ( ! [Z9: B] :
( ( ord_less @ B @ C3 @ Z9 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z9 @ ( F3 @ X2 ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_5239_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F4: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F4 )
= ( ! [Z9: B] :
( ( ord_less @ B @ Z9 @ C3 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ Z9 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_5240_tendsto__le,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F4: filter @ B,F3: B > A,X: A,G2: B > A,Y: A] :
( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
=> ( ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ A @ ( G2 @ X2 ) @ ( F3 @ X2 ) )
@ F4 )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ) ).
% tendsto_le
thf(fact_5241_tendsto__lowerbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: B > A,X: A,F4: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
=> ( ( eventually @ B
@ ^ [I4: B] : ( ord_less_eq @ A @ A3 @ ( F3 @ I4 ) )
@ F4 )
=> ( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ A3 @ X ) ) ) ) ) ).
% tendsto_lowerbound
thf(fact_5242_tendsto__upperbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: B > A,X: A,F4: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
=> ( ( eventually @ B
@ ^ [I4: B] : ( ord_less_eq @ A @ ( F3 @ I4 ) @ A3 )
@ F4 )
=> ( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ X @ A3 ) ) ) ) ) ).
% tendsto_upperbound
thf(fact_5243_metric__tendsto__imp__tendsto,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V7819770556892013058_space @ B )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F3: C > A,A3: A,F4: filter @ C,G2: C > B,B3: B] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( eventually @ C
@ ^ [X2: C] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G2 @ X2 ) @ B3 ) @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X2 ) @ A3 ) )
@ F4 )
=> ( filterlim @ C @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ B3 ) @ F4 ) ) ) ) ).
% metric_tendsto_imp_tendsto
thf(fact_5244_filterlim__at__infinity__imp__filterlim__at__top,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_infinity @ real ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_top
thf(fact_5245_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_infinity @ real ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( F3 @ X2 ) @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real @ F3 @ ( at_bot @ real ) @ F4 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_bot
thf(fact_5246_eventually__INF,axiom,
! [A: $tType,B: $tType,P: A > $o,F4: B > ( filter @ A ),B2: set @ B] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F4 @ B2 ) ) )
= ( ? [X8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X8 @ B2 )
& ( finite_finite2 @ B @ X8 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F4 @ X8 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_5247_continuous__arcosh__strong,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( arcosh @ real @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_arcosh_strong
thf(fact_5248_eventually__at__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A3: A,S: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ S ) )
= ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( ( ( X2 != A3 )
& ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A3 ) @ D5 ) )
=> ( P @ X2 ) ) ) ) ) ) ) ).
% eventually_at_le
thf(fact_5249_eventually__at__infinity__pos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P6: A > $o] :
( ( eventually @ A @ P6 @ ( at_infinity @ A ) )
= ( ? [B5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B5 )
& ! [X2: A] :
( ( ord_less_eq @ real @ B5 @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( P6 @ X2 ) ) ) ) ) ) ).
% eventually_at_infinity_pos
thf(fact_5250_Bseq__eq__bounded,axiom,
! [F3: nat > real,A3: real,B3: real] :
( ( ord_less_eq @ ( set @ real ) @ ( image2 @ nat @ real @ F3 @ ( top_top @ ( set @ nat ) ) ) @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) )
=> ( bfun @ nat @ real @ F3 @ ( at_top @ nat ) ) ) ).
% Bseq_eq_bounded
thf(fact_5251_tendsto__imp__filterlim__at__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F3: A > B,L5: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F3 @ X2 ) @ L5 )
@ F4 )
=> ( filterlim @ A @ B @ F3 @ ( topolo174197925503356063within @ B @ L5 @ ( set_ord_lessThan @ B @ L5 ) ) @ F4 ) ) ) ) ).
% tendsto_imp_filterlim_at_left
thf(fact_5252_tendstoD,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F4: filter @ B,E2: real] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X2 ) @ L ) @ E2 )
@ F4 ) ) ) ) ).
% tendstoD
thf(fact_5253_tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F4: filter @ B] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X2 ) @ L ) @ E )
@ F4 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ).
% tendstoI
thf(fact_5254_tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
= ( ! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X2 ) @ L ) @ E3 )
@ F4 ) ) ) ) ) ).
% tendsto_iff
thf(fact_5255_eventually__Inf,axiom,
! [A: $tType,P: A > $o,B2: set @ ( filter @ A )] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B2 ) )
= ( ? [X8: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X8 @ B2 )
& ( finite_finite2 @ ( filter @ A ) @ X8 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ X8 ) ) ) ) ) ).
% eventually_Inf
thf(fact_5256_summable__comparison__test__ev,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G2: nat > real] :
( ( eventually @ nat
@ ^ [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( G2 @ N2 ) )
@ ( at_top @ nat ) )
=> ( ( summable @ real @ G2 )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test_ev
thf(fact_5257_Bseq__def,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X4: nat > A] :
( ( bfun @ nat @ A @ X4 @ ( at_top @ nat ) )
= ( ? [K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
& ! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X4 @ N2 ) ) @ K5 ) ) ) ) ) ).
% Bseq_def
thf(fact_5258_BseqI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [K4: real,X4: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K4 )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X4 @ N3 ) ) @ K4 )
=> ( bfun @ nat @ A @ X4 @ ( at_top @ nat ) ) ) ) ) ).
% BseqI
thf(fact_5259_BseqE,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X4: nat > A] :
( ( bfun @ nat @ A @ X4 @ ( at_top @ nat ) )
=> ~ ! [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
=> ~ ! [N4: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X4 @ N4 ) ) @ K9 ) ) ) ) ).
% BseqE
thf(fact_5260_BseqD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X4: nat > A] :
( ( bfun @ nat @ A @ X4 @ ( at_top @ nat ) )
=> ? [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
& ! [N4: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X4 @ N4 ) ) @ K9 ) ) ) ) ).
% BseqD
thf(fact_5261_Bseq__iff1a,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X4: nat > A] :
( ( bfun @ nat @ A @ X4 @ ( at_top @ nat ) )
= ( ? [N5: nat] :
! [N2: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X4 @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ).
% Bseq_iff1a
thf(fact_5262_Bseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X4: nat > A] :
( ( bfun @ nat @ A @ X4 @ ( at_top @ nat ) )
= ( ? [N5: nat] :
! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X4 @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ).
% Bseq_iff
thf(fact_5263_Bseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( bfun @ nat @ real @ ( power_power @ real @ X ) @ ( at_top @ nat ) ) ) ) ).
% Bseq_realpow
thf(fact_5264_tendsto__arcosh__strong,axiom,
! [B: $tType,F3: B > real,A3: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ A3 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( arcosh @ real @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_arcosh_strong
thf(fact_5265_filterlim__at__top__at__left,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P: B > $o,G2: B > A,A3: A] :
( ! [X3: A,Y2: A] :
( ( Q @ X3 )
=> ( ( Q @ Y2 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F3 @ ( G2 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G2 @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) )
=> ( ! [B7: A] :
( ( Q @ B7 )
=> ( ord_less @ A @ B7 @ A3 ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_left
thf(fact_5266_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B2: set @ A,F4: A > ( filter @ B ),P: B > $o] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A7: A] :
( ( member @ A @ A7 @ B2 )
=> ! [B7: A] :
( ( member @ A @ B7 @ B2 )
=> ? [X5: A] :
( ( member @ A @ X5 @ B2 )
& ( ord_less_eq @ ( filter @ B ) @ ( F4 @ X5 ) @ ( inf_inf @ ( filter @ B ) @ ( F4 @ A7 ) @ ( F4 @ B7 ) ) ) ) ) )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F4 @ B2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( eventually @ B @ P @ ( F4 @ X2 ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_5267_tendsto__0__le,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F4: filter @ A,G2: A > C,K4: real] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G2 @ X2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) @ K4 ) )
@ F4 )
=> ( filterlim @ A @ C @ G2 @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F4 ) ) ) ) ).
% tendsto_0_le
thf(fact_5268_filterlim__at__withinI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,C3: A,F4: filter @ B,A4: set @ A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F4 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( member @ A @ ( F3 @ X2 ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ F4 )
=> ( filterlim @ B @ A @ F3 @ ( topolo174197925503356063within @ A @ C3 @ A4 ) @ F4 ) ) ) ) ).
% filterlim_at_withinI
thf(fact_5269_filterlim__at__infinity,axiom,
! [C: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C3: real,F3: C > A,F4: filter @ C] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ C @ A @ F3 @ ( at_infinity @ A ) @ F4 )
= ( ! [R5: real] :
( ( ord_less @ real @ C3 @ R5 )
=> ( eventually @ C
@ ^ [X2: C] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X2 ) ) )
@ F4 ) ) ) ) ) ) ).
% filterlim_at_infinity
thf(fact_5270_tendsto__zero__powrI,axiom,
! [A: $tType,F3: A > real,F4: filter @ A,G2: A > real,B3: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( filterlim @ A @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ) ) ).
% tendsto_zero_powrI
thf(fact_5271_tendsto__powr2,axiom,
! [A: $tType,F3: A > real,A3: real,F4: filter @ A,G2: A > real,B3: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( filterlim @ A @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A3 @ B3 ) )
@ F4 ) ) ) ) ) ).
% tendsto_powr2
thf(fact_5272_tendsto__powr_H,axiom,
! [A: $tType,F3: A > real,A3: real,F4: filter @ A,G2: A > real,B3: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( filterlim @ A @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( ( ( A3
!= ( zero_zero @ real ) )
| ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X2 ) )
@ F4 ) ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_powr'
thf(fact_5273_eventually__floor__less,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ B @ L ) ) @ ( F3 @ X2 ) )
@ F4 ) ) ) ) ).
% eventually_floor_less
thf(fact_5274_LIM__at__top__divide,axiom,
! [A: $tType,F3: A > real,A3: real,F4: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( filterlim @ A @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X2 ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ) ).
% LIM_at_top_divide
thf(fact_5275_eventually__less__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F3 @ X2 ) @ ( ring_1_of_int @ B @ ( archimedean_ceiling @ B @ L ) ) )
@ F4 ) ) ) ) ).
% eventually_less_ceiling
thf(fact_5276_filterlim__inverse__at__top,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( inverse_inverse @ real @ ( F3 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_inverse_at_top
thf(fact_5277_filterlim__inverse__at__top__iff,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( ( filterlim @ A @ real
@ ^ [X2: A] : ( inverse_inverse @ real @ ( F3 @ X2 ) )
@ ( at_top @ real )
@ F4 )
= ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% filterlim_inverse_at_top_iff
thf(fact_5278_filterlim__inverse__at__bot,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( F3 @ X2 ) @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( inverse_inverse @ real @ ( F3 @ X2 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ).
% filterlim_inverse_at_bot
thf(fact_5279_Bseq__iff3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X4: nat > A] :
( ( bfun @ nat @ A @ X4 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [N5: nat] :
! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X4 @ N2 ) @ ( uminus_uminus @ A @ ( X4 @ N5 ) ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff3
thf(fact_5280_Bseq__iff2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X4: nat > A] :
( ( bfun @ nat @ A @ X4 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [X2: A] :
! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X4 @ N2 ) @ ( uminus_uminus @ A @ X2 ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff2
thf(fact_5281_summable__Cauchy_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G2: nat > real] :
( ( eventually @ nat
@ ^ [M2: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) @ ( G2 @ M2 ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_Cauchy'
thf(fact_5282_cauchy__filter__metric,axiom,
! [A: $tType] :
( ( ( real_V768167426530841204y_dist @ A )
& ( topolo7287701948861334536_space @ A ) )
=> ( ( topolo6773858410816713723filter @ A )
= ( ^ [F8: filter @ A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [P3: A > $o] :
( ( eventually @ A @ P3 @ F8 )
& ! [X2: A,Y3: A] :
( ( ( P3 @ X2 )
& ( P3 @ Y3 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y3 ) @ E3 ) ) ) ) ) ) ) ).
% cauchy_filter_metric
thf(fact_5283_MVT,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ( differentiable @ real @ real @ F3 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [L7: real,Z3: real] :
( ( ord_less @ real @ A3 @ Z3 )
& ( ord_less @ real @ Z3 @ B3 )
& ( has_field_derivative @ real @ F3 @ L7 @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) )
& ( ( minus_minus @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) )
= ( times_times @ real @ ( minus_minus @ real @ B3 @ A3 ) @ L7 ) ) ) ) ) ) ).
% MVT
thf(fact_5284_eventually__all__finite,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [P: A > B > $o,Net: filter @ A] :
( ! [Y2: B] :
( eventually @ A
@ ^ [X2: A] : ( P @ X2 @ Y2 )
@ Net )
=> ( eventually @ A
@ ^ [X2: A] :
! [X8: B] : ( P @ X2 @ X8 )
@ Net ) ) ) ).
% eventually_all_finite
thf(fact_5285_continuous__on__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S3: set @ A,F3: A > B,T2: set @ A] :
( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S3 )
=> ( topolo81223032696312382ous_on @ A @ B @ T2 @ F3 ) ) ) ) ).
% continuous_on_subset
thf(fact_5286_IVT2_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F3: A > B,B3: A,Y: B,A3: A] :
( ( ord_less_eq @ B @ ( F3 @ B3 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F3 @ A3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F3 )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less_eq @ A @ X3 @ B3 )
& ( ( F3 @ X3 )
= Y ) ) ) ) ) ) ) ).
% IVT2'
thf(fact_5287_IVT_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F3: A > B,A3: A,Y: B,B3: A] :
( ( ord_less_eq @ B @ ( F3 @ A3 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F3 @ B3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F3 )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less_eq @ A @ X3 @ B3 )
& ( ( F3 @ X3 )
= Y ) ) ) ) ) ) ) ).
% IVT'
thf(fact_5288_continuous__on__empty,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B] : ( topolo81223032696312382ous_on @ A @ B @ ( bot_bot @ ( set @ A ) ) @ F3 ) ) ).
% continuous_on_empty
thf(fact_5289_continuous__on__sing,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X: A,F3: A > B] : ( topolo81223032696312382ous_on @ A @ B @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ F3 ) ) ).
% continuous_on_sing
thf(fact_5290_continuous__on__open__Un,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S3: set @ A,T2: set @ A,F3: A > B] :
( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ T2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ T2 @ F3 )
=> ( topolo81223032696312382ous_on @ A @ B @ ( sup_sup @ ( set @ A ) @ S3 @ T2 ) @ F3 ) ) ) ) ) ) ).
% continuous_on_open_Un
thf(fact_5291_continuous__image__closed__interval,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ? [C5: real,D6: real] :
( ( ( image2 @ real @ real @ F3 @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ real @ C5 @ D6 ) )
& ( ord_less_eq @ real @ C5 @ D6 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_5292_continuous__on__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S3: set @ A,F3: A > B,G2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S3 @ G2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( G2 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S3
@ ^ [X2: A] : ( divide_divide @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% continuous_on_divide
thf(fact_5293_continuous__on__compose2,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [T2: set @ A,G2: A > B,S3: set @ C,F3: C > A] :
( ( topolo81223032696312382ous_on @ A @ B @ T2 @ G2 )
=> ( ( topolo81223032696312382ous_on @ C @ A @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ F3 @ S3 ) @ T2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S3
@ ^ [X2: C] : ( G2 @ ( F3 @ X2 ) ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_5294_continuous__on__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S3: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( F3 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S3
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_on_inverse
thf(fact_5295_continuous__on__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S3: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( F3 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S3
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_on_sgn
thf(fact_5296_continuous__onI__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( dense_order @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,A4: set @ A] :
( ( topolo1002775350975398744n_open @ B @ ( image2 @ A @ B @ F3 @ A4 ) )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ A4 @ F3 ) ) ) ) ).
% continuous_onI_mono
thf(fact_5297_open__Collect__less,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,G2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G2 )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% open_Collect_less
thf(fact_5298_continuous__on__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S3: set @ A,F3: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S3 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( cos @ A @ ( F3 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S3
@ ^ [X2: A] : ( tan @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_on_tan
thf(fact_5299_open__Collect__less__Int,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F3: A > real,G2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S3 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ real @ S3 @ G2 )
=> ? [A9: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A9 )
& ( ( inf_inf @ ( set @ A ) @ A9 @ S3 )
= ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ord_less @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ) ) ).
% open_Collect_less_Int
thf(fact_5300_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X2: A] :
! [Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( P @ Y3 ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_5301_continuous__on__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S3: set @ A,F3: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S3 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( sin @ A @ ( F3 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S3
@ ^ [X2: A] : ( cot @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_on_cot
thf(fact_5302_continuous__on__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A4: set @ C,F3: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A4 @ F3 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A4 )
=> ( ( cosh @ A @ ( F3 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ C @ A @ A4
@ ^ [X2: C] : ( tanh @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_on_tanh
thf(fact_5303_finite__set__of__finite__funs,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,D2: B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F2: A > B] :
! [X2: A] :
( ( ( member @ A @ X2 @ A4 )
=> ( member @ B @ ( F2 @ X2 ) @ B2 ) )
& ( ~ ( member @ A @ X2 @ A4 )
=> ( ( F2 @ X2 )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_5304_continuous__on__arcosh_H,axiom,
! [A4: set @ real,F3: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A4 @ F3 )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ A4 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( F3 @ X3 ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A4
@ ^ [X2: real] : ( arcosh @ real @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_arcosh'
thf(fact_5305_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P3 @ X2 )
& ! [Y3: A] :
( ( P3 @ Y3 )
=> ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ) ) ).
% Least_def
thf(fact_5306_open__Collect__positive,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S3 @ F3 )
=> ? [A9: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A9 )
& ( ( inf_inf @ ( set @ A ) @ A9 @ S3 )
= ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X2 ) ) ) ) ) ) ) ) ).
% open_Collect_positive
thf(fact_5307_continuous__on__powr_H,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S3: set @ C,F3: C > real,G2: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S3 @ F3 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S3 @ G2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ S3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) )
& ( ( ( F3 @ X3 )
= ( zero_zero @ real ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X3 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S3
@ ^ [X2: C] : ( powr @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% continuous_on_powr'
thf(fact_5308_continuous__on__log,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F3: A > real,G2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S3 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ real @ S3 @ G2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( F3 @ X3 )
!= ( one_one @ real ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X3 ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S3
@ ^ [X2: A] : ( log @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_on_log
thf(fact_5309_continuous__on__arccos,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S3 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( F3 @ X3 ) )
& ( ord_less_eq @ real @ ( F3 @ X3 ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S3
@ ^ [X2: A] : ( arccos @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_on_arccos
thf(fact_5310_continuous__on__arcsin,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S3 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( F3 @ X3 ) )
& ( ord_less_eq @ real @ ( F3 @ X3 ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S3
@ ^ [X2: A] : ( arcsin @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_on_arcsin
thf(fact_5311_continuous__on__artanh,axiom,
! [A4: set @ real] :
( ( ord_less_eq @ ( set @ real ) @ A4 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A4 @ ( artanh @ real ) ) ) ).
% continuous_on_artanh
thf(fact_5312_DERIV__atLeastAtMost__imp__continuous__on,axiom,
! [A: $tType] :
( ( ( ord @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,B3: A,F3: A > A] :
( ! [X3: A] :
( ( ord_less_eq @ A @ A3 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ B3 )
=> ? [Y5: A] : ( has_field_derivative @ A @ F3 @ Y5 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F3 ) ) ) ).
% DERIV_atLeastAtMost_imp_continuous_on
thf(fact_5313_Rolle__deriv,axiom,
! [A3: real,B3: real,F3: real > real,F10: real > real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( ( F3 @ A3 )
= ( F3 @ B3 ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ( has_derivative @ real @ real @ F3 @ ( F10 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A3 @ Z3 )
& ( ord_less @ real @ Z3 @ B3 )
& ( ( F10 @ Z3 )
= ( ^ [V6: real] : ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% Rolle_deriv
thf(fact_5314_mvt,axiom,
! [A3: real,B3: real,F3: real > real,F10: real > real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ( has_derivative @ real @ real @ F3 @ ( F10 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ~ ! [Xi: real] :
( ( ord_less @ real @ A3 @ Xi )
=> ( ( ord_less @ real @ Xi @ B3 )
=> ( ( minus_minus @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) )
!= ( F10 @ Xi @ ( minus_minus @ real @ B3 @ A3 ) ) ) ) ) ) ) ) ).
% mvt
thf(fact_5315_nhds__imp__cauchy__filter,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [F4: filter @ A,X: A] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ ( topolo7230453075368039082e_nhds @ A @ X ) )
=> ( topolo6773858410816713723filter @ A @ F4 ) ) ) ).
% nhds_imp_cauchy_filter
thf(fact_5316_continuous__on__Icc__at__leftD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,B3: A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F3 )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ B3 ) ) @ ( topolo174197925503356063within @ A @ B3 @ ( set_ord_lessThan @ A @ B3 ) ) ) ) ) ) ).
% continuous_on_Icc_at_leftD
thf(fact_5317_DERIV__isconst__end,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ( has_field_derivative @ real @ F3 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ( F3 @ B3 )
= ( F3 @ A3 ) ) ) ) ) ).
% DERIV_isconst_end
thf(fact_5318_DERIV__neg__imp__decreasing__open,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ord_less @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) ) ) ) ) ).
% DERIV_neg_imp_decreasing_open
thf(fact_5319_DERIV__pos__imp__increasing__open,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ord_less @ real @ ( F3 @ A3 ) @ ( F3 @ B3 ) ) ) ) ) ).
% DERIV_pos_imp_increasing_open
thf(fact_5320_DERIV__isconst2,axiom,
! [A3: real,B3: real,F3: real > real,X: real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ( has_field_derivative @ real @ F3 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ X )
=> ( ( ord_less_eq @ real @ X @ B3 )
=> ( ( F3 @ X )
= ( F3 @ A3 ) ) ) ) ) ) ) ).
% DERIV_isconst2
thf(fact_5321_Rolle,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( ( F3 @ A3 )
= ( F3 @ B3 ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ( differentiable @ real @ real @ F3 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A3 @ Z3 )
& ( ord_less @ real @ Z3 @ B3 )
& ( has_field_derivative @ real @ F3 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% Rolle
thf(fact_5322_summable__bounded__partials,axiom,
! [A: $tType] :
( ( ( real_V8037385150606011577_space @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: nat > A,G2: nat > real] :
( ( eventually @ nat
@ ^ [X02: nat] :
! [A5: nat] :
( ( ord_less_eq @ nat @ X02 @ A5 )
=> ! [B5: nat] :
( ( ord_less @ nat @ A5 @ B5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or3652927894154168847AtMost @ nat @ A5 @ B5 ) ) ) @ ( G2 @ A5 ) ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_bounded_partials
thf(fact_5323_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P3 @ X2 )
& ! [Y3: A] :
( ( P3 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_5324_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq2: A > A > $o,P3: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P3 @ X2 )
& ! [Y3: A] :
( ( P3 @ Y3 )
=> ( Less_eq2 @ X2 @ Y3 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_5325_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% finite_greaterThanAtMost
thf(fact_5326_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_5327_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_5328_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_5329_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_5330_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% infinite_Ioc_iff
thf(fact_5331_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanAtMost
thf(fact_5332_cSup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_greaterThanAtMost
thf(fact_5333_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanAtMost
thf(fact_5334_cInf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_greaterThanAtMost
thf(fact_5335_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A3 @ B3 )
= ( set_or3652927894154168847AtMost @ A @ C3 @ D2 ) )
= ( ( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ D2 @ C3 ) )
| ( ( A3 = C3 )
& ( B3 = D2 ) ) ) ) ) ).
% Ioc_inj
thf(fact_5336_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_5337_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D2 ) )
= ( ( ord_less_eq @ A @ B3 @ A3 )
| ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_5338_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) ) ) ) ).
% infinite_Ioc
thf(fact_5339_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_5340_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_5341_GreatestI__ex__nat,axiom,
! [P: nat > $o,B3: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ B3 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_5342_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ B3 ) )
=> ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_5343_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ B3 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_5344_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D2 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B3 @ A3 )
| ( ord_less_eq @ A @ D2 @ C3 )
| ( ord_less_eq @ A @ B3 @ C3 )
| ( ord_less_eq @ A @ D2 @ A3 ) ) ) ) ).
% Ioc_disjoint
thf(fact_5345_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_5346_open__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S: set @ A,X: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( member @ A @ X @ S )
=> ( ( ord_less @ A @ Y @ X )
=> ? [B7: A] :
( ( ord_less @ A @ B7 @ X )
& ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ B7 @ X ) @ S ) ) ) ) ) ) ).
% open_left
thf(fact_5347_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_5348_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_5349_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_5350_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_5351_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_5352_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_5353_sum_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G2 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.head
thf(fact_5354_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G2 @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.head
thf(fact_5355_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_5356_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_5357_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_5358_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_5359_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A5: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A5 @ B5 ) @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_5360_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_5361_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_5362_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 ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_5363_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_5364_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 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_5365_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ I ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_5366_continuous__on__IccI,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,A3: A,B3: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ B3 ) ) @ ( topolo174197925503356063within @ A @ B3 @ ( set_ord_lessThan @ A @ B3 ) ) )
=> ( ! [X3: A] :
( ( ord_less @ A @ A3 @ X3 )
=> ( ( ord_less @ A @ X3 @ B3 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ X3 ) ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F3 ) ) ) ) ) ) ).
% continuous_on_IccI
thf(fact_5367_eventually__filtercomap__at__topological,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ B )
=> ! [P: A > $o,F3: A > B,A4: B,B2: set @ B] :
( ( eventually @ A @ P @ ( filtercomap @ A @ B @ F3 @ ( topolo174197925503356063within @ B @ A4 @ B2 ) ) )
= ( ? [S7: set @ B] :
( ( topolo1002775350975398744n_open @ B @ S7 )
& ( member @ B @ A4 @ S7 )
& ! [X2: A] :
( ( member @ B @ ( F3 @ X2 ) @ ( minus_minus @ ( set @ B ) @ ( inf_inf @ ( set @ B ) @ S7 @ B2 ) @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) )
=> ( P @ X2 ) ) ) ) ) ) ).
% eventually_filtercomap_at_topological
thf(fact_5368_at__within__eq,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [X2: A,S8: set @ A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ ( set @ A ) @ ( filter @ A )
@ ^ [S7: set @ A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S7 @ S8 ) @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [S7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S7 )
& ( member @ A @ X2 @ S7 ) ) ) ) ) ) ) ) ).
% at_within_eq
thf(fact_5369_principal__inject,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( principal @ A @ A4 )
= ( principal @ A @ B2 ) )
= ( A4 = B2 ) ) ).
% principal_inject
thf(fact_5370_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_5371_filterlim__filtercomap,axiom,
! [A: $tType,B: $tType,F3: A > B,F4: filter @ B] : ( filterlim @ A @ B @ F3 @ F4 @ ( filtercomap @ A @ B @ F3 @ F4 ) ) ).
% filterlim_filtercomap
thf(fact_5372_filtercomap__bot,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( filtercomap @ A @ B @ F3 @ ( bot_bot @ ( filter @ B ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ).
% filtercomap_bot
thf(fact_5373_eventually__filtercomapI,axiom,
! [B: $tType,A: $tType,P: A > $o,F4: filter @ A,F3: B > A] :
( ( eventually @ A @ P @ F4 )
=> ( eventually @ B
@ ^ [X2: B] : ( P @ ( F3 @ X2 ) )
@ ( filtercomap @ B @ A @ F3 @ F4 ) ) ) ).
% eventually_filtercomapI
thf(fact_5374_filtercomap__top,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( filtercomap @ A @ B @ F3 @ ( top_top @ ( filter @ B ) ) )
= ( top_top @ ( filter @ A ) ) ) ).
% filtercomap_top
thf(fact_5375_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X ) @ ( set_ord_greaterThan @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% greaterThan_subset_iff
thf(fact_5376_principal__le__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ ( principal @ A @ A4 ) @ ( principal @ A @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% principal_le_iff
thf(fact_5377_inf__principal,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( inf_inf @ ( filter @ A ) @ ( principal @ A @ A4 ) @ ( principal @ A @ B2 ) )
= ( principal @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% inf_principal
thf(fact_5378_sup__principal,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( sup_sup @ ( filter @ A ) @ ( principal @ A @ A4 ) @ ( principal @ A @ B2 ) )
= ( principal @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% sup_principal
thf(fact_5379_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X ) )
= ( top_top @ A ) ) ) ) ).
% Sup_greaterThanAtLeast
thf(fact_5380_SUP__principal,axiom,
! [A: $tType,B: $tType,A4: B > ( set @ A ),I5: set @ B] :
( ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [I4: B] : ( principal @ A @ ( A4 @ I4 ) )
@ I5 ) )
= ( principal @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I5 ) ) ) ) ).
% SUP_principal
thf(fact_5381_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less @ A @ L2 ) ) ) ) ) ).
% greaterThan_def
thf(fact_5382_filtercomap__filtercomap,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,G2: B > C,F4: filter @ C] :
( ( filtercomap @ A @ B @ F3 @ ( filtercomap @ B @ C @ G2 @ F4 ) )
= ( filtercomap @ A @ C
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) )
@ F4 ) ) ).
% filtercomap_filtercomap
thf(fact_5383_filtercomap__ident,axiom,
! [A: $tType,F4: filter @ A] :
( ( filtercomap @ A @ A
@ ^ [X2: A] : X2
@ F4 )
= F4 ) ).
% filtercomap_ident
thf(fact_5384_filtercomap__inf,axiom,
! [A: $tType,B: $tType,F3: A > B,F1: filter @ B,F22: filter @ B] :
( ( filtercomap @ A @ B @ F3 @ ( inf_inf @ ( filter @ B ) @ F1 @ F22 ) )
= ( inf_inf @ ( filter @ A ) @ ( filtercomap @ A @ B @ F3 @ F1 ) @ ( filtercomap @ A @ B @ F3 @ F22 ) ) ) ).
% filtercomap_inf
thf(fact_5385_infinite__Ioi,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_greaterThan @ A @ A3 ) ) ) ).
% infinite_Ioi
thf(fact_5386_greaterThan__non__empty,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
( ( set_ord_greaterThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% greaterThan_non_empty
thf(fact_5387_filtercomap__mono,axiom,
! [B: $tType,A: $tType,F4: filter @ A,F11: filter @ A,F3: B > A] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ F11 )
=> ( ord_less_eq @ ( filter @ B ) @ ( filtercomap @ B @ A @ F3 @ F4 ) @ ( filtercomap @ B @ A @ F3 @ F11 ) ) ) ).
% filtercomap_mono
thf(fact_5388_eventually__filtercomap,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: A > B,F4: filter @ B] :
( ( eventually @ A @ P @ ( filtercomap @ A @ B @ F3 @ F4 ) )
= ( ? [Q7: B > $o] :
( ( eventually @ B @ Q7 @ F4 )
& ! [X2: A] :
( ( Q7 @ ( F3 @ X2 ) )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_filtercomap
thf(fact_5389_eventually__principal,axiom,
! [A: $tType,P: A > $o,S: set @ A] :
( ( eventually @ A @ P @ ( principal @ A @ S ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( P @ X2 ) ) ) ) ).
% eventually_principal
thf(fact_5390_principal__eq__bot__iff,axiom,
! [A: $tType,X4: set @ A] :
( ( ( principal @ A @ X4 )
= ( bot_bot @ ( filter @ A ) ) )
= ( X4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% principal_eq_bot_iff
thf(fact_5391_bot__eq__principal__empty,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( principal @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bot_eq_principal_empty
thf(fact_5392_filterlim__iff__le__filtercomap,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F2: A > B,F8: filter @ B,G8: filter @ A] : ( ord_less_eq @ ( filter @ A ) @ G8 @ ( filtercomap @ A @ B @ F2 @ F8 ) ) ) ) ).
% filterlim_iff_le_filtercomap
thf(fact_5393_filtercomap__neq__bot,axiom,
! [A: $tType,B: $tType,F4: filter @ A,F3: B > A] :
( ! [P8: A > $o] :
( ( eventually @ A @ P8 @ F4 )
=> ? [X5: B] : ( P8 @ ( F3 @ X5 ) ) )
=> ( ( filtercomap @ B @ A @ F3 @ F4 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtercomap_neq_bot
thf(fact_5394_filterlim__principal,axiom,
! [B: $tType,A: $tType,F3: A > B,S: set @ B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( principal @ B @ S ) @ F4 )
= ( eventually @ A
@ ^ [X2: A] : ( member @ B @ ( F3 @ X2 ) @ S )
@ F4 ) ) ).
% filterlim_principal
thf(fact_5395_top__eq__principal__UNIV,axiom,
! [A: $tType] :
( ( top_top @ ( filter @ A ) )
= ( principal @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% top_eq_principal_UNIV
thf(fact_5396_nhds__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [X2: A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ real @ ( filter @ A )
@ ^ [E3: real] :
( principal @ A
@ ( collect @ A
@ ^ [Y3: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y3 @ X2 ) @ E3 ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% nhds_metric
thf(fact_5397_le__principal,axiom,
! [A: $tType,F4: filter @ A,A4: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ ( principal @ A @ A4 ) )
= ( eventually @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 )
@ F4 ) ) ).
% le_principal
thf(fact_5398_INT__greaterThan__UNIV,axiom,
( ( complete_Inf_Inf @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_greaterThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% INT_greaterThan_UNIV
thf(fact_5399_filterlim__If,axiom,
! [B: $tType,A: $tType,F3: A > B,G3: filter @ B,F4: filter @ A,P: A > $o,G2: A > B] :
( ( filterlim @ A @ B @ F3 @ G3 @ ( inf_inf @ ( filter @ A ) @ F4 @ ( principal @ A @ ( collect @ A @ P ) ) ) )
=> ( ( filterlim @ A @ B @ G2 @ G3
@ ( inf_inf @ ( filter @ A ) @ F4
@ ( principal @ A
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( if @ B @ ( P @ X2 ) @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ G3
@ F4 ) ) ) ).
% filterlim_If
thf(fact_5400_at__right__eq,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ X @ A5 ) )
@ ( set_ord_greaterThan @ A @ X ) ) ) ) ) ) ).
% at_right_eq
thf(fact_5401_eventually__inf__principal,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,S3: set @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F4 @ ( principal @ A @ S3 ) ) )
= ( eventually @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S3 )
=> ( P @ X2 ) )
@ F4 ) ) ).
% eventually_inf_principal
thf(fact_5402_filtercomap__sup,axiom,
! [A: $tType,B: $tType,F3: A > B,F1: filter @ B,F22: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( sup_sup @ ( filter @ A ) @ ( filtercomap @ A @ B @ F3 @ F1 ) @ ( filtercomap @ A @ B @ F3 @ F22 ) ) @ ( filtercomap @ A @ B @ F3 @ ( sup_sup @ ( filter @ B ) @ F1 @ F22 ) ) ) ).
% filtercomap_sup
thf(fact_5403_eventually__at__right__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ X @ B5 )
& ! [Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ( ord_less @ A @ Y3 @ B5 )
=> ( P @ Y3 ) ) ) ) ) ) ) ).
% eventually_at_right_field
thf(fact_5404_eventually__at__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,Y: A,P: A > $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ X @ B5 )
& ! [Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ( ord_less @ A @ Y3 @ B5 )
=> ( P @ Y3 ) ) ) ) ) ) ) ) ).
% eventually_at_right
thf(fact_5405_at__within__Icc__at__right,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( topolo174197925503356063within @ A @ A3 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ).
% at_within_Icc_at_right
thf(fact_5406_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_5407_filtercomap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,F4: C > ( filter @ B ),B2: set @ C] :
( ( filtercomap @ A @ B @ F3 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F4 @ B2 ) ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B5: C] : ( filtercomap @ A @ B @ F3 @ ( F4 @ B5 ) )
@ B2 ) ) ) ).
% filtercomap_INF
thf(fact_5408_nhds__discrete,axiom,
! [A: $tType] :
( ( topolo8865339358273720382pology @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [X2: A] : ( principal @ A @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% nhds_discrete
thf(fact_5409_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_5410_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_5411_eventually__at__right__less,axiom,
! [A: $tType] :
( ( ( no_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A] : ( eventually @ A @ ( ord_less @ A @ X ) @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) ) ) ).
% eventually_at_right_less
thf(fact_5412_eventually__filtercomap__at__top__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F3: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F3 @ ( at_top @ A ) ) )
= ( ? [N5: A] :
! [X2: B] :
( ( ord_less_eq @ A @ N5 @ ( F3 @ X2 ) )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_filtercomap_at_top_linorder
thf(fact_5413_eventually__filtercomap__at__top__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: B > $o,F3: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F3 @ ( at_top @ A ) ) )
= ( ? [N5: A] :
! [X2: B] :
( ( ord_less @ A @ N5 @ ( F3 @ X2 ) )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_filtercomap_at_top_dense
thf(fact_5414_filtercomap__neq__bot__surj,axiom,
! [A: $tType,B: $tType,F4: filter @ A,F3: B > A] :
( ( F4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( filtercomap @ B @ A @ F3 @ F4 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% filtercomap_neq_bot_surj
thf(fact_5415_eventually__filtercomap__at__bot__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F3: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F3 @ ( at_bot @ A ) ) )
= ( ? [N5: A] :
! [X2: B] :
( ( ord_less_eq @ A @ ( F3 @ X2 ) @ N5 )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_linorder
thf(fact_5416_eventually__filtercomap__at__bot__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: B > $o,F3: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F3 @ ( at_bot @ A ) ) )
= ( ? [N5: A] :
! [X2: B] :
( ( ord_less @ A @ ( F3 @ X2 ) @ N5 )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_dense
thf(fact_5417_filterlim__base,axiom,
! [B: $tType,A: $tType,E4: $tType,D: $tType,C: $tType,J5: set @ A,I: A > C,I5: set @ C,F4: C > ( set @ D ),F3: D > E4,G3: A > ( set @ E4 )] :
( ! [M4: A,X3: B] :
( ( member @ A @ M4 @ J5 )
=> ( member @ C @ ( I @ M4 ) @ I5 ) )
=> ( ! [M4: A,X3: D] :
( ( member @ A @ M4 @ J5 )
=> ( ( member @ D @ X3 @ ( F4 @ ( I @ M4 ) ) )
=> ( member @ E4 @ ( F3 @ X3 ) @ ( G3 @ M4 ) ) ) )
=> ( filterlim @ D @ E4 @ F3
@ ( complete_Inf_Inf @ ( filter @ E4 )
@ ( image2 @ A @ ( filter @ E4 )
@ ^ [J3: A] : ( principal @ E4 @ ( G3 @ J3 ) )
@ J5 ) )
@ ( complete_Inf_Inf @ ( filter @ D )
@ ( image2 @ C @ ( filter @ D )
@ ^ [I4: C] : ( principal @ D @ ( F4 @ I4 ) )
@ I5 ) ) ) ) ) ).
% filterlim_base
thf(fact_5418_less__separate,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [A7: A,B7: A] :
( ( member @ A @ X @ ( set_ord_lessThan @ A @ A7 ) )
& ( member @ A @ Y @ ( set_ord_greaterThan @ A @ B7 ) )
& ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A7 ) @ ( set_ord_greaterThan @ A @ B7 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% less_separate
thf(fact_5419_tendsto__principal__singleton,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,X: B] : ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( F3 @ X ) ) @ ( principal @ B @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% tendsto_principal_singleton
thf(fact_5420_eventually__at__rightI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B3: A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ).
% eventually_at_rightI
thf(fact_5421_nhds__discrete__open,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X: A] :
( ( topolo1002775350975398744n_open @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( topolo7230453075368039082e_nhds @ A @ X )
= ( principal @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% nhds_discrete_open
thf(fact_5422_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_5423_eventually__at__right__real,axiom,
! [A3: real,B3: real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( eventually @ real
@ ^ [X2: real] : ( member @ real @ X2 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) ) ) ).
% eventually_at_right_real
thf(fact_5424_filtercomap__SUP,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > C,F4: B > ( filter @ C ),B2: set @ B] :
( ord_less_eq @ ( filter @ A )
@ ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [B5: B] : ( filtercomap @ A @ C @ F3 @ ( F4 @ B5 ) )
@ B2 ) )
@ ( filtercomap @ A @ C @ F3 @ ( complete_Sup_Sup @ ( filter @ C ) @ ( image2 @ B @ ( filter @ C ) @ F4 @ B2 ) ) ) ) ).
% filtercomap_SUP
thf(fact_5425_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_ord_greaterThan @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K ) @ ( insert2 @ nat @ ( suc @ K ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% greaterThan_Suc
thf(fact_5426_at__bot__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A] :
( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atMost @ A @ K3 ) )
@ ( set_ord_atMost @ A @ C3 ) ) ) ) ) ).
% at_bot_sub
thf(fact_5427_filterlim__times__pos,axiom,
! [A: $tType,B: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: B > A,P6: A,F1: filter @ B,C3: A,L: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo174197925503356063within @ A @ P6 @ ( set_ord_greaterThan @ A @ P6 ) ) @ F1 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( L
= ( times_times @ A @ C3 @ P6 ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C3 @ ( F3 @ X2 ) )
@ ( topolo174197925503356063within @ A @ L @ ( set_ord_greaterThan @ A @ L ) )
@ F1 ) ) ) ) ) ).
% filterlim_times_pos
thf(fact_5428_at__within__order,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,S3: set @ A] :
( ( ( top_top @ ( set @ A ) )
!= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( topolo174197925503356063within @ A @ X @ S3 )
= ( inf_inf @ ( filter @ A )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A5 ) @ S3 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( set_ord_greaterThan @ A @ X ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A5 ) @ S3 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( set_ord_lessThan @ A @ X ) ) ) ) ) ) ) ).
% at_within_order
thf(fact_5429_tendsto__imp__filterlim__at__right,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F3: A > B,L5: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ L5 @ ( F3 @ X2 ) )
@ F4 )
=> ( filterlim @ A @ B @ F3 @ ( topolo174197925503356063within @ B @ L5 @ ( set_ord_greaterThan @ B @ L5 ) ) @ F4 ) ) ) ) ).
% tendsto_imp_filterlim_at_right
thf(fact_5430_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,I5: set @ A,F4: A > ( set @ B ),F3: B > C,G3: D > ( set @ C ),J5: set @ D] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F4 @ I2 ) @ ( F4 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F4 @ J2 ) @ ( F4 @ I2 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F3
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image2 @ D @ ( filter @ C )
@ ^ [J3: D] : ( principal @ C @ ( G3 @ J3 ) )
@ J5 ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [I4: A] : ( principal @ B @ ( F4 @ I4 ) )
@ I5 ) ) )
= ( ! [X2: D] :
( ( member @ D @ X2 @ J5 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ I5 )
& ! [Z6: B] :
( ( member @ B @ Z6 @ ( F4 @ Y3 ) )
=> ( member @ C @ ( F3 @ Z6 ) @ ( G3 @ X2 ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_5431_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X4: set @ A,F3: A > ( set @ B )] :
( ( finite_finite2 @ A @ X4 )
=> ( ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [X2: A] : ( principal @ B @ ( F3 @ X2 ) )
@ X4 ) )
= ( principal @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ X4 ) ) ) ) ) ).
% INF_principal_finite
thf(fact_5432_at__infinity__def,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( at_infinity @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ real @ ( filter @ A )
@ ^ [R5: real] :
( principal @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ X2 ) ) ) )
@ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% at_infinity_def
thf(fact_5433_at__bot__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atMost @ A @ K3 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_bot_def
thf(fact_5434_continuous__on__Icc__at__rightD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,B3: A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F3 )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ).
% continuous_on_Icc_at_rightD
thf(fact_5435_filterlim__at__bot__at__right,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P: B > $o,G2: B > A,A3: A] :
( ! [X3: A,Y2: A] :
( ( Q @ X3 )
=> ( ( Q @ Y2 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F3 @ ( G2 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G2 @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( ! [B7: A] :
( ( Q @ B7 )
=> ( ord_less @ A @ A3 @ B7 ) )
=> ( ( eventually @ B @ P @ ( at_bot @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_bot_at_right
thf(fact_5436_at__within__def,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [A5: A,S8: set @ A] : ( inf_inf @ ( filter @ A ) @ ( topolo7230453075368039082e_nhds @ A @ A5 ) @ ( principal @ A @ ( minus_minus @ ( set @ A ) @ S8 @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% at_within_def
thf(fact_5437_at__left__eq,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ A5 @ X ) )
@ ( set_ord_lessThan @ A @ X ) ) ) ) ) ) ).
% at_left_eq
thf(fact_5438_isCont__If__ge,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,G2: A > B,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) @ G2 )
=> ( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( G2 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( if @ B @ ( ord_less_eq @ A @ X2 @ A3 ) @ ( G2 @ X2 ) @ ( F3 @ X2 ) ) ) ) ) ) ).
% isCont_If_ge
thf(fact_5439_complete__uniform,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo2479028161051973599mplete @ A )
= ( ^ [S7: set @ A] :
! [F8: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F8 @ ( principal @ A @ S7 ) )
=> ( ( F8
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( topolo6773858410816713723filter @ A @ F8 )
=> ? [X2: A] :
( ( member @ A @ X2 @ S7 )
& ( ord_less_eq @ ( filter @ A ) @ F8 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) ) ) ) ) ) ) ) ) ).
% complete_uniform
thf(fact_5440_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A] :
( ! [A7: A,B7: A,X3: A] :
( ( member @ A @ A7 @ S )
=> ( ( member @ A @ B7 @ S )
=> ( ( ord_less_eq @ A @ A7 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ B7 )
=> ( member @ A @ X3 @ S ) ) ) ) )
=> ? [A7: A,B7: A] :
( ( S
= ( bot_bot @ ( set @ A ) ) )
| ( S
= ( top_top @ ( set @ A ) ) )
| ( S
= ( set_ord_lessThan @ A @ B7 ) )
| ( S
= ( set_ord_atMost @ A @ B7 ) )
| ( S
= ( set_ord_greaterThan @ A @ A7 ) )
| ( S
= ( set_ord_atLeast @ A @ A7 ) )
| ( S
= ( set_or5935395276787703475ssThan @ A @ A7 @ B7 ) )
| ( S
= ( set_or3652927894154168847AtMost @ A @ A7 @ B7 ) )
| ( S
= ( set_or7035219750837199246ssThan @ A @ A7 @ B7 ) )
| ( S
= ( set_or1337092689740270186AtMost @ A @ A7 @ B7 ) ) ) ) ) ).
% interval_cases
thf(fact_5441_sequentially__imp__eventually__at__right,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A3: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ! [F6: nat > A] :
( ! [N4: nat] : ( ord_less @ A @ A3 @ ( F6 @ N4 ) )
=> ( ! [N4: nat] : ( ord_less @ A @ ( F6 @ N4 ) @ B3 )
=> ( ( order_antimono @ nat @ A @ F6 )
=> ( ( filterlim @ nat @ A @ F6 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N2: nat] : ( P @ ( F6 @ N2 ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_right
thf(fact_5442_atLeast__0,axiom,
( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
= ( top_top @ ( set @ nat ) ) ) ).
% atLeast_0
thf(fact_5443_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_5444_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_5445_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X ) @ ( set_ord_atLeast @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% atLeast_subset_iff
thf(fact_5446_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H: A,L3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H ) @ ( set_ord_atLeast @ A @ L3 ) )
= ( ~ ( ord_less_eq @ A @ L @ H )
| ( ord_less_eq @ A @ L3 @ L ) ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_5447_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_5448_infinite__Ici,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_atLeast @ A @ A3 ) ) ) ).
% infinite_Ici
thf(fact_5449_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) ) ) ) ).
% antimonoD
thf(fact_5450_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) ) ) ) ).
% antimonoE
thf(fact_5451_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ Y2 ) @ ( F3 @ X3 ) ) )
=> ( order_antimono @ A @ B @ F3 ) ) ) ).
% antimonoI
thf(fact_5452_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F2: A > B] :
! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% antimono_def
thf(fact_5453_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less_eq @ A @ L2 ) ) ) ) ) ).
% atLeast_def
thf(fact_5454_atLeast__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ( set_ord_atLeast @ A @ X )
= ( top_top @ ( set @ A ) ) )
= ( X
= ( bot_bot @ A ) ) ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_5455_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_5456_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_5457_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_5458_not__Iic__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H: A,L3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_ord_atLeast @ A @ L3 ) ) ) ).
% not_Iic_le_Ici
thf(fact_5459_Ioi__le__Ico,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_atLeast @ A @ A3 ) ) ) ).
% Ioi_le_Ico
thf(fact_5460_decseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: nat > A,I: nat] :
( ( order_antimono @ nat @ A @ A4 )
=> ( ord_less_eq @ A @ ( A4 @ ( suc @ I ) ) @ ( A4 @ I ) ) ) ) ).
% decseq_SucD
thf(fact_5461_decseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) )
=> ( order_antimono @ nat @ A @ X4 ) ) ) ).
% decseq_SucI
thf(fact_5462_decseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F2: nat > A] :
! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) ) ) ) ) ).
% decseq_Suc_iff
thf(fact_5463_decseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X8: nat > A] :
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ A @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).
% decseq_def
thf(fact_5464_decseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,I: nat,J: nat] :
( ( order_antimono @ nat @ A @ F3 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( F3 @ J ) @ ( F3 @ I ) ) ) ) ) ).
% decseqD
thf(fact_5465_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_5466_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_5467_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_5468_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_5469_decseq__bounded,axiom,
! [X4: nat > real,B2: real] :
( ( order_antimono @ nat @ real @ X4 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ B2 @ ( X4 @ I2 ) )
=> ( bfun @ nat @ real @ X4 @ ( at_top @ nat ) ) ) ) ).
% decseq_bounded
thf(fact_5470_atMost__Int__atLeast,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ N ) @ ( set_ord_atLeast @ A @ N ) )
= ( insert2 @ A @ N @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atMost_Int_atLeast
thf(fact_5471_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_5472_ivl__disj__un__singleton_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A] :
( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_greaterThan @ A @ L ) )
= ( set_ord_atLeast @ A @ L ) ) ) ).
% ivl_disj_un_singleton(1)
thf(fact_5473_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_5474_decseq__ge,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X4: nat > A,L5: A,N: nat] :
( ( order_antimono @ nat @ A @ X4 )
=> ( ( filterlim @ nat @ A @ X4 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ L5 @ ( X4 @ N ) ) ) ) ) ).
% decseq_ge
thf(fact_5475_decseq__convergent,axiom,
! [X4: nat > real,B2: real] :
( ( order_antimono @ nat @ real @ X4 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ B2 @ ( X4 @ I2 ) )
=> ~ ! [L6: real] :
( ( filterlim @ nat @ real @ X4 @ ( topolo7230453075368039082e_nhds @ real @ L6 ) @ ( at_top @ nat ) )
=> ~ ! [I3: nat] : ( ord_less_eq @ real @ L6 @ ( X4 @ I3 ) ) ) ) ) ).
% decseq_convergent
thf(fact_5476_at__top__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A] :
( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K3 ) )
@ ( set_ord_atLeast @ A @ C3 ) ) ) ) ) ).
% at_top_sub
thf(fact_5477_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K ) @ ( insert2 @ nat @ K @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast_Suc
thf(fact_5478_continuous__on__arcosh,axiom,
! [A4: set @ real] :
( ( ord_less_eq @ ( set @ real ) @ A4 @ ( set_ord_atLeast @ real @ ( one_one @ real ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A4 @ ( arcosh @ real ) ) ) ).
% continuous_on_arcosh
thf(fact_5479_at__top__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K3 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_top_def
thf(fact_5480_tendsto__at__right__sequentially,axiom,
! [C: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A3: B,B3: B,X4: B > C,L5: C] :
( ( ord_less @ B @ A3 @ B3 )
=> ( ! [S2: nat > B] :
( ! [N4: nat] : ( ord_less @ B @ A3 @ ( S2 @ N4 ) )
=> ( ! [N4: nat] : ( ord_less @ B @ ( S2 @ N4 ) @ B3 )
=> ( ( order_antimono @ nat @ B @ S2 )
=> ( ( filterlim @ nat @ B @ S2 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C
@ ^ [N2: nat] : ( X4 @ ( S2 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L5 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ C @ X4 @ ( topolo7230453075368039082e_nhds @ C @ L5 ) @ ( topolo174197925503356063within @ B @ A3 @ ( set_ord_greaterThan @ B @ A3 ) ) ) ) ) ) ).
% tendsto_at_right_sequentially
thf(fact_5481_Gcd__eq__Max,axiom,
! [M5: set @ nat] :
( ( finite_finite2 @ nat @ M5 )
=> ( ( M5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( gcd_Gcd @ nat @ M5 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image2 @ nat @ ( set @ nat )
@ ^ [M2: nat] :
( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M2 ) )
@ M5 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_5482_continuous__at__Sup__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S: set @ A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Sup_Sup @ A @ S ) @ ( set_ord_lessThan @ A @ ( complete_Sup_Sup @ A @ S ) ) ) @ F3 )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S )
=> ( ( F3 @ ( complete_Sup_Sup @ A @ S ) )
= ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ S ) ) ) ) ) ) ) ) ).
% continuous_at_Sup_antimono
thf(fact_5483_continuous__at__Inf__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S: set @ A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Inf_Inf @ A @ S ) @ ( set_ord_greaterThan @ A @ ( complete_Inf_Inf @ A @ S ) ) ) @ F3 )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S )
=> ( ( F3 @ ( complete_Inf_Inf @ A @ S ) )
= ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ S ) ) ) ) ) ) ) ) ).
% continuous_at_Inf_antimono
thf(fact_5484_bdd__belowI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A,M: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ M @ X3 ) )
=> ( condit1013018076250108175_below @ A @ A4 ) ) ) ).
% bdd_belowI
thf(fact_5485_bdd__below_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A,M5: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ M5 @ X3 ) )
=> ( condit1013018076250108175_below @ A @ A4 ) ) ) ).
% bdd_below.I
thf(fact_5486_bdd__above_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A,M5: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ X3 @ M5 ) )
=> ( condit941137186595557371_above @ A @ A4 ) ) ) ).
% bdd_above.I
thf(fact_5487_bdd__below__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit1013018076250108175_below @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_below_empty
thf(fact_5488_bdd__above__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit941137186595557371_above @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_above_empty
thf(fact_5489_bdd__below__insert,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: A,A4: set @ A] :
( ( condit1013018076250108175_below @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( condit1013018076250108175_below @ A @ A4 ) ) ) ).
% bdd_below_insert
thf(fact_5490_bdd__above__insert,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: A,A4: set @ A] :
( ( condit941137186595557371_above @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( condit941137186595557371_above @ A @ A4 ) ) ) ).
% bdd_above_insert
thf(fact_5491_bdd__below__Un,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( condit1013018076250108175_below @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ( condit1013018076250108175_below @ A @ A4 )
& ( condit1013018076250108175_below @ A @ B2 ) ) ) ) ).
% bdd_below_Un
thf(fact_5492_bdd__above__Un,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( condit941137186595557371_above @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ( condit941137186595557371_above @ A @ A4 )
& ( condit941137186595557371_above @ A @ B2 ) ) ) ) ).
% bdd_above_Un
thf(fact_5493_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Max_singleton
thf(fact_5494_bdd__above__image__sup,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [F3: B > A,G2: B > A,A4: set @ B] :
( ( condit941137186595557371_above @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( sup_sup @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ A4 ) )
= ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
& ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) ) ) ) ).
% bdd_above_image_sup
thf(fact_5495_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_5496_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_5497_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_5498_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ B,C3: A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [Uu3: B] : C3
@ A4 ) )
= C3 ) ) ) ) ).
% Max_const
thf(fact_5499_bdd__below__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I5: set @ B,A4: B > ( set @ A )] :
( ( finite_finite2 @ B @ I5 )
=> ( ( condit1013018076250108175_below @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I5 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I5 )
=> ( condit1013018076250108175_below @ A @ ( A4 @ X2 ) ) ) ) ) ) ) ).
% bdd_below_UN
thf(fact_5500_bdd__above__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I5: set @ B,A4: B > ( set @ A )] :
( ( finite_finite2 @ B @ I5 )
=> ( ( condit941137186595557371_above @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I5 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I5 )
=> ( condit941137186595557371_above @ A @ ( A4 @ X2 ) ) ) ) ) ) ) ).
% bdd_above_UN
thf(fact_5501_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ) ).
% Max_insert
thf(fact_5502_cInf__le__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_5503_bdd__below_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A4: set @ B,M5: A,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ M5 @ ( F3 @ X3 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ).
% bdd_below.I2
thf(fact_5504_bdd__belowI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A4: set @ B,M: A,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ M @ ( F3 @ X3 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ).
% bdd_belowI2
thf(fact_5505_bdd__above_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A4: set @ B,F3: B > A,M5: A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ M5 ) )
=> ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ).
% bdd_above.I2
thf(fact_5506_Sup__fin__Max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% Sup_fin_Max
thf(fact_5507_bdd__above__nat,axiom,
( ( condit941137186595557371_above @ nat )
= ( finite_finite2 @ nat ) ) ).
% bdd_above_nat
thf(fact_5508_bdd__below__finite,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( condit1013018076250108175_below @ A @ A4 ) ) ) ).
% bdd_below_finite
thf(fact_5509_bdd__above__finite,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( condit941137186595557371_above @ A @ A4 ) ) ) ).
% bdd_above_finite
thf(fact_5510_bdd__below__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( condit1013018076250108175_below @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( condit1013018076250108175_below @ A @ A4 ) ) ) ) ).
% bdd_below_mono
thf(fact_5511_bdd__below_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A] :
( ( condit1013018076250108175_below @ A @ A4 )
=> ~ ! [M9: A] :
~ ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ A @ M9 @ X5 ) ) ) ) ).
% bdd_below.E
thf(fact_5512_bdd__below_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit1013018076250108175_below @ A )
= ( ^ [A6: set @ A] :
? [M8: A] :
! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less_eq @ A @ M8 @ X2 ) ) ) ) ) ).
% bdd_below.unfold
thf(fact_5513_bdd__above_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A] :
( ( condit941137186595557371_above @ A @ A4 )
=> ~ ! [M9: A] :
~ ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ A @ X5 @ M9 ) ) ) ) ).
% bdd_above.E
thf(fact_5514_bdd__above_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit941137186595557371_above @ A )
= ( ^ [A6: set @ A] :
? [M8: A] :
! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less_eq @ A @ X2 @ M8 ) ) ) ) ) ).
% bdd_above.unfold
thf(fact_5515_bdd__above__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( condit941137186595557371_above @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( condit941137186595557371_above @ A @ A4 ) ) ) ) ).
% bdd_above_mono
thf(fact_5516_cInf__lower,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X4: set @ A] :
( ( member @ A @ X @ X4 )
=> ( ( condit1013018076250108175_below @ A @ X4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X4 ) @ X ) ) ) ) ).
% cInf_lower
thf(fact_5517_cInf__lower2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X4: set @ A,Y: A] :
( ( member @ A @ X @ X4 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( condit1013018076250108175_below @ A @ X4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X4 ) @ Y ) ) ) ) ) ).
% cInf_lower2
thf(fact_5518_cSup__upper2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X4: set @ A,Y: A] :
( ( member @ A @ X @ X4 )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( ( condit941137186595557371_above @ A @ X4 )
=> ( ord_less_eq @ A @ Y @ ( complete_Sup_Sup @ A @ X4 ) ) ) ) ) ) ).
% cSup_upper2
thf(fact_5519_cSup__upper,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X4: set @ A] :
( ( member @ A @ X @ X4 )
=> ( ( condit941137186595557371_above @ A @ X4 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X4 ) ) ) ) ) ).
% cSup_upper
thf(fact_5520_Max_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ord_less_eq @ A @ A3 @ ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ).
% Max.coboundedI
thf(fact_5521_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ B2 )
& ( ord_less_eq @ A @ X3 @ Xa ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ A4 )
& ( ord_less_eq @ A @ X3 @ Xa ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( lattic643756798349783984er_Max @ A @ B2 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_5522_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ A4 )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( member @ A @ X @ A4 )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= X ) ) ) ) ) ).
% Max_eqI
thf(fact_5523_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ).
% Max_ge
thf(fact_5524_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ A4 ) ) ) ) ).
% Max_in
thf(fact_5525_Max_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ).
% Max.in_idem
thf(fact_5526_cINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A4: set @ B,X: B,U: A] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( member @ B @ X @ A4 )
=> ( ( ord_less_eq @ A @ ( F3 @ X ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ U ) ) ) ) ) ).
% cINF_lower2
thf(fact_5527_cINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A4: set @ B,X: B] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( member @ B @ X @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( F3 @ X ) ) ) ) ) ).
% cINF_lower
thf(fact_5528_cInf__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ! [B7: A] :
( ( member @ A @ B7 @ B2 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A4 )
& ( ord_less_eq @ A @ X5 @ B7 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ) ) ).
% cInf_mono
thf(fact_5529_le__cInf__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A,A3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S )
=> ( ( ord_less_eq @ A @ A3 @ ( complete_Inf_Inf @ A @ S ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( ord_less_eq @ A @ A3 @ X2 ) ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_5530_cSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: B,A4: set @ B,F3: B > A] :
( ( member @ B @ X @ A4 )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ) ).
% cSUP_upper
thf(fact_5531_cSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A4: set @ B,X: B,U: A] :
( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( member @ B @ X @ A4 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ X ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_5532_cInf__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A,Y: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X4 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X4 ) @ Y )
= ( ? [X2: A] :
( ( member @ A @ X2 @ X4 )
& ( ord_less @ A @ X2 @ Y ) ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_5533_cSup__le__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A,A3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S ) @ A3 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( ord_less_eq @ A @ X2 @ A3 ) ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_5534_cSup__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ! [B7: A] :
( ( member @ A @ B7 @ B2 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A4 )
& ( ord_less_eq @ A @ B7 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B2 ) @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_5535_less__cSup__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A,Y: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X4 )
=> ( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X4 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ X4 )
& ( ord_less @ A @ Y @ X2 ) ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_5536_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A7: A] :
( ( member @ A @ A7 @ A4 )
=> ( ord_less_eq @ A @ A7 @ X ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X ) ) ) ) ) ).
% Max.boundedI
thf(fact_5537_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X )
=> ! [A10: A] :
( ( member @ A @ A10 @ A4 )
=> ( ord_less_eq @ A @ A10 @ X ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_5538_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( ( member @ A @ M @ A4 )
& ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ A @ X2 @ M ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_5539_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_5540_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A4 )
= M )
= ( ( member @ A @ M @ A4 )
& ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ A @ X2 @ M ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_5541_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_5542_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [B7: A] :
( ( member @ A @ B7 @ A4 )
=> ( ord_less_eq @ A @ B7 @ A3 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ A3 @ A4 ) )
= A3 ) ) ) ) ).
% Max_insert2
thf(fact_5543_Max__Sup,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% Max_Sup
thf(fact_5544_cSup__eq__Max,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X4 )
= ( lattic643756798349783984er_Max @ A @ X4 ) ) ) ) ) ).
% cSup_eq_Max
thf(fact_5545_less__cINF__D,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A4: set @ B,Y: A,I: B] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ Y @ ( F3 @ I ) ) ) ) ) ) ).
% less_cINF_D
thf(fact_5546_cSUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A4: set @ B,Y: A,I: B] :
( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ Y )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ ( F3 @ I ) @ Y ) ) ) ) ) ).
% cSUP_lessD
thf(fact_5547_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F3: B > A,U: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ord_less_eq @ A @ U @ ( F3 @ X2 ) ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_5548_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B2: set @ B,F3: C > A,A4: set @ C,G2: B > A] :
( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ F3 @ A4 ) )
=> ( ! [M4: B] :
( ( member @ B @ M4 @ B2 )
=> ? [X5: C] :
( ( member @ C @ X5 @ A4 )
& ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G2 @ M4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F3 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ B2 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_5549_cInf__superset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B2 ) @ ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_5550_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,G2: C > A,B2: set @ C,F3: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ G2 @ B2 ) )
=> ( ! [N3: B] :
( ( member @ B @ N3 @ A4 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B2 )
& ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G2 @ X5 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G2 @ B2 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_5551_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F3: B > A,U: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ U )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ X2 ) @ U ) ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_5552_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_5553_cInf__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,A3: A] :
( ( condit1013018076250108175_below @ A @ X4 )
=> ( ( ( X4
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ X4 ) )
= A3 ) )
& ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ X4 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ X4 ) ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_5554_cInf__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,A3: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X4 )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ X4 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ X4 ) ) ) ) ) ) ).
% cInf_insert
thf(fact_5555_cSup__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,A3: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X4 )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ X4 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ X4 ) ) ) ) ) ) ).
% cSup_insert
thf(fact_5556_cSup__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,A3: A] :
( ( condit941137186595557371_above @ A @ X4 )
=> ( ( ( X4
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ X4 ) )
= A3 ) )
& ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ X4 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ X4 ) ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_5557_cInf__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B2 )
=> ( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_5558_cSup__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B2 )
=> ( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_5559_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ ( lattic643756798349783984er_Max @ A @ B2 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_5560_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M5: set @ A,N6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M5 @ N6 )
=> ( ( M5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M5 ) @ ( lattic643756798349783984er_Max @ A @ N6 ) ) ) ) ) ) ).
% Max_mono
thf(fact_5561_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H: A > A,N6: set @ A] :
( ! [X3: A,Y2: A] :
( ( H @ ( ord_max @ A @ X3 @ Y2 ) )
= ( ord_max @ A @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic643756798349783984er_Max @ A @ N6 ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ H @ N6 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_5562_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B2 ) @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ) ).
% Max.subset
thf(fact_5563_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_5564_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] : ( member @ A @ ( ord_max @ A @ X3 @ Y2 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Max.closed
thf(fact_5565_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A4: set @ B,F3: B > A,A3: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ A3 )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ ( F3 @ X2 ) @ A3 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_5566_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ ( lattic643756798349783984er_Max @ A @ B2 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_5567_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A4: set @ B,F3: B > A,A3: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ A3 @ ( F3 @ X2 ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_5568_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F3: B > A,G2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G2 @ A4 ) )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A5: B] : ( inf_inf @ A @ ( F3 @ A5 ) @ ( G2 @ A5 ) )
@ A4 ) ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_5569_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F3: B > A,G2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G2 @ A4 ) )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A5: B] : ( sup_sup @ A @ ( F3 @ A5 ) @ ( G2 @ A5 ) )
@ A4 ) ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_5570_Max_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A4 ) )
= ( finite_fold @ A @ A @ ( ord_max @ A ) @ X @ A4 ) ) ) ) ).
% Max.eq_fold
thf(fact_5571_card__le__Suc__Max,axiom,
! [S: set @ nat] :
( ( finite_finite2 @ nat @ S )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ S ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S ) ) ) ) ).
% card_le_Suc_Max
thf(fact_5572_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X8: set @ nat] :
( if @ nat
@ ( X8
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X8 ) ) ) ) ).
% Sup_nat_def
thf(fact_5573_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N2 ) @ M2 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_5574_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F3: B > A,K: A] :
( ( finite_finite2 @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F3 @ X2 ) @ K )
@ S ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image2 @ B @ A @ F3 @ S ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_5575_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,G2: B > A,B2: set @ B,F3: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G2 @ B2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B2 )
=> ( ord_less_eq @ A @ ( G2 @ X3 ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ B2 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_5576_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,G2: B > A,B2: set @ B,F3: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G2 @ B2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_5577_less__eq__cInf__inter,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( condit1013018076250108175_below @ A @ A4 )
=> ( ( condit1013018076250108175_below @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_5578_cINF__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F3: B > A,A3: B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( inf_inf @ A @ ( F3 @ A3 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ) ) ).
% cINF_insert
thf(fact_5579_cSup__inter__less__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( condit941137186595557371_above @ A @ A4 )
=> ( ( condit941137186595557371_above @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_5580_cSUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F3: B > A,A3: B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( sup_sup @ A @ ( F3 @ A3 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_5581_cINF__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F3: B > A,B2: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ B2 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ B2 ) ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_5582_cSUP__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F3: B > A,B2: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ B2 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ B2 ) ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_5583_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_5584_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_5585_cINF__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A4: set @ C,B2: C > ( set @ D ),F3: D > B] :
( ( A4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A4 )
=> ( ( B2 @ X3 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit1013018076250108175_below @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X2: C] : ( image2 @ D @ B @ F3 @ ( B2 @ X2 ) )
@ A4 ) ) )
=> ( ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F3 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B2 @ A4 ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F3 @ ( B2 @ X2 ) ) )
@ A4 ) ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_5586_cSUP__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A4: set @ C,B2: C > ( set @ D ),F3: D > B] :
( ( A4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A4 )
=> ( ( B2 @ X3 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit941137186595557371_above @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X2: C] : ( image2 @ D @ B @ F3 @ ( B2 @ X2 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F3 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B2 @ A4 ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F3 @ ( B2 @ X2 ) ) )
@ A4 ) ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_5587_sum__le__card__Max,axiom,
! [A: $tType,A4: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A4 ) @ ( times_times @ nat @ ( finite_card @ A @ A4 ) @ ( lattic643756798349783984er_Max @ nat @ ( image2 @ A @ nat @ F3 @ A4 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_5588_dual__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Min @ A
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 ) )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% dual_Min
thf(fact_5589_uniformity__dist,axiom,
! [A: $tType] :
( ( real_V768167426530841204y_dist @ A )
=> ( ( topolo7806501430040627800ormity @ A )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ A @ A ) )
@ ( image2 @ real @ ( filter @ ( product_prod @ A @ A ) )
@ ^ [E3: real] :
( principal @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y3: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y3 ) @ E3 ) ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ).
% uniformity_dist
thf(fact_5590_compactE__image,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,C2: set @ B,F3: B > ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ! [T5: B] :
( ( member @ B @ T5 @ C2 )
=> ( topolo1002775350975398744n_open @ A @ ( F3 @ T5 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ C2 ) ) )
=> ~ ! [C8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C8 @ C2 )
=> ( ( finite_finite2 @ B @ C8 )
=> ~ ( ord_less_eq @ ( set @ A ) @ S @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ C8 ) ) ) ) ) ) ) ) ) ).
% compactE_image
thf(fact_5591_compact__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo2193935891317330818ompact @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% compact_empty
thf(fact_5592_linorder_OMin_Ocong,axiom,
! [A: $tType] :
( ( lattices_Min @ A )
= ( lattices_Min @ A ) ) ).
% linorder.Min.cong
thf(fact_5593_compact__attains__inf,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ S )
& ! [Xa: A] :
( ( member @ A @ Xa @ S )
=> ( ord_less_eq @ A @ X3 @ Xa ) ) ) ) ) ) ).
% compact_attains_inf
thf(fact_5594_compact__attains__sup,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ S )
& ! [Xa: A] :
( ( member @ A @ Xa @ S )
=> ( ord_less_eq @ A @ Xa @ X3 ) ) ) ) ) ) ).
% compact_attains_sup
thf(fact_5595_continuous__attains__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [S3: set @ A,F3: A > B] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ S3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ S3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Xa ) ) ) ) ) ) ) ) ).
% continuous_attains_inf
thf(fact_5596_continuous__attains__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [S3: set @ A,F3: A > B] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ S3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ S3 )
=> ( ord_less_eq @ B @ ( F3 @ Xa ) @ ( F3 @ X3 ) ) ) ) ) ) ) ) ).
% continuous_attains_sup
thf(fact_5597_Cauchy__uniform__iff,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [P3: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ P3 @ ( topolo7806501430040627800ormity @ A ) )
=> ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ! [M2: nat] :
( ( ord_less_eq @ nat @ N5 @ M2 )
=> ( P3 @ ( product_Pair @ A @ A @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ) ) ) ).
% Cauchy_uniform_iff
thf(fact_5598_uniformity__complex__def,axiom,
( ( topolo7806501430040627800ormity @ complex )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ complex @ complex ) )
@ ( image2 @ real @ ( filter @ ( product_prod @ complex @ complex ) )
@ ^ [E3: real] :
( principal @ ( product_prod @ complex @ complex )
@ ( collect @ ( product_prod @ complex @ complex )
@ ( product_case_prod @ complex @ complex @ $o
@ ^ [X2: complex,Y3: complex] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ complex @ X2 @ Y3 ) @ E3 ) ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% uniformity_complex_def
thf(fact_5599_uniformity__real__def,axiom,
( ( topolo7806501430040627800ormity @ real )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ real @ real ) )
@ ( image2 @ real @ ( filter @ ( product_prod @ real @ real ) )
@ ^ [E3: real] :
( principal @ ( product_prod @ real @ real )
@ ( collect @ ( product_prod @ real @ real )
@ ( product_case_prod @ real @ real @ $o
@ ^ [X2: real,Y3: real] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ real @ X2 @ Y3 ) @ E3 ) ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% uniformity_real_def
thf(fact_5600_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 ) )
=> ? [X8: set @ A] :
( ( finite_finite2 @ A @ X8 )
& ! [X2: A] :
( ( member @ A @ X2 @ S7 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ X8 )
& ( E6 @ ( product_Pair @ A @ A @ Y3 @ X2 ) ) ) ) ) ) ) ) ) ).
% totally_bounded_def
thf(fact_5601_eventually__uniformity__metric,axiom,
! [A: $tType] :
( ( real_V768167426530841204y_dist @ A )
=> ! [P: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ P @ ( topolo7806501430040627800ormity @ A ) )
= ( ? [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
& ! [X2: A,Y3: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y3 ) @ E3 )
=> ( P @ ( product_Pair @ A @ A @ X2 @ Y3 ) ) ) ) ) ) ) ).
% eventually_uniformity_metric
thf(fact_5602_compact__eq__Heine__Borel,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo2193935891317330818ompact @ A )
= ( ^ [S7: set @ A] :
! [C4: set @ ( set @ A )] :
( ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C4 )
=> ( topolo1002775350975398744n_open @ A @ X2 ) )
& ( ord_less_eq @ ( set @ A ) @ S7 @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) ) )
=> ? [D7: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ D7 @ C4 )
& ( finite_finite2 @ ( set @ A ) @ D7 )
& ( ord_less_eq @ ( set @ A ) @ S7 @ ( complete_Sup_Sup @ ( set @ A ) @ D7 ) ) ) ) ) ) ) ).
% compact_eq_Heine_Borel
thf(fact_5603_compactI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A] :
( ! [C7: set @ ( set @ A )] :
( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C7 )
=> ( topolo1002775350975398744n_open @ A @ X5 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ C7 ) )
=> ? [C9: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C9 @ C7 )
& ( finite_finite2 @ ( set @ A ) @ C9 )
& ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ C9 ) ) ) ) )
=> ( topolo2193935891317330818ompact @ A @ S3 ) ) ) ).
% compactI
thf(fact_5604_compactE,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,T11: set @ ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ ( complete_Sup_Sup @ ( set @ A ) @ T11 ) )
=> ( ! [B4: set @ A] :
( ( member @ ( set @ A ) @ B4 @ T11 )
=> ( topolo1002775350975398744n_open @ A @ B4 ) )
=> ~ ! [T12: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ T12 @ T11 )
=> ( ( finite_finite2 @ ( set @ A ) @ T12 )
=> ~ ( ord_less_eq @ ( set @ A ) @ S @ ( complete_Sup_Sup @ ( set @ A ) @ T12 ) ) ) ) ) ) ) ) ).
% compactE
thf(fact_5605_prod__filter__INF2,axiom,
! [B: $tType,C: $tType,A: $tType,J5: set @ A,A4: filter @ B,B2: A > ( filter @ C )] :
( ( J5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ A4 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ B2 @ J5 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I4: A] : ( prod_filter @ B @ C @ A4 @ ( B2 @ I4 ) )
@ J5 ) ) ) ) ).
% prod_filter_INF2
thf(fact_5606_prod__filter__INF1,axiom,
! [B: $tType,C: $tType,A: $tType,I5: set @ A,A4: A > ( filter @ B ),B2: filter @ C] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ A4 @ I5 ) ) @ B2 )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I4: A] : ( prod_filter @ B @ C @ ( A4 @ I4 ) @ B2 )
@ I5 ) ) ) ) ).
% prod_filter_INF1
thf(fact_5607_prod__filter__INF,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,I5: set @ A,J5: set @ B,A4: A > ( filter @ C ),B2: B > ( filter @ D )] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( J5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( prod_filter @ C @ D @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ A4 @ I5 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ B2 @ J5 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [I4: A] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ B @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [J3: B] : ( prod_filter @ C @ D @ ( A4 @ I4 ) @ ( B2 @ J3 ) )
@ J5 ) )
@ I5 ) ) ) ) ) ).
% prod_filter_INF
thf(fact_5608_prod__filter__eq__bot,axiom,
! [A: $tType,B: $tType,A4: filter @ A,B2: filter @ B] :
( ( ( prod_filter @ A @ B @ A4 @ B2 )
= ( bot_bot @ ( filter @ ( product_prod @ A @ B ) ) ) )
= ( ( A4
= ( bot_bot @ ( filter @ A ) ) )
| ( B2
= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% prod_filter_eq_bot
thf(fact_5609_prod__filter__mono,axiom,
! [A: $tType,B: $tType,F4: filter @ A,F11: filter @ A,G3: filter @ B,G7: filter @ B] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ F11 )
=> ( ( ord_less_eq @ ( filter @ B ) @ G3 @ G7 )
=> ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ F4 @ G3 ) @ ( prod_filter @ A @ B @ F11 @ G7 ) ) ) ) ).
% prod_filter_mono
thf(fact_5610_eventually__prod__same,axiom,
! [A: $tType,P: ( product_prod @ A @ A ) > $o,F4: filter @ A] :
( ( eventually @ ( product_prod @ A @ A ) @ P @ ( prod_filter @ A @ A @ F4 @ F4 ) )
= ( ? [Q7: A > $o] :
( ( eventually @ A @ Q7 @ F4 )
& ! [X2: A,Y3: A] :
( ( Q7 @ X2 )
=> ( ( Q7 @ Y3 )
=> ( P @ ( product_Pair @ A @ A @ X2 @ Y3 ) ) ) ) ) ) ) ).
% eventually_prod_same
thf(fact_5611_eventually__prod__filter,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,F4: filter @ A,G3: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ P @ ( prod_filter @ A @ B @ F4 @ G3 ) )
= ( ? [Pf: A > $o,Pg: B > $o] :
( ( eventually @ A @ Pf @ F4 )
& ( eventually @ B @ Pg @ G3 )
& ! [X2: A,Y3: B] :
( ( Pf @ X2 )
=> ( ( Pg @ Y3 )
=> ( P @ ( product_Pair @ A @ B @ X2 @ Y3 ) ) ) ) ) ) ) ).
% eventually_prod_filter
thf(fact_5612_prod__filter__mono__iff,axiom,
! [A: $tType,B: $tType,A4: filter @ A,B2: filter @ B,C2: filter @ A,D3: filter @ B] :
( ( A4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( B2
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ A4 @ B2 ) @ ( prod_filter @ A @ B @ C2 @ D3 ) )
= ( ( ord_less_eq @ ( filter @ A ) @ A4 @ C2 )
& ( ord_less_eq @ ( filter @ B ) @ B2 @ D3 ) ) ) ) ) ).
% prod_filter_mono_iff
thf(fact_5613_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B,G3: filter @ B,F4: filter @ A,G2: A > C,H7: filter @ C] :
( ( filterlim @ A @ B @ F3 @ G3 @ F4 )
=> ( ( filterlim @ A @ C @ G2 @ H7 @ F4 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( prod_filter @ B @ C @ G3 @ H7 )
@ F4 ) ) ) ).
% filterlim_Pair
thf(fact_5614_cauchy__filter__def,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo6773858410816713723filter @ A )
= ( ^ [F8: filter @ A] : ( ord_less_eq @ ( filter @ ( product_prod @ A @ A ) ) @ ( prod_filter @ A @ A @ F8 @ F8 ) @ ( topolo7806501430040627800ormity @ A ) ) ) ) ) ).
% cauchy_filter_def
thf(fact_5615_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 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( P @ ( product_Pair @ nat @ nat @ N2 @ M2 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_5616_eventually__prod1,axiom,
! [A: $tType,B: $tType,B2: filter @ A,P: B > $o,A4: filter @ B] :
( ( B2
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ^ [X2: B,Y3: A] : ( P @ X2 ) )
@ ( prod_filter @ B @ A @ A4 @ B2 ) )
= ( eventually @ B @ P @ A4 ) ) ) ).
% eventually_prod1
thf(fact_5617_eventually__prod2,axiom,
! [A: $tType,B: $tType,A4: filter @ A,P: B > $o,B2: filter @ B] :
( ( A4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X2: A] : P )
@ ( prod_filter @ A @ B @ A4 @ B2 ) )
= ( eventually @ B @ P @ B2 ) ) ) ).
% eventually_prod2
thf(fact_5618_tendsto__at__iff__sequentially,axiom,
! [C: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > C,A3: C,X: A,S3: set @ A] :
( ( filterlim @ A @ C @ F3 @ ( topolo7230453075368039082e_nhds @ C @ A3 ) @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( ! [X8: nat > A] :
( ! [I4: nat] : ( member @ A @ ( X8 @ I4 ) @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C @ ( comp @ A @ C @ nat @ F3 @ X8 ) @ ( topolo7230453075368039082e_nhds @ C @ A3 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% tendsto_at_iff_sequentially
thf(fact_5619_sequentially__imp__eventually__at__left,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B3: A,A3: A,P: A > $o] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ! [F6: nat > A] :
( ! [N4: nat] : ( ord_less @ A @ B3 @ ( F6 @ N4 ) )
=> ( ! [N4: nat] : ( ord_less @ A @ ( F6 @ N4 ) @ A3 )
=> ( ( order_mono @ nat @ A @ F6 )
=> ( ( filterlim @ nat @ A @ F6 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N2: nat] : ( P @ ( F6 @ N2 ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_left
thf(fact_5620_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X2: B,Z6: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y3 ) )
@ Z6
@ B2 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_5621_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F3: B > A,X4: set @ B,G1: A > C,G22: A > C] :
( ( ( image2 @ B @ A @ F3 @ X4 )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ X4 )
=> ( ( comp @ A @ C @ B @ G1 @ F3 @ X3 )
= ( comp @ A @ C @ B @ G22 @ F3 @ X3 ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_5622_funpow_Osimps_I2_J,axiom,
! [A: $tType,N: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F3 )
= ( comp @ A @ A @ A @ F3 @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ).
% funpow.simps(2)
thf(fact_5623_funpow__Suc__right,axiom,
! [A: $tType,N: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F3 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ F3 ) ) ).
% funpow_Suc_right
thf(fact_5624_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F3: A > B,A4: A,B2: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( inf_inf @ A @ A4 @ B2 ) ) @ ( inf_inf @ B @ ( F3 @ A4 ) @ ( F3 @ B2 ) ) ) ) ) ).
% mono_inf
thf(fact_5625_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F3: A > B,A4: A,B2: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F3 @ A4 ) @ ( F3 @ B2 ) ) @ ( F3 @ ( sup_sup @ A @ A4 @ B2 ) ) ) ) ) ).
% mono_sup
thf(fact_5626_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mono_invE
thf(fact_5627_incseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [X8: nat > A] :
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ A @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) ) ) ) ) ).
% incseq_def
thf(fact_5628_incseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,I: nat,J: nat] :
( ( order_mono @ nat @ A @ F3 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( F3 @ I ) @ ( F3 @ J ) ) ) ) ) ).
% incseqD
thf(fact_5629_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F2: A > B] :
! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% mono_def
thf(fact_5630_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( order_mono @ A @ B @ F3 ) ) ) ).
% monoI
thf(fact_5631_monoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ).
% monoE
thf(fact_5632_monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ).
% monoD
thf(fact_5633_incseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: nat > A,I: nat] :
( ( order_mono @ nat @ A @ A4 )
=> ( ord_less_eq @ A @ ( A4 @ I ) @ ( A4 @ ( suc @ I ) ) ) ) ) ).
% incseq_SucD
thf(fact_5634_incseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
=> ( order_mono @ nat @ A @ X4 ) ) ) ).
% incseq_SucI
thf(fact_5635_incseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [F2: nat > A] :
! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) ) ) ) ) ).
% incseq_Suc_iff
thf(fact_5636_comp__fun__commute__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite4664212375090638736ute_on @ A @ B )
= ( ^ [S7: set @ A,F2: A > B > B] :
! [X2: A,Y3: A] :
( ( member @ A @ X2 @ S7 )
=> ( ( member @ A @ Y3 @ S7 )
=> ( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X2 ) )
= ( comp @ B @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_5637_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,Y: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( member @ A @ X @ S )
=> ( ( member @ A @ Y @ S )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y ) @ ( F3 @ X ) )
= ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_5638_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [A: $tType,B: $tType,C: $tType,S: set @ A,F3: A > B > B,X: A,Y: A,G2: C > B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( member @ A @ X @ S )
=> ( ( member @ A @ Y @ S )
=> ( ( comp @ B @ B @ C @ ( F3 @ Y ) @ ( comp @ B @ B @ C @ ( F3 @ X ) @ G2 ) )
= ( comp @ B @ B @ C @ ( F3 @ X ) @ ( comp @ B @ B @ C @ ( F3 @ Y ) @ G2 ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_5639_comp__fun__commute__on_Ointro,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B] :
( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ S )
=> ( ( member @ A @ Y2 @ S )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y2 ) @ ( F3 @ X3 ) )
= ( comp @ B @ B @ B @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) ) ) )
=> ( finite4664212375090638736ute_on @ A @ B @ S @ F3 ) ) ).
% comp_fun_commute_on.intro
thf(fact_5640_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ) ).
% mono_strict_invE
thf(fact_5641_mono__pow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,N: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( order_mono @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ) ).
% mono_pow
thf(fact_5642_comp__funpow,axiom,
! [B: $tType,A: $tType,N: nat,F3: A > A] :
( ( compow @ ( ( B > A ) > B > A ) @ N @ ( comp @ A @ A @ B @ F3 ) )
= ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ).
% comp_funpow
thf(fact_5643_max__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,M: A,N: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_max @ B @ ( F3 @ M ) @ ( F3 @ N ) )
= ( F3 @ ( ord_max @ A @ M @ N ) ) ) ) ) ).
% max_of_mono
thf(fact_5644_mono__Suc,axiom,
order_mono @ nat @ nat @ suc ).
% mono_Suc
thf(fact_5645_comp__fun__commute_Ointro,axiom,
! [B: $tType,A: $tType,F3: A > B > B] :
( ! [Y2: A,X3: A] :
( ( comp @ B @ B @ B @ ( F3 @ Y2 ) @ ( F3 @ X3 ) )
= ( comp @ B @ B @ B @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( finite6289374366891150609ommute @ A @ B @ F3 ) ) ).
% comp_fun_commute.intro
thf(fact_5646_comp__fun__commute_Ocomp__fun__commute,axiom,
! [B: $tType,A: $tType,F3: A > B > B,Y: A,X: A] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y ) @ ( F3 @ X ) )
= ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ).
% comp_fun_commute.comp_fun_commute
thf(fact_5647_comp__fun__commute__def,axiom,
! [B: $tType,A: $tType] :
( ( finite6289374366891150609ommute @ A @ B )
= ( ^ [F2: A > B > B] :
! [Y3: A,X2: A] :
( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X2 ) )
= ( comp @ B @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% comp_fun_commute_def
thf(fact_5648_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_5649_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A3 ) ) ) ).
% mono_add
thf(fact_5650_funpow__mono,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: A > A,A4: A,B2: A,N: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F3 @ A4 ) @ ( compow @ ( A > A ) @ N @ F3 @ B2 ) ) ) ) ) ).
% funpow_mono
thf(fact_5651_comp__fun__commute__const,axiom,
! [B: $tType,A: $tType,F3: B > B] :
( finite6289374366891150609ommute @ A @ B
@ ^ [Uu3: A] : F3 ) ).
% comp_fun_commute_const
thf(fact_5652_mono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_mono @ nat @ A
@ ^ [I4: nat] : ( compow @ ( A > A ) @ I4 @ Q @ ( bot_bot @ A ) ) ) ) ) ).
% mono_funpow
thf(fact_5653_funpow__add,axiom,
! [A: $tType,M: nat,N: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M @ N ) @ F3 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M @ F3 ) @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ).
% funpow_add
thf(fact_5654_filterlim__filtercomap__iff,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B,G2: B > C,G3: filter @ C,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( filtercomap @ B @ C @ G2 @ G3 ) @ F4 )
= ( filterlim @ A @ C @ ( comp @ B @ C @ A @ G2 @ F3 ) @ G3 @ F4 ) ) ).
% filterlim_filtercomap_iff
thf(fact_5655_cclfp__lowerbound,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F3: A > A,A4: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ A4 ) @ A4 )
=> ( ord_less_eq @ A @ ( order_532582986084564980_cclfp @ A @ F3 ) @ A4 ) ) ) ) ).
% cclfp_lowerbound
thf(fact_5656_comp__fun__commute__filter__fold,axiom,
! [A: $tType,P: A > $o] :
( finite6289374366891150609ommute @ A @ ( set @ A )
@ ^ [X2: A,A14: set @ A] : ( if @ ( set @ A ) @ ( P @ X2 ) @ ( insert2 @ A @ X2 @ A14 ) @ A14 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_5657_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
! [B: $tType,A: $tType,F3: A > B > B,G2: A > nat] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( finite6289374366891150609ommute @ A @ B
@ ^ [X2: A] : ( compow @ ( B > B ) @ ( G2 @ X2 ) @ ( F3 @ X2 ) ) ) ) ).
% comp_fun_commute.comp_fun_commute_funpow
thf(fact_5658_mono__times__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( order_mono @ nat @ nat @ ( times_times @ nat @ N ) ) ) ).
% mono_times_nat
thf(fact_5659_mono__mult,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( order_mono @ A @ A @ ( times_times @ A @ A3 ) ) ) ) ).
% mono_mult
thf(fact_5660_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [F3: A > B,M: A,N: A,M7: B,N7: B] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ( image2 @ A @ B @ F3 @ ( set_or7035219750837199246ssThan @ A @ M @ N ) )
= ( set_or7035219750837199246ssThan @ B @ M7 @ N7 ) )
=> ( ( ord_less @ A @ M @ N )
=> ( ( F3 @ M )
= M7 ) ) ) ) ) ).
% mono_image_least
thf(fact_5661_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H: B > A,G2: C > B,A4: set @ C] :
( ( ( H @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X3: B,Y2: B] :
( ( H @ ( plus_plus @ B @ X3 @ Y2 ) )
= ( plus_plus @ A @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H @ G2 ) @ A4 )
= ( H @ ( groups7311177749621191930dd_sum @ C @ B @ G2 @ A4 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_5662_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [F3: A > A,P6: A,K: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ P6 @ ( F3 @ P6 ) )
=> ( ord_less_eq @ A @ P6 @ ( compow @ ( A > A ) @ K @ F3 @ ( top_top @ A ) ) ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_5663_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F3: A > A,P6: A,K: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ P6 ) @ P6 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F3 @ ( bot_bot @ A ) ) @ P6 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_5664_funpow__mono2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: A > A,I: nat,J: nat,X: A,Y: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ ( F3 @ X ) )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ I @ F3 @ X ) @ ( compow @ ( A > A ) @ J @ F3 @ Y ) ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_5665_incseq__bounded,axiom,
! [X4: nat > real,B2: real] :
( ( order_mono @ nat @ real @ X4 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( X4 @ I2 ) @ B2 )
=> ( bfun @ nat @ real @ X4 @ ( at_top @ nat ) ) ) ) ).
% incseq_bounded
thf(fact_5666_mono__SUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A4: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( F3 @ ( A4 @ X2 ) )
@ I5 ) )
@ ( F3 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A4 @ I5 ) ) ) ) ) ) ).
% mono_SUP
thf(fact_5667_mono__Sup,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ A4 ) ) @ ( F3 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% mono_Sup
thf(fact_5668_mono__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ A4 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ A4 ) ) ) ) ) ).
% mono_Inf
thf(fact_5669_mono__INF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A4: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A4 @ I5 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( F3 @ ( A4 @ X2 ) )
@ I5 ) ) ) ) ) ).
% mono_INF
thf(fact_5670_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,S: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ? [X5: A] :
( ( member @ A @ X5 @ S )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ S )
=> ( ord_less_eq @ A @ X5 @ Xa2 ) ) )
=> ( ( ord_Least @ B
@ ^ [Y3: B] : ( member @ B @ Y3 @ ( image2 @ A @ B @ F3 @ S ) ) )
= ( F3
@ ( ord_Least @ A
@ ^ [X2: A] : ( member @ A @ X2 @ S ) ) ) ) ) ) ) ).
% Least_mono
thf(fact_5671_antimono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_antimono @ nat @ A
@ ^ [I4: nat] : ( compow @ ( A > A ) @ I4 @ Q @ ( top_top @ A ) ) ) ) ) ).
% antimono_funpow
thf(fact_5672_incseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X4: nat > A,L5: A,N: nat] :
( ( order_mono @ nat @ A @ X4 )
=> ( ( filterlim @ nat @ A @ X4 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( X4 @ N ) @ L5 ) ) ) ) ).
% incseq_le
thf(fact_5673_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,H: B > C,G2: C > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X3: B,Y2: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( member @ B @ Y2 @ A4 )
=> ( ( X3 != Y2 )
=> ( ( ( H @ X3 )
= ( H @ Y2 ) )
=> ( ( G2 @ ( H @ X3 ) )
= ( zero_zero @ A ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G2 @ ( image2 @ B @ C @ H @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G2 @ H ) @ A4 ) ) ) ) ) ).
% sum.reindex_nontrivial
thf(fact_5674_funpow__increasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [M: nat,N: nat,F3: A > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F3 @ ( top_top @ A ) ) @ ( compow @ ( A > A ) @ M @ F3 @ ( top_top @ A ) ) ) ) ) ) ).
% funpow_increasing
thf(fact_5675_funpow__decreasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [M: nat,N: nat,F3: A > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M @ F3 @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N @ F3 @ ( bot_bot @ A ) ) ) ) ) ) ).
% funpow_decreasing
thf(fact_5676_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,H: B > C,G2: C > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X3: B,Y2: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( member @ B @ Y2 @ A4 )
=> ( ( X3 != Y2 )
=> ( ( ( H @ X3 )
= ( H @ Y2 ) )
=> ( ( G2 @ ( H @ X3 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G2 @ ( image2 @ B @ C @ H @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G2 @ H ) @ A4 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_5677_comp__fun__commute__def_H,axiom,
! [B: $tType,A: $tType] :
( ( finite6289374366891150609ommute @ A @ B )
= ( finite4664212375090638736ute_on @ A @ B @ ( top_top @ ( set @ A ) ) ) ) ).
% comp_fun_commute_def'
thf(fact_5678_incseq__convergent,axiom,
! [X4: nat > real,B2: real] :
( ( order_mono @ nat @ real @ X4 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( X4 @ I2 ) @ B2 )
=> ~ ! [L6: real] :
( ( filterlim @ nat @ real @ X4 @ ( topolo7230453075368039082e_nhds @ real @ L6 ) @ ( at_top @ nat ) )
=> ~ ! [I3: nat] : ( ord_less_eq @ real @ ( X4 @ I3 ) @ L6 ) ) ) ) ).
% incseq_convergent
thf(fact_5679_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F3 @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( lattic643756798349783984er_Max @ B @ ( image2 @ A @ B @ F3 @ A4 ) ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_5680_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I5: set @ C,G2: A > B,F3: C > A] :
( ( finite_finite2 @ C @ I5 )
=> ( ! [I2: C] :
( ( member @ C @ I2 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G2 @ ( F3 @ I2 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G2 @ ( image2 @ C @ A @ F3 @ I5 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G2 @ F3 ) @ I5 ) ) ) ) ) ).
% sum_image_le
thf(fact_5681_sum_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G2 @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc_shift
thf(fact_5682_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G2 @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_lessThan_Suc_shift
thf(fact_5683_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_5684_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_5685_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% sum.atLeastLessThan_shift_0
thf(fact_5686_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% prod.atLeastLessThan_shift_0
thf(fact_5687_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A4: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ A4 @ I5 ) )
=> ( ( I5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( F3 @ ( A4 @ X2 ) )
@ I5 ) )
@ ( F3 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A4 @ I5 ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_5688_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ A4 ) ) @ ( F3 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ) ).
% mono_cSup
thf(fact_5689_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ A4 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ A4 ) ) ) ) ) ) ) ).
% mono_cInf
thf(fact_5690_mono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A4: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ A4 @ I5 ) )
=> ( ( I5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A4 @ I5 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( F3 @ ( A4 @ X2 ) )
@ I5 ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_5691_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G2
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_5692_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G2
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_5693_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G2
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_5694_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G2
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_5695_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_5696_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_5697_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_5698_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A] :
( ( finite_finite2 @ A @ ( image2 @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( order_mono @ nat @ A @ F3 )
=> ( ! [N3: nat] :
( ( ( F3 @ N3 )
= ( F3 @ ( suc @ N3 ) ) )
=> ( ( F3 @ ( suc @ N3 ) )
= ( F3 @ ( suc @ ( suc @ N3 ) ) ) ) )
=> ? [N9: nat] :
( ! [N4: nat] :
( ( ord_less_eq @ nat @ N4 @ N9 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N9 )
=> ( ( ord_less @ nat @ M3 @ N4 )
=> ( ord_less @ A @ ( F3 @ M3 ) @ ( F3 @ N4 ) ) ) ) )
& ! [N4: nat] :
( ( ord_less_eq @ nat @ N9 @ N4 )
=> ( ( F3 @ N9 )
= ( F3 @ N4 ) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_5699_tendsto__at__left__sequentially,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [B3: B,A3: B,X4: B > A,L5: A] :
( ( ord_less @ B @ B3 @ A3 )
=> ( ! [S2: nat > B] :
( ! [N4: nat] : ( ord_less @ B @ ( S2 @ N4 ) @ A3 )
=> ( ! [N4: nat] : ( ord_less @ B @ B3 @ ( S2 @ N4 ) )
=> ( ( order_mono @ nat @ B @ S2 )
=> ( ( filterlim @ nat @ B @ S2 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( X4 @ ( S2 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L5 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ A @ X4 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( topolo174197925503356063within @ B @ A3 @ ( set_ord_lessThan @ B @ A3 ) ) ) ) ) ) ).
% tendsto_at_left_sequentially
thf(fact_5700_lim__at__infinity__0,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,L: A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) )
= ( filterlim @ A @ A @ ( comp @ A @ A @ A @ F3 @ ( inverse_inverse @ A ) ) @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% lim_at_infinity_0
thf(fact_5701_continuous__at__Sup__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Sup_Sup @ A @ S ) @ ( set_ord_lessThan @ A @ ( complete_Sup_Sup @ A @ S ) ) ) @ F3 )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S )
=> ( ( F3 @ ( complete_Sup_Sup @ A @ S ) )
= ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ S ) ) ) ) ) ) ) ) ).
% continuous_at_Sup_mono
thf(fact_5702_continuous__at__Inf__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Inf_Inf @ A @ S ) @ ( set_ord_greaterThan @ A @ ( complete_Inf_Inf @ A @ S ) ) ) @ F3 )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S )
=> ( ( F3 @ ( complete_Inf_Inf @ A @ S ) )
= ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ S ) ) ) ) ) ) ) ) ).
% continuous_at_Inf_mono
thf(fact_5703_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G2: B > A,Y6: set @ B,X4: set @ A,F4: filter @ B,F3: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G2 @ Y6 ) @ X4 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( member @ B @ X2 @ Y6 )
@ F4 )
=> ( ( map_filter_on @ A @ C @ X4 @ F3 @ ( map_filter_on @ B @ A @ Y6 @ G2 @ F4 ) )
= ( map_filter_on @ B @ C @ Y6 @ ( comp @ A @ C @ B @ F3 @ G2 ) @ F4 ) ) ) ) ).
% map_filter_on_comp
thf(fact_5704_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= Ys2 )
= ( ? [F2: nat > nat] :
( ( order_mono @ nat @ nat @ F2 )
& ( ( image2 @ nat @ nat @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I4 )
= ( nth @ A @ Ys2 @ ( F2 @ I4 ) ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I4 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) )
= ( ( F2 @ I4 )
= ( F2 @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_5705_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 ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R )
@ ( collect @ nat
@ ^ [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_5706_finite__relpow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),N: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% finite_relpow
thf(fact_5707_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F3: A > C,G2: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu3: C] : ( bot_bot @ ( set @ B ) )
@ F3 )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image2 @ D @ B @ G2 )
@ ^ [Uu3: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_5708_mono__Un,axiom,
! [B: $tType,A: $tType,F3: ( set @ A ) > ( set @ B ),A4: set @ A,B2: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F3 )
=> ( ord_less_eq @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F3 @ A4 ) @ ( F3 @ B2 ) ) @ ( F3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% mono_Un
thf(fact_5709_mono__Int,axiom,
! [B: $tType,A: $tType,F3: ( set @ A ) > ( set @ B ),A4: set @ A,B2: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F3 )
=> ( ord_less_eq @ ( set @ B ) @ ( F3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) @ ( inf_inf @ ( set @ B ) @ ( F3 @ A4 ) @ ( F3 @ B2 ) ) ) ) ).
% mono_Int
thf(fact_5710_comp__fun__commute_Ocomp__comp__fun__commute,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > B > B,G2: C > A] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( finite6289374366891150609ommute @ C @ B @ ( comp @ A @ ( B > B ) @ C @ F3 @ G2 ) ) ) ).
% comp_fun_commute.comp_comp_fun_commute
thf(fact_5711_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_5712_relpow__0__I,axiom,
! [A: $tType,X: A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R ) ) ).
% relpow_0_I
thf(fact_5713_relpow__0__E,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R ) )
=> ( X = Y ) ) ).
% relpow_0_E
thf(fact_5714_relpow__E,axiom,
! [A: $tType,X: A,Z: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z ) )
=> ~ ! [Y2: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M4 @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z ) @ R ) ) ) ) ) ).
% relpow_E
thf(fact_5715_relpow__E2,axiom,
! [A: $tType,X: A,Z: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z ) )
=> ~ ! [Y2: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M4 @ R ) ) ) ) ) ) ).
% relpow_E2
thf(fact_5716_eventually__map__filter__on,axiom,
! [B: $tType,A: $tType,X4: set @ A,F4: filter @ A,P: B > $o,F3: A > B] :
( ( eventually @ A
@ ^ [X2: A] : ( member @ A @ X2 @ X4 )
@ F4 )
=> ( ( eventually @ B @ P @ ( map_filter_on @ A @ B @ X4 @ F3 @ F4 ) )
= ( eventually @ A
@ ^ [X2: A] :
( ( P @ ( F3 @ X2 ) )
& ( member @ A @ X2 @ X4 ) )
@ F4 ) ) ) ).
% eventually_map_filter_on
thf(fact_5717_relpow__empty,axiom,
! [A: $tType,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% relpow_empty
thf(fact_5718_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B2: set @ ( set @ B ),G2: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ B2 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [A16: set @ B] :
( ( member @ ( set @ B ) @ A16 @ B2 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B2 )
=> ( ( A16 != A25 )
=> ! [X3: B] :
( ( member @ B @ X3 @ A16 )
=> ( ( member @ B @ X3 @ A25 )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( complete_Sup_Sup @ ( set @ B ) @ B2 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G2 @ B2 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_5719_sum_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups7311177749621191930dd_sum @ B @ A )
= ( ^ [G: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G ) @ ( zero_zero @ A ) ) ) ) ) ).
% sum.eq_fold
thf(fact_5720_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
! [B: $tType,A: $tType,C: $tType,S: set @ A,F3: A > B > B,G2: C > A,R: set @ C] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G2 @ ( top_top @ ( set @ C ) ) ) @ S )
=> ( finite4664212375090638736ute_on @ C @ B @ R @ ( comp @ A @ ( B > B ) @ C @ F3 @ G2 ) ) ) ) ).
% comp_fun_commute_on.comp_comp_fun_commute_on
thf(fact_5721_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_5722_infinite__int__iff__infinite__nat__abs,axiom,
! [S: set @ int] :
( ( ~ ( finite_finite2 @ int @ S ) )
= ( ~ ( finite_finite2 @ nat @ ( image2 @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ S ) ) ) ) ).
% infinite_int_iff_infinite_nat_abs
thf(fact_5723_relpow__fun__conv,axiom,
! [A: $tType,A3: A,B3: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
= ( ? [F2: nat > A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= A3 )
& ( ( F2 @ N )
= B3 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ I4 ) @ ( F2 @ ( suc @ I4 ) ) ) @ R ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_5724_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: set @ ( set @ B ),G2: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C2 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C2 )
=> ! [Xa2: set @ B] :
( ( member @ ( set @ B ) @ Xa2 @ C2 )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf @ ( set @ B ) @ X3 @ Xa2 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( complete_Sup_Sup @ ( set @ B ) @ C2 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G2 @ C2 ) ) ) ) ) ).
% prod.Union_disjoint
thf(fact_5725_sup__SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B2: A,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( sup_sup @ A @ B2 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F3 ) @ B2 @ A4 ) ) ) ) ).
% sup_SUP_fold_sup
thf(fact_5726_inf__INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B2: A,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( inf_inf @ A @ B2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F3 ) @ B2 @ A4 ) ) ) ) ).
% inf_INF_fold_inf
thf(fact_5727_sum_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B2: set @ ( set @ B ),G2: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ B2 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [A16: set @ B] :
( ( member @ ( set @ B ) @ A16 @ B2 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B2 )
=> ( ( A16 != A25 )
=> ! [X3: B] :
( ( member @ B @ X3 @ A16 )
=> ( ( member @ B @ X3 @ A25 )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( complete_Sup_Sup @ ( set @ B ) @ B2 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G2 @ B2 ) ) ) ) ) ).
% sum.Union_comp
thf(fact_5728_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_5729_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 ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R )
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ).
% relpow_finite_bounded
thf(fact_5730_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F3 ) @ ( bot_bot @ A ) @ A4 ) ) ) ) ).
% SUP_fold_sup
thf(fact_5731_INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F3 ) @ ( top_top @ A ) @ A4 ) ) ) ) ).
% INF_fold_inf
thf(fact_5732_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C2: set @ ( set @ B ),G2: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C2 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C2 )
=> ! [Xa2: set @ B] :
( ( member @ ( set @ B ) @ Xa2 @ C2 )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf @ ( set @ B ) @ X3 @ Xa2 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( complete_Sup_Sup @ ( set @ B ) @ C2 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G2 @ C2 ) ) ) ) ) ).
% sum.Union_disjoint
thf(fact_5733_comp__fun__commute__Pow__fold,axiom,
! [A: $tType] :
( finite6289374366891150609ommute @ A @ ( set @ ( set @ A ) )
@ ^ [X2: A,A6: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A6 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X2 ) @ A6 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_5734_filterlim__at__top__iff__inverse__0,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
= ( filterlim @ A @ real @ ( comp @ real @ real @ A @ ( inverse_inverse @ real ) @ F3 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% filterlim_at_top_iff_inverse_0
thf(fact_5735_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N2: nat,R6: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [I4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I4 @ R6 )
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I4 )
& ( ord_less_eq @ nat @ I4 @ ( suc @ N2 ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_5736_trancl__finite__eq__relpow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( transitive_trancl @ A @ R )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R )
@ ( collect @ nat
@ ^ [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_5737_nonneg__incseq__Bseq__subseq__iff,axiom,
! [F3: nat > real,G2: nat > nat] :
( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) )
=> ( ( order_mono @ nat @ real @ F3 )
=> ( ( order_strict_mono @ nat @ nat @ G2 )
=> ( ( bfun @ nat @ real
@ ^ [X2: nat] : ( F3 @ ( G2 @ X2 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ real @ F3 @ ( at_top @ nat ) ) ) ) ) ) ).
% nonneg_incseq_Bseq_subseq_iff
thf(fact_5738_ntrancl__Zero,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( zero_zero @ nat ) @ R )
= R ) ).
% ntrancl_Zero
thf(fact_5739_strict__mono__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( order_mono @ A @ B @ F3 ) ) ) ).
% strict_mono_mono
thf(fact_5740_infinite__enumerate,axiom,
! [S: set @ nat] :
( ~ ( finite_finite2 @ nat @ S )
=> ? [R3: nat > nat] :
( ( order_strict_mono @ nat @ nat @ R3 )
& ! [N4: nat] : ( member @ nat @ ( R3 @ N4 ) @ S ) ) ) ).
% infinite_enumerate
thf(fact_5741_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less @ A @ X @ Y )
=> ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ).
% strict_monoD
thf(fact_5742_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( order_strict_mono @ A @ B @ F3 ) ) ) ).
% strict_monoI
thf(fact_5743_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F2: A > B] :
! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ B @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_5744_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% strict_mono_less
thf(fact_5745_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ) ).
% strict_mono_eq
thf(fact_5746_strict__mono__imp__increasing,axiom,
! [F3: nat > nat,N: nat] :
( ( order_strict_mono @ nat @ nat @ F3 )
=> ( ord_less_eq @ nat @ N @ ( F3 @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_5747_strict__mono__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [R2: A > B,M: A,N: A] :
( ( order_strict_mono @ A @ B @ R2 )
=> ( ( ord_less_eq @ A @ M @ N )
=> ( ord_less_eq @ B @ ( R2 @ M ) @ ( R2 @ N ) ) ) ) ) ).
% strict_mono_leD
thf(fact_5748_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% strict_mono_less_eq
thf(fact_5749_trancl__mono,axiom,
! [A: $tType,P6: product_prod @ A @ A,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P6 @ ( transitive_trancl @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 )
=> ( member @ ( product_prod @ A @ A ) @ P6 @ ( transitive_trancl @ A @ S3 ) ) ) ) ).
% trancl_mono
thf(fact_5750_strict__mono__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_strict_mono @ nat @ A )
= ( ^ [F2: nat > A] :
! [N2: nat] : ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) ) ) ) ) ).
% strict_mono_Suc_iff
thf(fact_5751_trancl__power,axiom,
! [A: $tType,P6: product_prod @ A @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P6 @ ( transitive_trancl @ A @ R ) )
= ( ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( member @ ( product_prod @ A @ A ) @ P6 @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ) ).
% trancl_power
thf(fact_5752_strict__mono__enumerate,axiom,
! [S: set @ nat] :
( ~ ( finite_finite2 @ nat @ S )
=> ( order_strict_mono @ nat @ nat @ ( infini527867602293511546merate @ nat @ S ) ) ) ).
% strict_mono_enumerate
thf(fact_5753_summable__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G2: nat > nat,F3: nat > A] :
( ( order_strict_mono @ nat @ nat @ G2 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ ( image2 @ nat @ nat @ G2 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( summable @ A
@ ^ [N2: nat] : ( F3 @ ( G2 @ N2 ) ) )
= ( summable @ A @ F3 ) ) ) ) ) ).
% summable_mono_reindex
thf(fact_5754_sums__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G2: nat > nat,F3: nat > A,C3: A] :
( ( order_strict_mono @ nat @ nat @ G2 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ ( image2 @ nat @ nat @ G2 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [N2: nat] : ( F3 @ ( G2 @ N2 ) )
@ C3 )
= ( sums @ A @ F3 @ C3 ) ) ) ) ) ).
% sums_mono_reindex
thf(fact_5755_suminf__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G2: nat > nat,F3: nat > A] :
( ( order_strict_mono @ nat @ nat @ G2 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ ( image2 @ nat @ nat @ G2 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( F3 @ ( G2 @ N2 ) ) )
= ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_mono_reindex
thf(fact_5756_increasing__Bseq__subseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G2: nat > nat] :
( ! [X3: nat,Y2: nat] :
( ( ord_less_eq @ nat @ X3 @ Y2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X3 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ Y2 ) ) ) )
=> ( ( order_strict_mono @ nat @ nat @ G2 )
=> ( ( bfun @ nat @ A
@ ^ [X2: nat] : ( F3 @ ( G2 @ X2 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ) ) ).
% increasing_Bseq_subseq_iff
thf(fact_5757_coinduct3__mono__lemma,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [F3: A > ( set @ B ),X4: set @ B,B2: set @ B] :
( ( order_mono @ A @ ( set @ B ) @ F3 )
=> ( order_mono @ A @ ( set @ B )
@ ^ [X2: A] : ( sup_sup @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F3 @ X2 ) @ X4 ) @ B2 ) ) ) ) ).
% coinduct3_mono_lemma
thf(fact_5758_compact__imp__fip__image,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,I5: set @ B,F3: B > ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( topolo7761053866217962861closed @ A @ ( F3 @ I2 ) ) )
=> ( ! [I8: set @ B] :
( ( finite_finite2 @ B @ I8 )
=> ( ( ord_less_eq @ ( set @ B ) @ I8 @ I5 )
=> ( ( inf_inf @ ( set @ A ) @ S3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ I8 ) ) )
!= ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ S3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ I5 ) ) )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% compact_imp_fip_image
thf(fact_5759_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( inj_on @ real @ real
@ ^ [Y3: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y3 ) @ ( power_power @ real @ ( abs_abs @ real @ Y3 ) @ N ) )
@ ( top_top @ ( set @ real ) ) ) ) ).
% inj_sgn_power
thf(fact_5760_inj__on__empty,axiom,
! [B: $tType,A: $tType,F3: A > B] : ( inj_on @ A @ B @ F3 @ ( bot_bot @ ( set @ A ) ) ) ).
% inj_on_empty
thf(fact_5761_closed__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo7761053866217962861closed @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% closed_empty
thf(fact_5762_closed__Un,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,T4: set @ A] :
( ( topolo7761053866217962861closed @ A @ S )
=> ( ( topolo7761053866217962861closed @ A @ T4 )
=> ( topolo7761053866217962861closed @ A @ ( sup_sup @ ( set @ A ) @ S @ T4 ) ) ) ) ) ).
% closed_Un
thf(fact_5763_closed__singleton,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [A3: A] : ( topolo7761053866217962861closed @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% closed_singleton
thf(fact_5764_inj__mult__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_mult_left
thf(fact_5765_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A
@ ^ [B5: A] : ( divide_divide @ A @ B5 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_5766_closed__UN,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A4: set @ B,B2: B > ( set @ A )] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( topolo7761053866217962861closed @ A @ ( B2 @ X3 ) ) )
=> ( topolo7761053866217962861closed @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A4 ) ) ) ) ) ) ).
% closed_UN
thf(fact_5767_inj__on__insert,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: A,A4: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert2 @ A @ A3 @ A4 ) )
= ( ( inj_on @ A @ B @ F3 @ A4 )
& ~ ( member @ B @ ( F3 @ A3 ) @ ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_5768_subset__image__inj,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: B > A,T4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( image2 @ B @ A @ F3 @ T4 ) )
= ( ? [U5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U5 @ T4 )
& ( inj_on @ B @ A @ F3 @ U5 )
& ( S
= ( image2 @ B @ A @ F3 @ U5 ) ) ) ) ) ).
% subset_image_inj
thf(fact_5769_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,B2: set @ A,A3: A,A4: set @ A] :
( ( inj_on @ A @ B @ F3 @ B2 )
=> ( ( member @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ B @ ( F3 @ A3 ) @ ( image2 @ A @ B @ F3 @ A4 ) )
= ( member @ A @ A3 @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_5770_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,C2: set @ A,A4: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F3 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ( ( image2 @ A @ B @ F3 @ A4 )
= ( image2 @ A @ B @ F3 @ B2 ) )
= ( A4 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_5771_finite__image__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F3 @ A4 )
=> ( ( finite_finite2 @ B @ ( image2 @ A @ B @ F3 @ A4 ) )
= ( finite_finite2 @ A @ A4 ) ) ) ).
% finite_image_iff
thf(fact_5772_finite__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ( inj_on @ B @ A @ F3 @ A4 )
=> ( finite_finite2 @ B @ A4 ) ) ) ).
% finite_imageD
thf(fact_5773_inj__img__insertE,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A,X: B,B2: set @ B] :
( ( inj_on @ A @ B @ F3 @ A4 )
=> ( ~ ( member @ B @ X @ B2 )
=> ( ( ( insert2 @ B @ X @ B2 )
= ( image2 @ A @ B @ F3 @ A4 ) )
=> ~ ! [X16: A,A8: set @ A] :
( ~ ( member @ A @ X16 @ A8 )
=> ( ( A4
= ( insert2 @ A @ X16 @ A8 ) )
=> ( ( X
= ( F3 @ X16 ) )
=> ( B2
!= ( image2 @ A @ B @ F3 @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_5774_inj__on__Un__image__eq__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ( ( ( image2 @ A @ B @ F3 @ A4 )
= ( image2 @ A @ B @ F3 @ B2 ) )
= ( A4 = B2 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_5775_card__image,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F3 @ A4 )
=> ( ( finite_card @ B @ ( image2 @ A @ B @ F3 @ A4 ) )
= ( finite_card @ A @ A4 ) ) ) ).
% card_image
thf(fact_5776_inj__on__image__Fpow,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F3 @ A4 )
=> ( inj_on @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F3 ) @ ( finite_Fpow @ A @ A4 ) ) ) ).
% inj_on_image_Fpow
thf(fact_5777_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,B2: set @ A,A4: set @ A] :
( ( inj_on @ A @ B @ F3 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ ( image2 @ A @ B @ F3 @ B2 ) ) ) ) ).
% inj_on_strict_subset
thf(fact_5778_inj__on__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F3 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( inj_on @ A @ B @ F3 @ B2 ) ) ) ).
% inj_on_subset
thf(fact_5779_subset__inj__on,axiom,
! [B: $tType,A: $tType,F3: A > B,B2: set @ A,A4: set @ A] :
( ( inj_on @ A @ B @ F3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( inj_on @ A @ B @ F3 @ A4 ) ) ) ).
% subset_inj_on
thf(fact_5780_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A4: set @ A,F3: A > B] :
( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( ( F3 @ X3 )
!= ( F3 @ Y2 ) ) ) ) )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X3 ) ) ) )
=> ( inj_on @ A @ B @ F3 @ A4 ) ) ) ) ).
% linorder_inj_onI
thf(fact_5781_continuous__on__closed__Un,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S3: set @ A,T2: set @ A,F3: A > B] :
( ( topolo7761053866217962861closed @ A @ S3 )
=> ( ( topolo7761053866217962861closed @ A @ T2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ T2 @ F3 )
=> ( topolo81223032696312382ous_on @ A @ B @ ( sup_sup @ ( set @ A ) @ S3 @ T2 ) @ F3 ) ) ) ) ) ) ).
% continuous_on_closed_Un
thf(fact_5782_finite__inverse__image__gen,axiom,
! [A: $tType,B: $tType,A4: set @ A,F3: B > A,D3: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( inj_on @ B @ A @ F3 @ D3 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ D3 )
& ( member @ A @ ( F3 @ J3 ) @ A4 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_5783_finite__imp__closed,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( topolo7761053866217962861closed @ A @ S ) ) ) ).
% finite_imp_closed
thf(fact_5784_closed__insert,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [S: set @ A,A3: A] :
( ( topolo7761053866217962861closed @ A @ S )
=> ( topolo7761053866217962861closed @ A @ ( insert2 @ A @ A3 @ S ) ) ) ) ).
% closed_insert
thf(fact_5785_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F3: A > B] :
( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( F3 @ X3 )
!= ( F3 @ Y2 ) ) )
=> ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_5786_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,A4: set @ A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ A4 ) ) ) ).
% inj_on_mult
thf(fact_5787_inj__fn,axiom,
! [A: $tType,F3: A > A,N: nat] :
( ( inj_on @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fn
thf(fact_5788_finite__inverse__image,axiom,
! [A: $tType,B: $tType,A4: set @ A,F3: B > A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( inj_on @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] : ( member @ A @ ( F3 @ J3 ) @ A4 ) ) ) ) ) ).
% finite_inverse_image
thf(fact_5789_continuous__on__If,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S3: set @ A,T2: set @ A,F3: A > B,G2: A > B,P: A > $o] :
( ( topolo7761053866217962861closed @ A @ S3 )
=> ( ( topolo7761053866217962861closed @ A @ T2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ T2 @ G2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ~ ( P @ X3 )
=> ( ( F3 @ X3 )
= ( G2 @ X3 ) ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ T2 )
=> ( ( P @ X3 )
=> ( ( F3 @ X3 )
= ( G2 @ X3 ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ ( sup_sup @ ( set @ A ) @ S3 @ T2 )
@ ^ [X2: A] : ( if @ B @ ( P @ X2 ) @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% continuous_on_If
thf(fact_5790_continuous__on__cases,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S3: set @ A,T2: set @ A,F3: A > B,G2: A > B,P: A > $o] :
( ( topolo7761053866217962861closed @ A @ S3 )
=> ( ( topolo7761053866217962861closed @ A @ T2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ T2 @ G2 )
=> ( ! [X3: A] :
( ( ( ( member @ A @ X3 @ S3 )
& ~ ( P @ X3 ) )
| ( ( member @ A @ X3 @ T2 )
& ( P @ X3 ) ) )
=> ( ( F3 @ X3 )
= ( G2 @ X3 ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ ( sup_sup @ ( set @ A ) @ S3 @ T2 )
@ ^ [X2: A] : ( if @ B @ ( P @ X2 ) @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_on_cases
thf(fact_5791_finite__UNIV__inj__surj,axiom,
! [A: $tType,F3: A > A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( inj_on @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_5792_finite__UNIV__surj__inj,axiom,
! [A: $tType,F3: A > A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image2 @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_5793_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ ( image2 @ A @ B @ F3 @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_5794_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A4: set @ A,A17: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F2: A > B] :
( ( inj_on @ A @ B @ F2 @ A4 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ A17 ) ) )
= ( ? [G: B > A] :
( ( image2 @ B @ A @ G @ A17 )
= A4 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_5795_finite__surj__inj,axiom,
! [A: $tType,A4: set @ A,F3: A > A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( image2 @ A @ A @ F3 @ A4 ) )
=> ( inj_on @ A @ A @ F3 @ A4 ) ) ) ).
% finite_surj_inj
thf(fact_5796_inj__on__finite,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A,B2: set @ B] :
( ( inj_on @ A @ B @ F3 @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ B2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ A @ A4 ) ) ) ) ).
% inj_on_finite
thf(fact_5797_endo__inj__surj,axiom,
! [A: $tType,A4: set @ A,F3: A > A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ A @ A @ F3 @ A4 ) @ A4 )
=> ( ( inj_on @ A @ A @ F3 @ A4 )
=> ( ( image2 @ A @ A @ F3 @ A4 )
= A4 ) ) ) ) ).
% endo_inj_surj
thf(fact_5798_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F3: A > B,C2: set @ A,A4: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F3 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ( image2 @ A @ B @ F3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ ( image2 @ A @ B @ F3 @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_5799_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F3: A > B,C2: set @ A,A4: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F3 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ ( image2 @ A @ B @ F3 @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_5800_inj__on__iff__eq__card,axiom,
! [B: $tType,A: $tType,A4: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( inj_on @ A @ B @ F3 @ A4 )
= ( ( finite_card @ B @ ( image2 @ A @ B @ F3 @ A4 ) )
= ( finite_card @ A @ A4 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_5801_eq__card__imp__inj__on,axiom,
! [B: $tType,A: $tType,A4: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( finite_card @ B @ ( image2 @ A @ B @ F3 @ A4 ) )
= ( finite_card @ A @ A4 ) )
=> ( inj_on @ A @ B @ F3 @ A4 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_5802_pigeonhole,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B] :
( ( ord_less @ nat @ ( finite_card @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( finite_card @ B @ A4 ) )
=> ~ ( inj_on @ B @ A @ F3 @ A4 ) ) ).
% pigeonhole
thf(fact_5803_continuous__inj__imp__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo8458572112393995274pology @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A3: A,X: A,B3: A,F3: A > B] :
( ( ord_less @ A @ A3 @ X )
=> ( ( ord_less @ A @ X @ B3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F3 )
=> ( ( inj_on @ A @ B @ F3 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
=> ( ( ( ord_less @ B @ ( F3 @ A3 ) @ ( F3 @ X ) )
& ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ B3 ) ) )
| ( ( ord_less @ B @ ( F3 @ B3 ) @ ( F3 @ X ) )
& ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ A3 ) ) ) ) ) ) ) ) ) ).
% continuous_inj_imp_mono
thf(fact_5804_fold__image,axiom,
! [C: $tType,B: $tType,A: $tType,G2: A > B,A4: set @ A,F3: B > C > C,Z: C] :
( ( inj_on @ A @ B @ G2 @ A4 )
=> ( ( finite_fold @ B @ C @ F3 @ Z @ ( image2 @ A @ B @ G2 @ A4 ) )
= ( finite_fold @ A @ C @ ( comp @ B @ ( C > C ) @ A @ F3 @ G2 ) @ Z @ A4 ) ) ) ).
% fold_image
thf(fact_5805_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A,X: B,B2: set @ A] :
( ( inj_on @ A @ B @ F3 @ A4 )
=> ( ( member @ B @ X @ ( image2 @ A @ B @ F3 @ A4 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A4 @ F3 @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_5806_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X8: $o > A,Y7: $o > A] :
( ( ord_less_eq @ A @ ( X8 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq @ A @ ( X8 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_5807_inj__on__UNION__chain,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,A4: A > ( set @ B ),F3: B > C] :
( ! [I2: A,J2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ( member @ A @ J2 @ I5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A4 @ I2 ) @ ( A4 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A4 @ J2 ) @ ( A4 @ I2 ) ) ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( inj_on @ B @ C @ F3 @ ( A4 @ I2 ) ) )
=> ( inj_on @ B @ C @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I5 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_5808_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,F3: B > C,A4: A > ( set @ B )] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( inj_on @ B @ C @ F3 @ ( A4 @ I2 ) ) )
=> ( inj_on @ B @ C @ F3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I5 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_5809_closed__Collect__le,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,G2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G2 )
=> ( topolo7761053866217962861closed @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% closed_Collect_le
thf(fact_5810_card__bij__eq,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ A,B2: set @ B,G2: B > A] :
( ( inj_on @ A @ B @ F3 @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ B2 )
=> ( ( inj_on @ B @ A @ G2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G2 @ B2 ) @ A4 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( finite_card @ A @ A4 )
= ( finite_card @ B @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_5811_surjective__iff__injective__gen,axiom,
! [B: $tType,A: $tType,S: set @ A,T4: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ( ( finite_card @ A @ S )
= ( finite_card @ B @ T4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ S ) @ T4 )
=> ( ( ! [X2: B] :
( ( member @ B @ X2 @ T4 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ S )
& ( ( F3 @ Y3 )
= X2 ) ) ) )
= ( inj_on @ A @ B @ F3 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_5812_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_5813_t3__space,axiom,
! [A: $tType] :
( ( topological_t3_space @ A )
=> ! [S: set @ A,Y: A] :
( ( topolo7761053866217962861closed @ A @ S )
=> ( ~ ( member @ A @ Y @ S )
=> ? [U6: set @ A,V5: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U6 )
& ( topolo1002775350975398744n_open @ A @ V5 )
& ( member @ A @ Y @ U6 )
& ( ord_less_eq @ ( set @ A ) @ S @ V5 )
& ( ( inf_inf @ ( set @ A ) @ U6 @ V5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% t3_space
thf(fact_5814_t4__space,axiom,
! [A: $tType] :
( ( topological_t4_space @ A )
=> ! [S: set @ A,T4: set @ A] :
( ( topolo7761053866217962861closed @ A @ S )
=> ( ( topolo7761053866217962861closed @ A @ T4 )
=> ( ( ( inf_inf @ ( set @ A ) @ S @ T4 )
= ( bot_bot @ ( set @ A ) ) )
=> ? [U6: set @ A,V5: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U6 )
& ( topolo1002775350975398744n_open @ A @ V5 )
& ( ord_less_eq @ ( set @ A ) @ S @ U6 )
& ( ord_less_eq @ ( set @ A ) @ T4 @ V5 )
& ( ( inf_inf @ ( set @ A ) @ U6 @ V5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% t4_space
thf(fact_5815_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A,G2: A > B,B2: set @ A] :
( ( inj_on @ A @ B @ F3 @ A4 )
=> ( ( inj_on @ A @ B @ G2 @ B2 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ ( image2 @ A @ B @ G2 @ B2 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ A4 ) @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_5816_image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > B,C2: set @ A,A4: set @ C,B2: C > ( set @ A ),J: C] :
( ( inj_on @ A @ B @ F3 @ C2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( B2 @ X3 ) @ C2 ) )
=> ( ( member @ C @ J @ A4 )
=> ( ( image2 @ A @ B @ F3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B2 @ A4 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X2: C] : ( image2 @ A @ B @ F3 @ ( B2 @ X2 ) )
@ A4 ) ) ) ) ) ) ).
% image_INT
thf(fact_5817_nhds__closed,axiom,
! [A: $tType] :
( ( topological_t3_space @ A )
=> ! [X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( topolo1002775350975398744n_open @ A @ A4 )
=> ? [A8: set @ A] :
( ( member @ A @ X @ A8 )
& ( topolo7761053866217962861closed @ A @ A8 )
& ( ord_less_eq @ ( set @ A ) @ A8 @ A4 )
& ( eventually @ A
@ ^ [Y3: A] : ( member @ A @ Y3 @ A8 )
@ ( topolo7230453075368039082e_nhds @ A @ X ) ) ) ) ) ) ).
% nhds_closed
thf(fact_5818_continuous__on__closed__Union,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [I5: set @ A,U3: A > ( set @ B ),F3: B > C] :
( ( finite_finite2 @ A @ I5 )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( topolo7761053866217962861closed @ B @ ( U3 @ I2 ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( topolo81223032696312382ous_on @ B @ C @ ( U3 @ I2 ) @ F3 ) )
=> ( topolo81223032696312382ous_on @ B @ C @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ U3 @ I5 ) ) @ F3 ) ) ) ) ) ).
% continuous_on_closed_Union
thf(fact_5819_card__le__inj,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B2 ) )
=> ? [F6: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F6 @ A4 ) @ B2 )
& ( inj_on @ A @ B @ F6 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_5820_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ A,B2: set @ B] :
( ( inj_on @ A @ B @ F3 @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ B2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_5821_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ? [F2: A > B] :
( ( inj_on @ A @ B @ F2 @ A4 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B2 ) ) )
= ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_5822_inj__on__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ( inj_on @ A @ B @ F3 @ A4 )
& ( inj_on @ A @ B @ F3 @ B2 )
& ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) @ ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_5823_log__inj,axiom,
! [B3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( inj_on @ real @ real @ ( log @ B3 ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ).
% log_inj
thf(fact_5824_compact__imp__fip,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F4: set @ ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ! [T5: set @ A] :
( ( member @ ( set @ A ) @ T5 @ F4 )
=> ( topolo7761053866217962861closed @ A @ T5 ) )
=> ( ! [F15: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F15 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F15 @ F4 )
=> ( ( inf_inf @ ( set @ A ) @ S @ ( complete_Inf_Inf @ ( set @ A ) @ F15 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ S @ ( complete_Inf_Inf @ ( set @ A ) @ F4 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% compact_imp_fip
thf(fact_5825_compact__fip,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo2193935891317330818ompact @ A )
= ( ^ [U5: set @ A] :
! [A6: set @ ( set @ A )] :
( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A6 )
=> ( topolo7761053866217962861closed @ A @ X2 ) )
=> ( ! [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_5826_pos__deriv__imp__strict__mono,axiom,
! [F3: real > real,F10: real > real] :
( ! [X3: real] : ( has_field_derivative @ real @ F3 @ ( F10 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ! [X3: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F10 @ X3 ) )
=> ( order_strict_mono @ real @ real @ F3 ) ) ) ).
% pos_deriv_imp_strict_mono
thf(fact_5827_all__subset__image__inj,axiom,
! [A: $tType,B: $tType,F3: B > A,S: set @ B,P: ( set @ A ) > $o] :
( ( ! [T10: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T10 @ ( image2 @ B @ A @ F3 @ S ) )
=> ( P @ T10 ) ) )
= ( ! [T10: set @ B] :
( ( ( ord_less_eq @ ( set @ B ) @ T10 @ S )
& ( inj_on @ B @ A @ F3 @ T10 ) )
=> ( P @ ( image2 @ B @ A @ F3 @ T10 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_5828_ex__subset__image__inj,axiom,
! [A: $tType,B: $tType,F3: B > A,S: set @ B,P: ( set @ A ) > $o] :
( ( ? [T10: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T10 @ ( image2 @ B @ A @ F3 @ S ) )
& ( P @ T10 ) ) )
= ( ? [T10: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ T10 @ S )
& ( inj_on @ B @ A @ F3 @ T10 )
& ( P @ ( image2 @ B @ A @ F3 @ T10 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_5829_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G2: A > B,C2: set @ A,B2: set @ A,X: A] :
( ( inj_on @ A @ B @ G2 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ C2 @ ( sup_sup @ ( set @ A ) @ B2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ ( B > A )
@ ^ [I4: B] : ( if @ A @ ( member @ B @ I4 @ ( image2 @ A @ B @ G2 @ C2 ) ) @ ( the_inv_into @ A @ B @ C2 @ G2 @ I4 ) @ X )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_5830_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_5831_inj__Suc,axiom,
! [N6: set @ nat] : ( inj_on @ nat @ nat @ suc @ N6 ) ).
% inj_Suc
thf(fact_5832_inj__singleton,axiom,
! [A: $tType,A4: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X2: A] : ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
@ A4 ) ).
% inj_singleton
thf(fact_5833_inj__on__diff__nat,axiom,
! [N6: set @ nat,K: nat] :
( ! [N3: nat] :
( ( member @ nat @ N3 @ N6 )
=> ( ord_less_eq @ nat @ K @ N3 ) )
=> ( inj_on @ nat @ nat
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ K )
@ N6 ) ) ).
% inj_on_diff_nat
thf(fact_5834_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_5835_inj__graph,axiom,
! [B: $tType,A: $tType] :
( inj_on @ ( A > B ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [F2: A > B] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X2: A,Y3: B] :
( Y3
= ( F2 @ X2 ) ) ) )
@ ( top_top @ ( set @ ( A > B ) ) ) ) ).
% inj_graph
thf(fact_5836_range__inj__infinite,axiom,
! [A: $tType,F3: nat > A] :
( ( inj_on @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) )
=> ~ ( finite_finite2 @ A @ ( image2 @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% range_inj_infinite
thf(fact_5837_inj__enumerate,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( inj_on @ nat @ A @ ( infini527867602293511546merate @ A @ S ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% inj_enumerate
thf(fact_5838_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [N3: nat,F6: nat > A] :
( ( A4
= ( image2 @ nat @ A @ F6
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N3 ) ) ) )
& ( inj_on @ nat @ A @ F6
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N3 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_5839_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [F6: A > nat,N3: nat] :
( ( ( image2 @ A @ nat @ F6 @ A4 )
= ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N3 ) ) )
& ( inj_on @ A @ nat @ F6 @ A4 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_5840_ge__eq__refl,axiom,
! [A: $tType,R: A > A > $o,X: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y4: A,Z2: A] : Y4 = Z2
@ R )
=> ( R @ X @ X ) ) ).
% ge_eq_refl
thf(fact_5841_refl__ge__eq,axiom,
! [A: $tType,R: A > A > $o] :
( ! [X3: A] : ( R @ X3 @ X3 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y4: A,Z2: A] : Y4 = Z2
@ R ) ) ).
% refl_ge_eq
thf(fact_5842_inj__on__nth,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ! [X3: nat] :
( ( member @ nat @ X3 @ I5 )
=> ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( inj_on @ nat @ A @ ( nth @ A @ Xs ) @ I5 ) ) ) ).
% inj_on_nth
thf(fact_5843_infinite__iff__countable__subset,axiom,
! [A: $tType,S: set @ A] :
( ( ~ ( finite_finite2 @ A @ S ) )
= ( ? [F2: nat > A] :
( ( inj_on @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_5844_infinite__countable__subset,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ? [F6: nat > A] :
( ( inj_on @ nat @ A @ F6 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ F6 @ ( top_top @ ( set @ nat ) ) ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_5845_summable__reindex,axiom,
! [F3: nat > real,G2: nat > nat] :
( ( summable @ real @ F3 )
=> ( ( inj_on @ nat @ nat @ G2 @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) )
=> ( summable @ real @ ( comp @ nat @ real @ nat @ F3 @ G2 ) ) ) ) ) ).
% summable_reindex
thf(fact_5846_inj__on__funpow__least,axiom,
! [A: $tType,N: nat,F3: A > A,S3: A] :
( ( ( compow @ ( A > A ) @ N @ F3 @ S3 )
= S3 )
=> ( ! [M4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ( ord_less @ nat @ M4 @ N )
=> ( ( compow @ ( A > A ) @ M4 @ F3 @ S3 )
!= S3 ) ) )
=> ( inj_on @ nat @ A
@ ^ [K3: nat] : ( compow @ ( A > A ) @ K3 @ F3 @ S3 )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% inj_on_funpow_least
thf(fact_5847_suminf__reindex__mono,axiom,
! [F3: nat > real,G2: nat > nat] :
( ( summable @ real @ F3 )
=> ( ( inj_on @ nat @ nat @ G2 @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) )
=> ( ord_less_eq @ real @ ( suminf @ real @ ( comp @ nat @ real @ nat @ F3 @ G2 ) ) @ ( suminf @ real @ F3 ) ) ) ) ) ).
% suminf_reindex_mono
thf(fact_5848_suminf__reindex,axiom,
! [F3: nat > real,G2: nat > nat] :
( ( summable @ real @ F3 )
=> ( ( inj_on @ nat @ nat @ G2 @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) )
=> ( ! [X3: nat] :
( ~ ( member @ nat @ X3 @ ( image2 @ nat @ nat @ G2 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F3 @ X3 )
= ( zero_zero @ real ) ) )
=> ( ( suminf @ real @ ( comp @ nat @ real @ nat @ F3 @ G2 ) )
= ( suminf @ real @ F3 ) ) ) ) ) ) ).
% suminf_reindex
thf(fact_5849_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F16: B > A,A18: set @ B,B1: set @ A,F25: C > D,B22: set @ C,A26: set @ D] :
( ( ( image2 @ B @ A @ F16 @ A18 )
= B1 )
=> ( ( inj_on @ C @ D @ F25 @ B22 )
=> ( ( ord_less_eq @ ( set @ D ) @ ( image2 @ C @ D @ F25 @ B22 ) @ A26 )
=> ( ( ( B22
= ( bot_bot @ ( set @ C ) ) )
=> ( A26
= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( bNF_Wellorder_Func @ C @ A @ B22 @ B1 )
= ( image2 @ ( D > B ) @ ( C > A ) @ ( bNF_We4925052301507509544nc_map @ C @ B @ A @ D @ B22 @ F16 @ F25 ) @ ( bNF_Wellorder_Func @ D @ B @ A26 @ A18 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_5850_Func__non__emp,axiom,
! [A: $tType,B: $tType,B2: set @ A,A4: set @ B] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bNF_Wellorder_Func @ B @ A @ A4 @ B2 )
!= ( bot_bot @ ( set @ ( B > A ) ) ) ) ) ).
% Func_non_emp
thf(fact_5851_Func__is__emp,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( ( bNF_Wellorder_Func @ A @ B @ A4 @ B2 )
= ( bot_bot @ ( set @ ( A > B ) ) ) )
= ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Func_is_emp
thf(fact_5852_Func__map,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,G2: A > B,A26: set @ A,A18: set @ B,F16: B > C,B1: set @ C,F25: D > A,B22: set @ D] :
( ( member @ ( A > B ) @ G2 @ ( bNF_Wellorder_Func @ A @ B @ A26 @ A18 ) )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ F16 @ A18 ) @ B1 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ D @ A @ F25 @ B22 ) @ A26 )
=> ( member @ ( D > C ) @ ( bNF_We4925052301507509544nc_map @ D @ B @ C @ A @ B22 @ F16 @ F25 @ G2 ) @ ( bNF_Wellorder_Func @ D @ C @ B22 @ B1 ) ) ) ) ) ).
% Func_map
thf(fact_5853_rtrancl__finite__eq__relpow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( transitive_rtrancl @ A @ R )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R )
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ).
% rtrancl_finite_eq_relpow
thf(fact_5854_has__derivative__power__int,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,X: C,F10: C > A,S: set @ C,N: int] :
( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F3 @ F10 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( power_int @ A @ ( F3 @ X2 ) @ N )
@ ^ [H2: C] : ( times_times @ A @ ( F10 @ H2 ) @ ( times_times @ A @ ( ring_1_of_int @ A @ N ) @ ( power_int @ A @ ( F3 @ X ) @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ).
% has_derivative_power_int
thf(fact_5855_has__derivative__power__int_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,N: int,S: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A
@ ^ [X2: A] : ( power_int @ A @ X2 @ N )
@ ^ [Y3: A] : ( times_times @ A @ Y3 @ ( times_times @ A @ ( ring_1_of_int @ A @ N ) @ ( power_int @ A @ X @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_power_int'
thf(fact_5856_power__int__eq__0__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( ( power_int @ A @ X @ N )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( N
!= ( zero_zero @ int ) ) ) ) ) ).
% power_int_eq_0_iff
thf(fact_5857_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_5858_power__int__mono__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ B3 @ N ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ) ).
% power_int_mono_iff
thf(fact_5859_power__int__not__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_int_not_zero
thf(fact_5860_zero__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% zero_less_power_int
thf(fact_5861_rtrancl__subset__rtrancl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( transitive_rtrancl @ A @ S3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R2 ) @ ( transitive_rtrancl @ A @ S3 ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_5862_rtrancl__subset,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ R ) )
=> ( ( transitive_rtrancl @ A @ S )
= ( transitive_rtrancl @ A @ R ) ) ) ) ).
% rtrancl_subset
thf(fact_5863_rtrancl__mono,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R2 ) @ ( transitive_rtrancl @ A @ S3 ) ) ) ).
% rtrancl_mono
thf(fact_5864_zero__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% zero_le_power_int
thf(fact_5865_rtrancl__Un__subset,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( transitive_rtrancl @ A @ S ) ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) ) ).
% rtrancl_Un_subset
thf(fact_5866_continuous__on__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S3: set @ A,F3: A > B,N: int] :
( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( F3 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S3
@ ^ [X2: A] : ( power_int @ B @ ( F3 @ X2 ) @ N ) ) ) ) ) ).
% continuous_on_power_int
thf(fact_5867_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_5868_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less_eq @ int @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% power_int_increasing
thf(fact_5869_power__int__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less @ int @ N @ N6 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% power_int_strict_increasing
thf(fact_5870_power__int__diff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,M: int,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M != N ) )
=> ( ( power_int @ A @ X @ ( minus_minus @ int @ M @ N ) )
= ( divide_divide @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% power_int_diff
thf(fact_5871_tendsto__power__int,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > A,A3: A,F4: filter @ B,N: int] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( power_int @ A @ ( F3 @ X2 ) @ N )
@ ( topolo7230453075368039082e_nhds @ A @ ( power_int @ A @ A3 @ N ) )
@ F4 ) ) ) ) ).
% tendsto_power_int
thf(fact_5872_continuous__at__within__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A3: A,S3: set @ A,F3: A > B,N: int] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S3 ) @ F3 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S3 )
@ ^ [X2: A] : ( power_int @ B @ ( F3 @ X2 ) @ N ) ) ) ) ) ).
% continuous_at_within_power_int
thf(fact_5873_differentiable__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,X: A,S3: set @ A,N: int] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( power_int @ B @ ( F3 @ X2 ) @ N )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ).
% differentiable_power_int
thf(fact_5874_continuous__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F4: filter @ A,F3: A > B,N: int] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( power_int @ B @ ( F3 @ X2 ) @ N ) ) ) ) ) ).
% continuous_power_int
thf(fact_5875_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less @ int @ N @ N6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_5876_power__int__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,N: int] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( power_int @ A @ Y @ N ) ) ) ) ) ) ).
% power_int_mono
thf(fact_5877_power__int__strict__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,N: int] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less @ A @ ( power_int @ A @ B3 @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_strict_antimono
thf(fact_5878_one__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% one_le_power_int
thf(fact_5879_one__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% one_less_power_int
thf(fact_5880_power__int__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( ( plus_plus @ int @ M @ N )
!= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ N ) )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% power_int_add
thf(fact_5881_power__int__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,N: int] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ B3 @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_antimono
thf(fact_5882_power__int__strict__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,N: int] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ B3 @ N ) ) ) ) ) ) ).
% power_int_strict_mono
thf(fact_5883_power__int__le__one,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( one_one @ A ) ) ) ) ) ) ).
% power_int_le_one
thf(fact_5884_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less_eq @ int @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( A3
!= ( zero_zero @ A ) )
| ( N6
!= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_5885_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ M @ N ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_5886_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ int @ M @ N ) ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_5887_power__int__minus__mult,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( N
!= ( zero_zero @ int ) ) )
=> ( ( times_times @ A @ ( power_int @ A @ X @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) @ X )
= ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_minus_mult
thf(fact_5888_power__int__add__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ X ) ) ) ) ).
% power_int_add_1
thf(fact_5889_power__int__add__1_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ X @ ( power_int @ A @ X @ M ) ) ) ) ) ).
% power_int_add_1'
thf(fact_5890_power__int__def,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ( ( power_int @ A )
= ( ^ [X2: A,N2: int] : ( if @ A @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) @ ( power_power @ A @ X2 @ ( nat2 @ N2 ) ) @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ ( nat2 @ ( uminus_uminus @ int @ N2 ) ) ) ) ) ) ) ).
% power_int_def
thf(fact_5891_powr__real__of__int_H,axiom,
! [X: real,N: int] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( X
!= ( zero_zero @ real ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ N ) )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N ) )
= ( power_int @ real @ X @ N ) ) ) ) ).
% powr_real_of_int'
thf(fact_5892_DERIV__power__int,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D2: A,X: A,S3: set @ A,N: int] :
( ( has_field_derivative @ A @ F3 @ D2 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( power_int @ A @ ( F3 @ X2 ) @ N )
@ ( times_times @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ N ) @ ( power_int @ A @ ( F3 @ X ) @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) ) @ D2 )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ).
% DERIV_power_int
thf(fact_5893_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lex @ A @ R2 ) )
=> ~ ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ( ( take @ A @ I2 @ Xs )
= ( take @ A @ I2 @ Ys2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Ys2 @ I2 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_5894_prod__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( prod_filter @ A @ B )
= ( ^ [F8: filter @ A,G8: filter @ B] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) @ ( filter @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ ( filter @ ( product_prod @ A @ B ) )
@ ^ [P3: A > $o,Q7: B > $o] :
( principal @ ( product_prod @ A @ B )
@ ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X2: A,Y3: B] :
( ( P3 @ X2 )
& ( Q7 @ Y3 ) ) ) ) ) )
@ ( collect @ ( product_prod @ ( A > $o ) @ ( B > $o ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ $o
@ ^ [P3: A > $o,Q7: B > $o] :
( ( eventually @ A @ P3 @ F8 )
& ( eventually @ B @ Q7 @ G8 ) ) ) ) ) ) ) ) ).
% prod_filter_def
thf(fact_5895_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa3 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa3
= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( Deg2 = Xa3 )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X3 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma2: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ( Mi2 != Ma2 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma2 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_5896_ball__empty,axiom,
! [A: $tType,P: A > $o,X5: A] :
( ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X5 ) ) ).
% ball_empty
thf(fact_5897_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
@ ^ [X2: B] :
? [Y3: A] :
( ( P @ Y3 )
& ( Q @ X2 @ Y3 ) ) ) )
= ( ! [Y3: A] :
( ( P @ Y3 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] : ( Q @ X2 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_5898_ball__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) )
=> ( P @ X2 ) ) )
= ( ! [X8: A] : ( P @ X8 ) ) ) ).
% ball_UNIV
thf(fact_5899_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_5900_eventually__ex,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
? [X8: B] : ( P @ X2 @ X8 )
@ F4 )
= ( ? [Y7: A > B] :
( eventually @ A
@ ^ [X2: A] : ( P @ X2 @ ( Y7 @ X2 ) )
@ F4 ) ) ) ).
% eventually_ex
thf(fact_5901_Ball__fold,axiom,
! [A: $tType,A4: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( P @ X2 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K3: A,S8: $o] :
( S8
& ( P @ K3 ) )
@ $true
@ A4 ) ) ) ).
% Ball_fold
thf(fact_5902_finite__image__set,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: A > B] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Uu3: B] :
? [X2: A] :
( ( Uu3
= ( F3 @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_5903_finite__image__set2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > $o,Q: B > $o,F3: A > B > C] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ B @ ( collect @ B @ Q ) )
=> ( finite_finite2 @ C
@ ( collect @ C
@ ^ [Uu3: C] :
? [X2: A,Y3: B] :
( ( Uu3
= ( F3 @ X2 @ Y3 ) )
& ( P @ X2 )
& ( Q @ Y3 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_5904_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A6: set @ A,P3: A > $o] :
! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( P3 @ X2 ) ) ) ) ).
% Ball_def
thf(fact_5905_finite_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( set @ A ) > $o ) @ ( ( set @ A ) > $o )
@ ^ [P5: ( set @ A ) > $o,X2: set @ A] :
( ( X2
= ( bot_bot @ ( set @ A ) ) )
| ? [A6: set @ A,A5: A] :
( ( X2
= ( insert2 @ A @ A5 @ A6 ) )
& ( P5 @ A6 ) ) ) ) ).
% finite.mono
thf(fact_5906_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X2: B] :
( ( Uu3
= ( F3 @ X2 ) )
& ( member @ B @ X2 @ A4 ) ) )
= ( image2 @ B @ A @ F3 @ A4 ) ) ).
% Setcompr_eq_image
thf(fact_5907_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F3: B > A,P: B > $o] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X2: B] :
( ( Uu3
= ( F3 @ X2 ) )
& ( P @ X2 ) ) )
= ( image2 @ B @ A @ F3 @ ( collect @ B @ P ) ) ) ).
% setcompr_eq_image
thf(fact_5908_closed__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X2: A,Y3: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X2 @ Y3 ) )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% closed_superdiagonal
thf(fact_5909_closed__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X2: A,Y3: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X2 @ Y3 ) )
& ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ).
% closed_subdiagonal
thf(fact_5910_open__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X2: A,Y3: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X2 @ Y3 ) )
& ( ord_less @ A @ Y3 @ X2 ) ) ) ) ) ).
% open_superdiagonal
thf(fact_5911_open__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X2: A,Y3: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X2 @ Y3 ) )
& ( ord_less @ A @ X2 @ Y3 ) ) ) ) ) ).
% open_subdiagonal
thf(fact_5912_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F3: B > A] :
( ( collect @ A
@ ^ [U2: A] :
? [X2: B] :
( U2
= ( F3 @ X2 ) ) )
= ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% full_SetCompr_eq
thf(fact_5913_eventually__ball__finite,axiom,
! [A: $tType,B: $tType,A4: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( eventually @ B
@ ^ [Y3: B] : ( P @ Y3 @ X3 )
@ Net ) )
=> ( eventually @ B
@ ^ [X2: B] :
! [Y3: A] :
( ( member @ A @ Y3 @ A4 )
=> ( P @ X2 @ Y3 ) )
@ Net ) ) ) ).
% eventually_ball_finite
thf(fact_5914_eventually__ball__finite__distrib,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( eventually @ B
@ ^ [X2: B] :
! [Y3: A] :
( ( member @ A @ Y3 @ A4 )
=> ( P @ X2 @ Y3 ) )
@ Net )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( eventually @ B
@ ^ [Y3: B] : ( P @ Y3 @ X2 )
@ Net ) ) ) ) ) ).
% eventually_ball_finite_distrib
thf(fact_5915_Sup__eq__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A )
= ( ^ [A6: set @ A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [B5: A] :
! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less_eq @ A @ X2 @ B5 ) ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_5916_Inf__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A )
= ( ^ [A6: set @ A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [B5: A] :
! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less_eq @ A @ B5 @ X2 ) ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_5917_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs3: list @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [I4: nat] :
( ( Uu3
= ( nth @ A @ Xs3 @ I4 ) )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_5918_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A] :
( ( Uu3
= ( inf_inf @ A @ X @ A5 ) )
& ( member @ A @ A5 @ A4 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_5919_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ B2 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A,B5: A] :
( ( Uu3
= ( inf_inf @ A @ A5 @ B5 ) )
& ( member @ A @ A5 @ A4 )
& ( member @ A @ B5 @ B2 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_5920_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A] :
( ( Uu3
= ( sup_sup @ A @ X @ A5 ) )
& ( member @ A @ A5 @ A4 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_5921_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ ( lattic7752659483105999362nf_fin @ A @ B2 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A,B5: A] :
( ( Uu3
= ( sup_sup @ A @ A5 @ B5 ) )
& ( member @ A @ A5 @ A4 )
& ( member @ A @ B5 @ B2 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_5922_cInf__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S )
=> ( ( complete_Inf_Inf @ A @ S )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [X2: A] :
! [Y3: A] :
( ( member @ A @ Y3 @ S )
=> ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_5923_cSup__cInf,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S )
=> ( ( complete_Sup_Sup @ A @ S )
= ( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [X2: A] :
! [Y3: A] :
( ( member @ A @ Y3 @ S )
=> ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_5924_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
= ( ( Deg = Deg4 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma2: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ( Mi2 != Ma2 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ Ma2 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ X2 )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_5925_funpow__inj__finite,axiom,
! [A: $tType,P6: A > A,X: A] :
( ( inj_on @ A @ A @ P6 @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y3: A] :
? [N2: nat] :
( Y3
= ( compow @ ( A > A ) @ N2 @ P6 @ X ) ) ) )
=> ~ ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( compow @ ( A > A ) @ N3 @ P6 @ X )
!= X ) ) ) ) ).
% funpow_inj_finite
thf(fact_5926_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa3 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Y
= ( Xa3
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( Y
= ( ~ ( ( Deg2 = Xa3 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma2: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ( Mi2 != Ma2 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma2 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_5927_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_VEBT_valid @ X @ Xa3 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa3
!= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
=> ~ ( ( Deg2 = Xa3 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma2: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ( Mi2 != Ma2 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma2 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_5928_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa3 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa3 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Y
= ( Xa3
= ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa3 ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( Y
= ( ( Deg2 = Xa3 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma2: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ( Mi2 != Ma2 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma2 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) @ Xa3 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_5929_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_VEBT_valid @ X @ Xa3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa3 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa3 ) )
=> ( Xa3
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) @ Xa3 ) )
=> ~ ( ( Deg2 = Xa3 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma2: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ( Mi2 != Ma2 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma2 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_5930_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa3 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa3 ) )
=> ( Xa3
= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) @ Xa3 ) )
=> ( ( Deg2 = Xa3 )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X3 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma2: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ( Mi2 != Ma2 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma2 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_5931_Sup__Inf__le,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ ( set @ A )] :
( ord_less_eq @ A
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F2: ( set @ A ) > A] :
( ( Uu3
= ( image2 @ ( set @ A ) @ A @ F2 @ A4 ) )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A4 )
=> ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) )
@ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A4 ) ) ) ) ).
% Sup_Inf_le
thf(fact_5932_Inf__Sup__le,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A4: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A4 ) )
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F2: ( set @ A ) > A] :
( ( Uu3
= ( image2 @ ( set @ A ) @ A @ F2 @ A4 ) )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A4 )
=> ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_5933_finite__Inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A4: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A4 ) )
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F2: ( set @ A ) > A] :
( ( Uu3
= ( image2 @ ( set @ A ) @ A @ F2 @ A4 ) )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A4 )
=> ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% finite_Inf_Sup
thf(fact_5934_mono__compose,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ C ) )
=> ! [Q: A > B > C,F3: D > B] :
( ( order_mono @ A @ ( B > C ) @ Q )
=> ( order_mono @ A @ ( D > C )
@ ^ [I4: A,X2: D] : ( Q @ I4 @ ( F3 @ X2 ) ) ) ) ) ).
% mono_compose
thf(fact_5935_Pow__Compl,axiom,
! [A: $tType,A4: set @ A] :
( ( pow2 @ A @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [B6: set @ A] :
( ( Uu3
= ( uminus_uminus @ ( set @ A ) @ B6 ) )
& ( member @ ( set @ A ) @ A4 @ ( pow2 @ A @ B6 ) ) ) ) ) ).
% Pow_Compl
thf(fact_5936_Sup__real__def,axiom,
( ( complete_Sup_Sup @ real )
= ( ^ [X8: set @ real] :
( ord_Least @ real
@ ^ [Z6: real] :
! [X2: real] :
( ( member @ real @ X2 @ X8 )
=> ( ord_less_eq @ real @ X2 @ Z6 ) ) ) ) ) ).
% Sup_real_def
thf(fact_5937_Sup__int__def,axiom,
( ( complete_Sup_Sup @ int )
= ( ^ [X8: set @ int] :
( the @ int
@ ^ [X2: int] :
( ( member @ int @ X2 @ X8 )
& ! [Y3: int] :
( ( member @ int @ Y3 @ X8 )
=> ( ord_less_eq @ int @ Y3 @ X2 ) ) ) ) ) ) ).
% Sup_int_def
thf(fact_5938_Union__maximal__sets,axiom,
! [A: $tType,F17: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F17 )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [T10: set @ A] :
( ( member @ ( set @ A ) @ T10 @ F17 )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ F17 )
=> ~ ( ord_less @ ( set @ A ) @ T10 @ X2 ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ F17 ) ) ) ).
% Union_maximal_sets
thf(fact_5939_Inf__filter__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( filter @ A ) )
= ( ^ [S7: set @ ( filter @ A )] :
( complete_Sup_Sup @ ( filter @ A )
@ ( collect @ ( filter @ A )
@ ^ [F8: filter @ A] :
! [X2: filter @ A] :
( ( member @ ( filter @ A ) @ X2 @ S7 )
=> ( ord_less_eq @ ( filter @ A ) @ F8 @ X2 ) ) ) ) ) ) ).
% Inf_filter_def
thf(fact_5940_iteratesp_Omono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F3: A > A] :
( order_mono @ ( A > $o ) @ ( A > $o )
@ ^ [P5: A > $o,X2: A] :
( ? [Y3: A] :
( ( X2
= ( F3 @ Y3 ) )
& ( P5 @ Y3 ) )
| ? [M8: set @ A] :
( ( X2
= ( complete_Sup_Sup @ A @ M8 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ M8 )
=> ( P5 @ Y3 ) ) ) ) ) ) ).
% iteratesp.mono
thf(fact_5941_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R2 ) )
= ( ? [Y3: A,N2: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N2 ) @ Y3 ) @ R2 )
& ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys2
= ( list_update @ A @ Xs @ N2 @ Y3 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_5942_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lex @ A @ R5 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_5943_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_5944_listrel1__mono,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R2 ) @ ( listrel1 @ A @ S3 ) ) ) ).
% listrel1_mono
thf(fact_5945_chain__subset,axiom,
! [A: $tType,Ord: A > A > $o,A4: set @ A,B2: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( comple1602240252501008431_chain @ A @ Ord @ B2 ) ) ) ).
% chain_subset
thf(fact_5946_chain__empty,axiom,
! [A: $tType,Ord: A > A > $o] : ( comple1602240252501008431_chain @ A @ Ord @ ( bot_bot @ ( set @ A ) ) ) ).
% chain_empty
thf(fact_5947_ccpo__Sup__least,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A4: set @ A,Z: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ Z ) ) ) ) ).
% ccpo_Sup_least
thf(fact_5948_ccpo__Sup__upper,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A4: set @ A,X: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_5949_chain__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A] : ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% chain_singleton
thf(fact_5950_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_5951_in__chain__finite,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A4: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A4 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A4 ) @ A4 ) ) ) ) ) ).
% in_chain_finite
thf(fact_5952_set__nths,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs @ I5 ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [I4: nat] :
( ( Uu3
= ( nth @ A @ Xs @ I4 ) )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I4 @ I5 ) ) ) ) ).
% set_nths
thf(fact_5953_finite__subsets__at__top__finite,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite5375528669736107172at_top @ A @ A4 )
= ( principal @ ( set @ A ) @ ( insert2 @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ).
% finite_subsets_at_top_finite
thf(fact_5954_eventually__finite__subsets__at__top__weakI,axiom,
! [A: $tType,A4: set @ A,P: ( set @ A ) > $o] :
( ! [X9: set @ A] :
( ( finite_finite2 @ A @ X9 )
=> ( ( ord_less_eq @ ( set @ A ) @ X9 @ A4 )
=> ( P @ X9 ) ) )
=> ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A4 ) ) ) ).
% eventually_finite_subsets_at_top_weakI
thf(fact_5955_nths__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( nths @ A @ Xs @ ( bot_bot @ ( set @ nat ) ) )
= ( nil @ A ) ) ).
% nths_empty
thf(fact_5956_eventually__finite__subsets__at__top__finite,axiom,
! [A: $tType,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A4 ) )
= ( P @ A4 ) ) ) ).
% eventually_finite_subsets_at_top_finite
thf(fact_5957_finite__subsets__at__top__neq__bot,axiom,
! [A: $tType,A4: set @ A] :
( ( finite5375528669736107172at_top @ A @ A4 )
!= ( bot_bot @ ( filter @ ( set @ A ) ) ) ) ).
% finite_subsets_at_top_neq_bot
thf(fact_5958_set__nths__subset,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( nths @ A @ Xs @ I5 ) ) @ ( set2 @ A @ Xs ) ) ).
% set_nths_subset
thf(fact_5959_nths__all,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ nat @ I2 @ I5 ) )
=> ( ( nths @ A @ Xs @ I5 )
= Xs ) ) ).
% nths_all
thf(fact_5960_eventually__finite__subsets__at__top,axiom,
! [A: $tType,P: ( set @ A ) > $o,A4: set @ A] :
( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A4 ) )
= ( ? [X8: set @ A] :
( ( finite_finite2 @ A @ X8 )
& ( ord_less_eq @ ( set @ A ) @ X8 @ A4 )
& ! [Y7: set @ A] :
( ( ( finite_finite2 @ A @ Y7 )
& ( ord_less_eq @ ( set @ A ) @ X8 @ Y7 )
& ( ord_less_eq @ ( set @ A ) @ Y7 @ A4 ) )
=> ( P @ Y7 ) ) ) ) ) ).
% eventually_finite_subsets_at_top
thf(fact_5961_length__nths,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I5 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I4 @ I5 ) ) ) ) ) ).
% length_nths
thf(fact_5962_finite__subsets__at__top__def,axiom,
! [A: $tType] :
( ( finite5375528669736107172at_top @ A )
= ( ^ [A6: set @ A] :
( complete_Inf_Inf @ ( filter @ ( set @ A ) )
@ ( image2 @ ( set @ A ) @ ( filter @ ( set @ A ) )
@ ^ [X8: set @ A] :
( principal @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Y7: set @ A] :
( ( finite_finite2 @ A @ Y7 )
& ( ord_less_eq @ ( set @ A ) @ X8 @ Y7 )
& ( ord_less_eq @ ( set @ A ) @ Y7 @ A6 ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [X8: set @ A] :
( ( finite_finite2 @ A @ X8 )
& ( ord_less_eq @ ( set @ A ) @ X8 @ A6 ) ) ) ) ) ) ) ).
% finite_subsets_at_top_def
thf(fact_5963_filterlim__lessThan__at__top,axiom,
filterlim @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( finite5375528669736107172at_top @ nat @ ( top_top @ ( set @ nat ) ) ) @ ( at_top @ nat ) ).
% filterlim_lessThan_at_top
thf(fact_5964_filterlim__atMost__at__top,axiom,
filterlim @ nat @ ( set @ nat ) @ ( set_ord_atMost @ nat ) @ ( finite5375528669736107172at_top @ nat @ ( top_top @ ( set @ nat ) ) ) @ ( at_top @ nat ) ).
% filterlim_atMost_at_top
thf(fact_5965_filterlim__finite__subsets__at__top,axiom,
! [A: $tType,B: $tType,F3: A > ( set @ B ),A4: set @ B,F4: filter @ A] :
( ( filterlim @ A @ ( set @ B ) @ F3 @ ( finite5375528669736107172at_top @ B @ A4 ) @ F4 )
= ( ! [X8: set @ B] :
( ( ( finite_finite2 @ B @ X8 )
& ( ord_less_eq @ ( set @ B ) @ X8 @ A4 ) )
=> ( eventually @ A
@ ^ [Y3: A] :
( ( finite_finite2 @ B @ ( F3 @ Y3 ) )
& ( ord_less_eq @ ( set @ B ) @ X8 @ ( F3 @ Y3 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F3 @ Y3 ) @ A4 ) )
@ F4 ) ) ) ) ).
% filterlim_finite_subsets_at_top
thf(fact_5966_flat__lub__def,axiom,
! [A: $tType] :
( ( partial_flat_lub @ A )
= ( ^ [B5: A,A6: set @ A] :
( if @ A @ ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert2 @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5
@ ( the @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_5967_Rats__eq__int__div__nat,axiom,
( ( field_char_0_Rats @ real )
= ( collect @ real
@ ^ [Uu3: real] :
? [I4: int,N2: nat] :
( ( Uu3
= ( divide_divide @ real @ ( ring_1_of_int @ real @ I4 ) @ ( semiring_1_of_nat @ real @ N2 ) ) )
& ( N2
!= ( zero_zero @ nat ) ) ) ) ) ).
% Rats_eq_int_div_nat
thf(fact_5968_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( ord_max @ A @ X2 ) @ Y3 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_5969_Rats__no__top__le,axiom,
! [X: real] :
? [X3: real] :
( ( member @ real @ X3 @ ( field_char_0_Rats @ real ) )
& ( ord_less_eq @ real @ X @ X3 ) ) ).
% Rats_no_top_le
thf(fact_5970_Rats__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_0
thf(fact_5971_Rats__no__bot__less,axiom,
! [X: real] :
? [X3: real] :
( ( member @ real @ X3 @ ( field_char_0_Rats @ real ) )
& ( ord_less @ real @ X3 @ X ) ) ).
% Rats_no_bot_less
thf(fact_5972_Rats__dense__in__real,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ? [X3: real] :
( ( member @ real @ X3 @ ( field_char_0_Rats @ real ) )
& ( ord_less @ real @ X @ X3 )
& ( ord_less @ real @ X3 @ Y ) ) ) ).
% Rats_dense_in_real
thf(fact_5973_Rats__infinite,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ~ ( finite_finite2 @ A @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_infinite
thf(fact_5974_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_5975_Max_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Max.infinite
thf(fact_5976_Inf__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Inf_fin.infinite
thf(fact_5977_Sup__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Sup_fin.infinite
thf(fact_5978_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( inf_inf @ A @ X2 ) @ Y3 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_5979_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( sup_sup @ A @ X2 ) @ Y3 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_5980_take__bit__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ) ).
% take_bit_numeral_numeral
thf(fact_5981_Nats__altdef1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [Uu3: A] :
? [N2: int] :
( ( Uu3
= ( ring_1_of_int @ A @ N2 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) ) ) ) ) ).
% Nats_altdef1
thf(fact_5982_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_5983_Nats__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_1
thf(fact_5984_Nats__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A3: A,B3: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B3 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_add
thf(fact_5985_Nats__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_0
thf(fact_5986_Nats__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A3: A,B3: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B3 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B3 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_mult
thf(fact_5987_of__nat__in__Nats,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_Nats @ A ) ) ) ).
% of_nat_in_Nats
thf(fact_5988_Nats__induct,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: A,P: A > $o] :
( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
=> ( ! [N3: nat] : ( P @ ( semiring_1_of_nat @ A @ N3 ) )
=> ( P @ X ) ) ) ) ).
% Nats_induct
thf(fact_5989_Nats__cases,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
=> ~ ! [N3: nat] :
( X
!= ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% Nats_cases
thf(fact_5990_Nats__diff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B3: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B3 @ ( semiring_1_Nats @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( member @ A @ ( minus_minus @ A @ A3 @ B3 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ) ).
% Nats_diff
thf(fact_5991_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_5992_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_5993_take__bit__num__eq__None__imp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: num] :
( ( ( bit_take_bit_num @ M @ N )
= ( none @ num ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_num_eq_None_imp
thf(fact_5994_Nats__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( image2 @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% Nats_def
thf(fact_5995_Nats__altdef2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [N2: A] :
( ( member @ A @ N2 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ N2 ) ) ) ) ) ).
% Nats_altdef2
thf(fact_5996_comp__fun__idem__on_Ofold__insert__idem,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,A4: set @ A,Z: B] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert2 @ A @ X @ A4 ) )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ A4 ) ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem
thf(fact_5997_comp__fun__idem__on_Ofold__insert__idem2,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,A4: set @ A,Z: B] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert2 @ A @ X @ A4 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X @ Z ) @ A4 ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem2
thf(fact_5998_comp__fun__idem__on_Oaxioms_I1_J,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F3 )
=> ( finite4664212375090638736ute_on @ A @ B @ S @ F3 ) ) ).
% comp_fun_idem_on.axioms(1)
thf(fact_5999_comp__fun__idem__on_Ofun__left__idem,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,X: A,Z: B] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F3 )
=> ( ( member @ A @ X @ S )
=> ( ( F3 @ X @ ( F3 @ X @ Z ) )
= ( F3 @ X @ Z ) ) ) ) ).
% comp_fun_idem_on.fun_left_idem
thf(fact_6000_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F3 )
=> ( ( member @ A @ X @ S )
=> ( ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ X ) )
= ( F3 @ X ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_6001_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
! [B: $tType,A: $tType,C: $tType,S: set @ A,F3: A > B > B,G2: C > A,R: set @ C] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G2 @ ( top_top @ ( set @ C ) ) ) @ S )
=> ( finite673082921795544331dem_on @ C @ B @ R @ ( comp @ A @ ( B > B ) @ C @ F3 @ G2 ) ) ) ) ).
% comp_fun_idem_on.comp_comp_fun_idem_on
thf(fact_6002_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N2: nat,M2: num] :
( if @ ( option @ num )
@ ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( numeral_numeral @ nat @ M2 ) )
= ( zero_zero @ nat ) )
@ ( none @ num )
@ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( numeral_numeral @ nat @ M2 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_6003_Sup__filter__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( filter @ A ) )
= ( ^ [S7: set @ ( filter @ A )] :
( abs_filter @ A
@ ^ [P3: A > $o] :
! [X2: filter @ A] :
( ( member @ ( filter @ A ) @ X2 @ S7 )
=> ( eventually @ A @ P3 @ X2 ) ) ) ) ) ).
% Sup_filter_def
thf(fact_6004_num__of__nat__numeral__eq,axiom,
! [Q5: num] :
( ( num_of_nat @ ( numeral_numeral @ nat @ Q5 ) )
= Q5 ) ).
% num_of_nat_numeral_eq
thf(fact_6005_bot__filter__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P3: A > $o] : $true ) ) ).
% bot_filter_def
thf(fact_6006_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ ( zero_zero @ nat ) )
= one2 ) ).
% num_of_nat.simps(1)
thf(fact_6007_sup__filter__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( filter @ A ) )
= ( ^ [F8: filter @ A,F9: filter @ A] :
( abs_filter @ A
@ ^ [P3: A > $o] :
( ( eventually @ A @ P3 @ F8 )
& ( eventually @ A @ P3 @ F9 ) ) ) ) ) ).
% sup_filter_def
thf(fact_6008_principal__def,axiom,
! [A: $tType] :
( ( principal @ A )
= ( ^ [S7: set @ A] : ( abs_filter @ A @ ( ball @ A @ S7 ) ) ) ) ).
% principal_def
thf(fact_6009_numeral__num__of__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( numeral_numeral @ nat @ ( num_of_nat @ N ) )
= N ) ) ).
% numeral_num_of_nat
thf(fact_6010_num__of__nat__One,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( one_one @ nat ) )
=> ( ( num_of_nat @ N )
= one2 ) ) ).
% num_of_nat_One
thf(fact_6011_map__filter__on__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter_on @ A @ B )
= ( ^ [X8: set @ A,F2: A > B,F8: filter @ A] :
( abs_filter @ B
@ ^ [P3: B > $o] :
( eventually @ A
@ ^ [X2: A] :
( ( P3 @ ( F2 @ X2 ) )
& ( member @ A @ X2 @ X8 ) )
@ F8 ) ) ) ) ).
% map_filter_on_def
thf(fact_6012_top__filter__def,axiom,
! [A: $tType] :
( ( top_top @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P2: A > $o] :
! [X6: A] : ( P2 @ X6 ) ) ) ).
% top_filter_def
thf(fact_6013_filtercomap__def,axiom,
! [B: $tType,A: $tType] :
( ( filtercomap @ A @ B )
= ( ^ [F2: A > B,F8: filter @ B] :
( abs_filter @ A
@ ^ [P3: A > $o] :
? [Q7: B > $o] :
( ( eventually @ B @ Q7 @ F8 )
& ! [X2: A] :
( ( Q7 @ ( F2 @ X2 ) )
=> ( P3 @ X2 ) ) ) ) ) ) ).
% filtercomap_def
thf(fact_6014_numeral__num__of__nat__unfold,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% numeral_num_of_nat_unfold
thf(fact_6015_inf__filter__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( filter @ A ) )
= ( ^ [F8: filter @ A,F9: filter @ A] :
( abs_filter @ A
@ ^ [P3: A > $o] :
? [Q7: A > $o,R6: A > $o] :
( ( eventually @ A @ Q7 @ F8 )
& ( eventually @ A @ R6 @ F9 )
& ! [X2: A] :
( ( ( Q7 @ X2 )
& ( R6 @ X2 ) )
=> ( P3 @ X2 ) ) ) ) ) ) ).
% inf_filter_def
thf(fact_6016_num__of__nat__double,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( plus_plus @ nat @ N @ N ) )
= ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% num_of_nat_double
thf(fact_6017_num__of__nat__plus__distrib,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_6018_num__of__nat_Osimps_I2_J,axiom,
! [N: nat] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= ( inc @ ( num_of_nat @ N ) ) ) )
& ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= one2 ) ) ) ).
% num_of_nat.simps(2)
thf(fact_6019_dual__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( min @ A
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 ) )
= ( ord_max @ A ) ) ) ).
% dual_min
thf(fact_6020_add__neg__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_6021_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_6022_diff__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_6023_diff__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ).
% diff_numeral_special(6)
thf(fact_6024_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one2 )
=> ( ! [X3: num] :
( ( P @ X3 )
=> ( P @ ( inc @ X3 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_6025_ord_Omin_Ocong,axiom,
! [A: $tType] :
( ( min @ A )
= ( min @ A ) ) ).
% ord.min.cong
thf(fact_6026_ord_Omin__def,axiom,
! [A: $tType] :
( ( min @ A )
= ( ^ [Less_eq2: A > A > $o,A5: A,B5: A] : ( if @ A @ ( Less_eq2 @ A5 @ B5 ) @ A5 @ B5 ) ) ) ).
% ord.min_def
thf(fact_6027_add__inc,axiom,
! [X: num,Y: num] :
( ( plus_plus @ num @ X @ ( inc @ Y ) )
= ( inc @ ( plus_plus @ num @ X @ Y ) ) ) ).
% add_inc
thf(fact_6028_inc_Osimps_I1_J,axiom,
( ( inc @ one2 )
= ( bit0 @ one2 ) ) ).
% inc.simps(1)
thf(fact_6029_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_6030_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_6031_add__One,axiom,
! [X: num] :
( ( plus_plus @ num @ X @ one2 )
= ( inc @ X ) ) ).
% add_One
thf(fact_6032_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_6033_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_6034_mult__inc,axiom,
! [X: num,Y: num] :
( ( times_times @ num @ X @ ( inc @ Y ) )
= ( plus_plus @ num @ ( times_times @ num @ X @ Y ) @ X ) ) ).
% mult_inc
thf(fact_6035_numeral__inc,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X: num] :
( ( numeral_numeral @ A @ ( inc @ X ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% numeral_inc
thf(fact_6036_sub__inc__One__eq,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( neg_numeral_sub @ A @ ( inc @ N ) @ one2 )
= ( numeral_numeral @ A @ N ) ) ) ).
% sub_inc_One_eq
thf(fact_6037_card__Min__le__sum,axiom,
! [A: $tType,A4: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A4 ) @ ( lattic643756798350308766er_Min @ nat @ ( image2 @ A @ nat @ F3 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A4 ) ) ) ).
% card_Min_le_sum
thf(fact_6038_total__on__singleton,axiom,
! [A: $tType,X: A] : ( total_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_6039_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Min_singleton
thf(fact_6040_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_6041_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_6042_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ B,C3: A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [Uu3: B] : C3
@ A4 ) )
= C3 ) ) ) ) ).
% Min_const
thf(fact_6043_minus__Min__eq__Max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S ) ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_6044_minus__Max__eq__Min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S ) ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_6045_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ A4 ) ) ) ) ).
% Min_in
thf(fact_6046_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_6047_total__onI,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( ( X3 != Y2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on @ A @ A4 @ R2 ) ) ).
% total_onI
thf(fact_6048_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A6: set @ A,R5: set @ ( product_prod @ A @ A )] :
! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( ( X2 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R5 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_6049_Inf__fin__Min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% Inf_fin_Min
thf(fact_6050_Min_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ A3 ) ) ) ) ).
% Min.coboundedI
thf(fact_6051_Min__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ A4 )
=> ( ord_less_eq @ A @ X @ Y2 ) )
=> ( ( member @ A @ X @ A4 )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= X ) ) ) ) ) ).
% Min_eqI
thf(fact_6052_Min__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ X ) ) ) ) ).
% Min_le
thf(fact_6053_Least__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ? [X_12: A] : ( P @ X_12 )
=> ( ( ord_Least @ A @ P )
= ( lattic643756798350308766er_Min @ A @ ( collect @ A @ P ) ) ) ) ) ) ).
% Least_Min
thf(fact_6054_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798350308766er_Min @ A @ A4 )
= M )
= ( ( member @ A @ M @ A4 )
& ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ A @ M @ X2 ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_6055_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ X )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_6056_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( ( member @ A @ M @ A4 )
& ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ A @ M @ X2 ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_6057_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) )
=> ! [A10: A] :
( ( member @ A @ A10 @ A4 )
=> ( ord_less_eq @ A @ X @ A10 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_6058_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A7: A] :
( ( member @ A @ A7 @ A4 )
=> ( ord_less_eq @ A @ X @ A7 ) )
=> ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_6059_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ X )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_6060_Min__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [B7: A] :
( ( member @ A @ B7 @ A4 )
=> ( ord_less_eq @ A @ A3 @ B7 ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ A3 @ A4 ) )
= A3 ) ) ) ) ).
% Min_insert2
thf(fact_6061_Min__Inf,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ).
% Min_Inf
thf(fact_6062_cInf__eq__Min,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X4 )
= ( lattic643756798350308766er_Min @ A @ X4 ) ) ) ) ) ).
% cInf_eq_Min
thf(fact_6063_Min_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Min.infinite
thf(fact_6064_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M5: set @ A,N6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M5 @ N6 )
=> ( ( M5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N6 ) @ ( lattic643756798350308766er_Min @ A @ M5 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_6065_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B2 ) @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_6066_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F3 @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F3 @ A4 ) ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_6067_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F3: B > A,K: A] :
( ( finite_finite2 @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F3 @ X2 ) @ K )
@ S ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image2 @ B @ A @ F3 @ S ) ) @ K ) ) ) ) ) ).
% Min_add_commute
thf(fact_6068_dual__Max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Max @ A
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 ) )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% dual_Max
thf(fact_6069_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( ord_min @ A @ X2 ) @ Y3 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Min.eq_fold'
thf(fact_6070_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min @ nat @ M @ N ) ) ) ).
% min_Suc_Suc
thf(fact_6071_min__0R,axiom,
! [N: nat] :
( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_6072_min__0L,axiom,
! [N: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_6073_min_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_min @ A @ ( ord_min @ A @ A3 @ B3 ) @ B3 )
= ( ord_min @ A @ A3 @ B3 ) ) ) ).
% min.right_idem
thf(fact_6074_min_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_min @ A @ A3 @ ( ord_min @ A @ A3 @ B3 ) )
= ( ord_min @ A @ A3 @ B3 ) ) ) ).
% min.left_idem
thf(fact_6075_min_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A] :
( ( ord_min @ A @ A3 @ A3 )
= A3 ) ) ).
% min.idem
thf(fact_6076_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B3 @ C3 ) )
= ( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% min.bounded_iff
thf(fact_6077_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= B3 ) ) ) ).
% min.absorb2
thf(fact_6078_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= A3 ) ) ) ).
% min.absorb1
thf(fact_6079_min_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= A3 ) ) ) ).
% min.absorb3
thf(fact_6080_min_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= B3 ) ) ) ).
% min.absorb4
thf(fact_6081_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ ( ord_min @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z @ X )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% min_less_iff_conj
thf(fact_6082_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% min_top2
thf(fact_6083_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% min_top
thf(fact_6084_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_6085_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_6086_max__min__same_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ( ord_max @ A @ Y @ ( ord_min @ A @ X @ Y ) )
= Y ) ) ).
% max_min_same(4)
thf(fact_6087_max__min__same_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X @ Y ) @ Y )
= Y ) ) ).
% max_min_same(3)
thf(fact_6088_max__min__same_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X @ Y ) @ X )
= X ) ) ).
% max_min_same(2)
thf(fact_6089_max__min__same_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ X @ ( ord_min @ A @ X @ Y ) )
= X ) ) ).
% max_min_same(1)
thf(fact_6090_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_6091_min__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(3)
thf(fact_6092_min__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(4)
thf(fact_6093_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_6094_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_6095_min__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( one_one @ A ) ) ) ).
% min_0_1(5)
thf(fact_6096_min__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% min_0_1(6)
thf(fact_6097_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_6098_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_6099_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_6100_Min__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min_insert
thf(fact_6101_Min_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ).
% Min.in_idem
thf(fact_6102_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C3: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ( ord_less_eq @ A @ B3 @ D2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B3 ) @ ( ord_min @ A @ C3 @ D2 ) ) ) ) ) ).
% min.mono
thf(fact_6103_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3
= ( ord_min @ A @ A3 @ B3 ) ) ) ) ).
% min.orderE
thf(fact_6104_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( ord_min @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% min.orderI
thf(fact_6105_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B3 @ C3 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B3 )
=> ~ ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% min.boundedE
thf(fact_6106_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B3 @ C3 ) ) ) ) ) ).
% min.boundedI
thf(fact_6107_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( A5
= ( ord_min @ A @ A5 @ B5 ) ) ) ) ) ).
% min.order_iff
thf(fact_6108_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B3 ) @ A3 ) ) ).
% min.cobounded1
thf(fact_6109_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B3 ) @ B3 ) ) ).
% min.cobounded2
thf(fact_6110_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_min @ A @ A5 @ B5 )
= A5 ) ) ) ) ).
% min.absorb_iff1
thf(fact_6111_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_min @ A @ A5 @ B5 )
= B5 ) ) ) ) ).
% min.absorb_iff2
thf(fact_6112_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% min.coboundedI1
thf(fact_6113_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% min.coboundedI2
thf(fact_6114_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X @ Z )
| ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% min_le_iff_disj
thf(fact_6115_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_min @ A @ X @ Y )
= Y ) ) ) ).
% min_absorb2
thf(fact_6116_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_min @ A @ X @ Y )
= X ) ) ) ).
% min_absorb1
thf(fact_6117_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B5 ) @ A5 @ B5 ) ) ) ) ).
% min_def
thf(fact_6118_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B5 ) @ A5 @ B5 ) ) ) ) ).
% min_def_raw
thf(fact_6119_max__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_max @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( F3 @ ( ord_min @ A @ X @ Y ) ) ) ) ) ).
% max_of_antimono
thf(fact_6120_min__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_min @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( F3 @ ( ord_max @ A @ X @ Y ) ) ) ) ) ).
% min_of_antimono
thf(fact_6121_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_min_eq_max
thf(fact_6122_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_max_eq_min
thf(fact_6123_inf__min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( inf_inf @ A )
= ( ord_min @ A ) ) ) ).
% inf_min
thf(fact_6124_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Z ) @ ( plus_plus @ A @ Y @ Z ) ) ) ) ).
% min_add_distrib_left
thf(fact_6125_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ X @ ( ord_min @ A @ Y @ Z ) )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z ) ) ) ) ).
% min_add_distrib_right
thf(fact_6126_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q5: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M @ N ) @ Q5 )
= ( ord_min @ nat @ ( times_times @ nat @ M @ Q5 ) @ ( times_times @ nat @ N @ Q5 ) ) ) ).
% nat_mult_min_left
thf(fact_6127_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q5: nat] :
( ( times_times @ nat @ M @ ( ord_min @ nat @ N @ Q5 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q5 ) ) ) ).
% nat_mult_min_right
thf(fact_6128_linorder_OMax_Ocong,axiom,
! [A: $tType] :
( ( lattices_Max @ A )
= ( lattices_Max @ A ) ) ).
% linorder.Max.cong
thf(fact_6129_min_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_min @ A @ B3 @ ( ord_min @ A @ A3 @ C3 ) )
= ( ord_min @ A @ A3 @ ( ord_min @ A @ B3 @ C3 ) ) ) ) ).
% min.left_commute
thf(fact_6130_min_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B5: A] : ( ord_min @ A @ B5 @ A5 ) ) ) ) ).
% min.commute
thf(fact_6131_min_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_min @ A @ ( ord_min @ A @ A3 @ B3 ) @ C3 )
= ( ord_min @ A @ A3 @ ( ord_min @ A @ B3 @ C3 ) ) ) ) ).
% min.assoc
thf(fact_6132_of__nat__min,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X @ Y ) )
= ( ord_min @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_min
thf(fact_6133_min__max__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_min @ A @ A3 @ ( ord_max @ A @ B3 @ C3 ) )
= ( ord_max @ A @ ( ord_min @ A @ A3 @ B3 ) @ ( ord_min @ A @ A3 @ C3 ) ) ) ) ).
% min_max_distrib2
thf(fact_6134_min__max__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_min @ A @ ( ord_max @ A @ B3 @ C3 ) @ A3 )
= ( ord_max @ A @ ( ord_min @ A @ B3 @ A3 ) @ ( ord_min @ A @ C3 @ A3 ) ) ) ) ).
% min_max_distrib1
thf(fact_6135_max__min__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_max @ A @ A3 @ ( ord_min @ A @ B3 @ C3 ) )
= ( ord_min @ A @ ( ord_max @ A @ A3 @ B3 ) @ ( ord_max @ A @ A3 @ C3 ) ) ) ) ).
% max_min_distrib2
thf(fact_6136_max__min__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_max @ A @ ( ord_min @ A @ B3 @ C3 ) @ A3 )
= ( ord_min @ A @ ( ord_max @ A @ B3 @ A3 ) @ ( ord_max @ A @ C3 @ A3 ) ) ) ) ).
% max_min_distrib1
thf(fact_6137_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ord_min @ A @ ( minus_minus @ A @ X @ Z ) @ ( minus_minus @ A @ Y @ Z ) ) ) ) ).
% min_diff_distrib_left
thf(fact_6138_min__diff,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_min @ nat @ ( minus_minus @ nat @ M @ I ) @ ( minus_minus @ nat @ N @ I ) )
= ( minus_minus @ nat @ ( ord_min @ nat @ M @ N ) @ I ) ) ).
% min_diff
thf(fact_6139_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ( ord_less @ A @ X @ Z )
| ( ord_less @ A @ Y @ Z ) ) ) ) ).
% min_less_iff_disj
thf(fact_6140_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ ( ord_min @ A @ B3 @ C3 ) )
=> ~ ( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% min.strict_boundedE
thf(fact_6141_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] :
( ( A5
= ( ord_min @ A @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_6142_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ A3 @ C3 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% min.strict_coboundedI1
thf(fact_6143_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% min.strict_coboundedI2
thf(fact_6144_min__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,M: A,N: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_min @ B @ ( F3 @ M ) @ ( F3 @ N ) )
= ( F3 @ ( ord_min @ A @ M @ N ) ) ) ) ) ).
% min_of_mono
thf(fact_6145_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P6: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P6 )
= ( ord_min @ A @ ( times_times @ A @ X @ P6 ) @ ( times_times @ A @ Y @ P6 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P6 )
= ( ord_max @ A @ ( times_times @ A @ X @ P6 ) @ ( times_times @ A @ Y @ P6 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_6146_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P6: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P6 )
= ( ord_max @ A @ ( times_times @ A @ X @ P6 ) @ ( times_times @ A @ Y @ P6 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P6 )
= ( ord_min @ A @ ( times_times @ A @ X @ P6 ) @ ( times_times @ A @ Y @ P6 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_6147_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P6: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ P6 @ ( ord_min @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P6 @ X ) @ ( times_times @ A @ P6 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ P6 @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P6 @ X ) @ ( times_times @ A @ P6 @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_6148_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P6: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ P6 @ ( ord_max @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P6 @ X ) @ ( times_times @ A @ P6 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ P6 @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P6 @ X ) @ ( times_times @ A @ P6 @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_6149_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P6: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P6 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P6 ) @ ( divide_divide @ A @ Y @ P6 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P6 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P6 ) @ ( divide_divide @ A @ Y @ P6 ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_6150_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P6: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P6 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P6 ) @ ( divide_divide @ A @ Y @ P6 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P6 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P6 ) @ ( divide_divide @ A @ Y @ P6 ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_6151_Inf__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A,X: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( S
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ X @ S ) )
= X ) )
& ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ X @ S ) )
= ( ord_min @ A @ X @ ( complete_Inf_Inf @ A @ S ) ) ) ) ) ) ) ).
% Inf_insert_finite
thf(fact_6152_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H: A > A,N6: set @ A] :
( ! [X3: A,Y2: A] :
( ( H @ ( ord_min @ A @ X3 @ Y2 ) )
= ( ord_min @ A @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic643756798350308766er_Min @ A @ N6 ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ H @ N6 ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_6153_Min_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B2 ) @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min.subset
thf(fact_6154_Min_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] : ( member @ A @ ( ord_min @ A @ X3 @ Y2 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Min.closed
thf(fact_6155_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_6156_Min_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ ( lattic643756798350308766er_Min @ A @ B2 ) ) ) ) ) ) ) ) ).
% Min.union
thf(fact_6157_Min_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A4 ) )
= ( finite_fold @ A @ A @ ( ord_min @ A ) @ X @ A4 ) ) ) ) ).
% Min.eq_fold
thf(fact_6158_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_6159_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_6160_lexord__take__index__conv,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y )
= X ) )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) ) )
& ( ( take @ A @ I4 @ X )
= ( take @ A @ I4 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I4 ) @ ( nth @ A @ Y @ I4 ) ) @ R2 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_6161_dual__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( max @ A
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 ) )
= ( ord_min @ A ) ) ) ).
% dual_max
thf(fact_6162_inf__nat__def,axiom,
( ( inf_inf @ nat )
= ( ord_min @ nat ) ) ).
% inf_nat_def
thf(fact_6163_ord_Omax__def,axiom,
! [A: $tType] :
( ( max @ A )
= ( ^ [Less_eq2: A > A > $o,A5: A,B5: A] : ( if @ A @ ( Less_eq2 @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% ord.max_def
thf(fact_6164_ord_Omax_Ocong,axiom,
! [A: $tType] :
( ( max @ A )
= ( max @ A ) ) ).
% ord.max.cong
thf(fact_6165_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_6166_find__Some__iff2,axiom,
! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
( ( ( some @ A @ X )
= ( find @ A @ P @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I4 ) )
& ( X
= ( nth @ A @ Xs @ I4 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I4 )
=> ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_6167_hd__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_6168_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_6169_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_6170_find__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I4 ) )
& ( X
= ( nth @ A @ Xs @ I4 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I4 )
=> ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_6171_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType] :
( ( size_list @ A )
= ( ^ [F2: A > nat,Xs3: list @ A] :
( if @ nat
@ ( Xs3
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( plus_plus @ nat @ ( F2 @ ( hd @ A @ Xs3 ) ) @ ( size_list @ A @ F2 @ ( tl @ A @ Xs3 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_6172_Arg__bounded,axiom,
! [Z: complex] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z ) )
& ( ord_less_eq @ real @ ( arg @ Z ) @ pi ) ) ).
% Arg_bounded
thf(fact_6173_Nitpick_Osize__list__simp_I2_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( ^ [Xs3: list @ A] :
( if @ nat
@ ( Xs3
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_6174_nth__tl,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
=> ( ( nth @ A @ ( tl @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_6175_Arg__correct,axiom,
! [Z: complex] :
( ( Z
!= ( zero_zero @ complex ) )
=> ( ( ( sgn_sgn @ complex @ Z )
= ( cis @ ( arg @ Z ) ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z ) )
& ( ord_less_eq @ real @ ( arg @ Z ) @ pi ) ) ) ).
% Arg_correct
thf(fact_6176_nth__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A,M: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate @ A @ M @ Xs ) @ N )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_6177_rotate__length01,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate @ A @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_6178_rotate__id,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
= ( zero_zero @ nat ) )
=> ( ( rotate @ A @ N @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_6179_cis__Arg__unique,axiom,
! [Z: complex,X: real] :
( ( ( sgn_sgn @ complex @ Z )
= ( cis @ X ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( arg @ Z )
= X ) ) ) ) ).
% cis_Arg_unique
thf(fact_6180_bij__betw__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( bij_betw @ nat @ complex
@ ^ [K3: nat] : ( cis @ ( divide_divide @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( semiring_1_of_nat @ real @ K3 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
@ ( set_ord_lessThan @ nat @ N )
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= ( one_one @ complex ) ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_6181_bij__betw__nth__root__unity,axiom,
! [C3: complex,N: nat] :
( ( C3
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( bij_betw @ complex @ complex @ ( times_times @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( root @ N @ ( real_V7770717601297561774m_norm @ complex @ C3 ) ) ) @ ( cis @ ( divide_divide @ real @ ( arg @ C3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) )
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= ( one_one @ complex ) ) )
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= C3 ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_6182_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ A,A17: set @ B,B2: set @ A,B13: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A4 @ A17 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( ( image2 @ A @ B @ F3 @ B2 )
= B13 )
=> ( bij_betw @ A @ B @ F3 @ B2 @ B13 ) ) ) ) ).
% bij_betw_subset
thf(fact_6183_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A4: set @ A,F10: B > A,F3: A > B,A17: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( F10 @ ( F3 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A17 )
=> ( ( F3 @ ( F10 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ A17 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F10 @ A17 ) @ A4 )
=> ( bij_betw @ A @ B @ F3 @ A4 @ A17 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_6184_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ B] :
( ( bij_betw @ A @ B @ F3 @ ( bot_bot @ ( set @ A ) ) @ A4 )
=> ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ).
% bij_betw_empty1
thf(fact_6185_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A] :
( ( bij_betw @ A @ B @ F3 @ A4 @ ( bot_bot @ ( set @ B ) ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% bij_betw_empty2
thf(fact_6186_bij__betw__same__card,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ A,B2: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A4 @ B2 )
=> ( ( finite_card @ A @ A4 )
= ( finite_card @ B @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_6187_bij__betw__funpow,axiom,
! [A: $tType,F3: A > A,S: set @ A,N: nat] :
( ( bij_betw @ A @ A @ F3 @ S @ S )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ S @ S ) ) ).
% bij_betw_funpow
thf(fact_6188_bij__betw__iff__card,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ? [F2: A > B] : ( bij_betw @ A @ B @ F2 @ A4 @ B2 ) )
= ( ( finite_card @ A @ A4 )
= ( finite_card @ B @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_6189_finite__same__card__bij,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ( finite_card @ A @ A4 )
= ( finite_card @ B @ B2 ) )
=> ? [H6: A > B] : ( bij_betw @ A @ B @ H6 @ A4 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_6190_bij__fn,axiom,
! [A: $tType,F3: A > A,N: nat] :
( ( bij_betw @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_fn
thf(fact_6191_bij__betw__finite,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ A,B2: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A4 @ B2 )
=> ( ( finite_finite2 @ A @ A4 )
= ( finite_finite2 @ B @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_6192_bij__betwI_H,axiom,
! [A: $tType,B: $tType,X4: set @ A,F3: A > B,Y6: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ X4 )
=> ( ( ( F3 @ X3 )
= ( F3 @ Y2 ) )
= ( X3 = Y2 ) ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( member @ B @ ( F3 @ X3 ) @ Y6 ) )
=> ( ! [Y2: B] :
( ( member @ B @ Y2 @ Y6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ X4 )
& ( Y2
= ( F3 @ X5 ) ) ) )
=> ( bij_betw @ A @ B @ F3 @ X4 @ Y6 ) ) ) ) ).
% bij_betwI'
thf(fact_6193_bij__betw__comp__iff2,axiom,
! [C: $tType,A: $tType,B: $tType,F10: A > B,A17: set @ A,A19: set @ B,F3: C > A,A4: set @ C] :
( ( bij_betw @ A @ B @ F10 @ A17 @ A19 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ F3 @ A4 ) @ A17 )
=> ( ( bij_betw @ C @ A @ F3 @ A4 @ A17 )
= ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F10 @ F3 ) @ A4 @ A19 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_6194_Schroeder__Bernstein,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ A,B2: set @ B,G2: B > A] :
( ( inj_on @ A @ B @ F3 @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ B2 )
=> ( ( inj_on @ B @ A @ G2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G2 @ B2 ) @ A4 )
=> ? [H6: A > B] : ( bij_betw @ A @ B @ H6 @ A4 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_6195_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B3: A,A4: set @ A,F3: A > B,A17: set @ B] :
( ~ ( member @ A @ B3 @ A4 )
=> ( ~ ( member @ B @ ( F3 @ B3 ) @ A17 )
=> ( ( bij_betw @ A @ B @ F3 @ A4 @ A17 )
=> ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A17 @ ( insert2 @ B @ ( F3 @ B3 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_6196_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B3: A,A4: set @ A,F3: A > B,A17: set @ B] :
( ~ ( member @ A @ B3 @ A4 )
=> ( ~ ( member @ B @ ( F3 @ B3 ) @ A17 )
=> ( ( bij_betw @ A @ B @ F3 @ A4 @ A17 )
= ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A17 @ ( insert2 @ B @ ( F3 @ B3 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_6197_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ A,B2: set @ B,C2: set @ A,D3: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A4 @ B2 )
=> ( ( bij_betw @ A @ B @ F3 @ C2 @ D3 )
=> ( ( ( inf_inf @ ( set @ B ) @ B2 @ D3 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A4 @ C2 ) @ ( sup_sup @ ( set @ B ) @ B2 @ D3 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_6198_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ A,C2: set @ A,B2: set @ B,D3: set @ B] :
( ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A4 @ C2 ) @ ( sup_sup @ ( set @ B ) @ B2 @ D3 ) )
=> ( ( bij_betw @ A @ B @ F3 @ C2 @ D3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ C2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ B2 @ D3 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F3 @ A4 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_6199_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ A,C2: set @ B,G2: A > B,B2: set @ A,D3: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A4 @ C2 )
=> ( ( bij_betw @ A @ B @ G2 @ B2 @ D3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C2 @ D3 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ A4 ) @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( sup_sup @ ( set @ A ) @ A4 @ B2 )
@ ( sup_sup @ ( set @ B ) @ C2 @ D3 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_6200_bij__betw__UNION__chain,axiom,
! [B: $tType,C: $tType,A: $tType,I5: set @ A,A4: A > ( set @ B ),F3: B > C,A17: A > ( set @ C )] :
( ! [I2: A,J2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ( member @ A @ J2 @ I5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A4 @ I2 ) @ ( A4 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A4 @ J2 ) @ ( A4 @ I2 ) ) ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( bij_betw @ B @ C @ F3 @ ( A4 @ I2 ) @ ( A17 @ I2 ) ) )
=> ( bij_betw @ B @ C @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I5 ) ) @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ A17 @ I5 ) ) ) ) ) ).
% bij_betw_UNION_chain
thf(fact_6201_infinite__imp__bij__betw2,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ? [H6: A > A] : ( bij_betw @ A @ A @ H6 @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_6202_infinite__imp__bij__betw,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ? [H6: A > A] : ( bij_betw @ A @ A @ H6 @ A4 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_6203_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M5: set @ A] :
( ( finite_finite2 @ A @ M5 )
=> ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M5 ) ) @ M5 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_6204_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M5: set @ A] :
( ( finite_finite2 @ A @ M5 )
=> ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M5 ) ) @ M5 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_6205_finite__bij__enumerate,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( bij_betw @ nat @ A @ ( infini527867602293511546merate @ A @ S ) @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S ) ) @ S ) ) ) ).
% finite_bij_enumerate
thf(fact_6206_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S6: set @ B,T7: set @ C,H: B > C,S: set @ B,T4: set @ C,G2: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T7 )
=> ( ( bij_betw @ B @ C @ H @ ( minus_minus @ ( set @ B ) @ S @ S6 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T7 ) )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ S6 )
=> ( ( G2 @ ( H @ A7 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ T7 )
=> ( ( G2 @ B7 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( G2 @ ( H @ X2 ) )
@ S )
= ( groups7311177749621191930dd_sum @ C @ A @ G2 @ T4 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_6207_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S6: set @ B,T7: set @ C,H: B > C,S: set @ B,T4: set @ C,G2: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T7 )
=> ( ( bij_betw @ B @ C @ H @ ( minus_minus @ ( set @ B ) @ S @ S6 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T7 ) )
=> ( ! [A7: B] :
( ( member @ B @ A7 @ S6 )
=> ( ( G2 @ ( H @ A7 ) )
= ( one_one @ A ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ T7 )
=> ( ( G2 @ B7 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G2 @ ( H @ X2 ) )
@ S )
= ( groups7121269368397514597t_prod @ C @ A @ G2 @ T4 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_6208_ex__bij__betw__strict__mono__card,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M5: set @ A] :
( ( finite_finite2 @ A @ M5 )
=> ~ ! [H6: nat > A] :
( ( bij_betw @ nat @ A @ H6 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ M5 ) ) @ M5 )
=> ~ ( strict_mono_on @ nat @ A @ H6 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ M5 ) ) ) ) ) ) ).
% ex_bij_betw_strict_mono_card
thf(fact_6209_Arg__def,axiom,
( arg
= ( ^ [Z6: complex] :
( if @ real
@ ( Z6
= ( zero_zero @ complex ) )
@ ( zero_zero @ real )
@ ( fChoice @ real
@ ^ [A5: real] :
( ( ( sgn_sgn @ complex @ Z6 )
= ( cis @ A5 ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ A5 )
& ( ord_less_eq @ real @ A5 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_6210_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( linord4507533701916653071of_set @ A @ A4 )
= ( cons @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_6211_bij__betw__Suc,axiom,
! [M5: set @ nat,N6: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M5 @ N6 )
= ( ( image2 @ nat @ nat @ suc @ M5 )
= N6 ) ) ).
% bij_betw_Suc
thf(fact_6212_list_Osimps_I15_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( set2 @ A @ ( cons @ A @ X21 @ X22 ) )
= ( insert2 @ A @ X21 @ ( set2 @ A @ X22 ) ) ) ).
% list.simps(15)
thf(fact_6213_nth__Cons__0,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) )
= X ) ).
% nth_Cons_0
thf(fact_6214_n__lists__Nil,axiom,
! [A: $tType,N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( nil @ ( list @ A ) ) ) ) ) ).
% n_lists_Nil
thf(fact_6215_nths__singleton,axiom,
! [A: $tType,A4: set @ nat,X: A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A4 )
= ( cons @ A @ X @ ( nil @ A ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A4 )
= ( nil @ A ) ) ) ) ).
% nths_singleton
thf(fact_6216_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_6217_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( Xs
!= ( cons @ A @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_6218_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_6219_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Y: B,Ys2: list @ B] :
( ( ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X @ ( cons @ B @ Y @ Ys2 ) )
= ( cons @ B @ X @ ( cons @ B @ Y @ Ys2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X @ ( cons @ B @ Y @ Ys2 ) )
= ( cons @ B @ Y @ ( linorder_insort_key @ B @ A @ F3 @ X @ Ys2 ) ) ) ) ) ) ).
% insort_key.simps(2)
thf(fact_6220_list__update__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) @ Y )
= ( cons @ A @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_6221_some__in__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
@ A4 )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_6222_Cons__shuffles__subset2,axiom,
! [A: $tType,Y: A,Xs: list @ A,Ys2: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ Xs @ Ys2 ) ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys2 ) ) ) ).
% Cons_shuffles_subset2
thf(fact_6223_Cons__shuffles__subset1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ Ys2 ) ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys2 ) ) ).
% Cons_shuffles_subset1
thf(fact_6224_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [X2: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ X2 @ Ys3 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_6225_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,F3: B > A,A3: B] :
( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ord_less_eq @ A @ ( F3 @ A3 ) @ ( F3 @ X3 ) ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ A3 @ Xs )
= ( cons @ B @ A3 @ Xs ) ) ) ) ).
% insort_is_Cons
thf(fact_6226_bij__enumerate,axiom,
! [S: set @ nat] :
( ~ ( finite_finite2 @ nat @ S )
=> ( bij_betw @ nat @ nat @ ( infini527867602293511546merate @ nat @ S ) @ ( top_top @ ( set @ nat ) ) @ S ) ) ).
% bij_enumerate
thf(fact_6227_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_6228_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M5: set @ A] :
( ( finite_finite2 @ A @ M5 )
=> ? [H6: A > nat] : ( bij_betw @ A @ nat @ H6 @ M5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M5 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_6229_the__elem__set,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% the_elem_set
thf(fact_6230_lexordp_Omono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P5: ( list @ A ) > ( list @ A ) > $o,X17: list @ A,X25: list @ A] :
( ? [Y3: A,Ys3: list @ A] :
( ( X17
= ( nil @ A ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) ) )
| ? [X2: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( X17
= ( cons @ A @ X2 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ord_less @ A @ X2 @ Y3 ) )
| ? [X2: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( X17
= ( cons @ A @ X2 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( ord_less @ A @ X2 @ Y3 )
& ~ ( ord_less @ A @ Y3 @ X2 )
& ( P5 @ Xs3 @ Ys3 ) ) ) ) ) ).
% lexordp.mono
thf(fact_6231_list_Osize_I4_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X22 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size(4)
thf(fact_6232_nth__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= X ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_6233_remdups__adj__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_replicate
thf(fact_6234_list_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X21: A,X22: list @ A] :
( ( size_list @ A @ X @ ( cons @ A @ X21 @ X22 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X21 ) @ ( size_list @ A @ X @ X22 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_6235_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_6236_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_6237_nth__equal__first__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= X )
= ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_6238_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A,N: nat] :
( ( X != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= Y )
= ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_6239_take__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_6240_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat,Y: A] :
( ( ( cons @ A @ X @ Xs )
= ( replicate @ A @ N @ Y ) )
= ( ( X = Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_6241_Cons__lenlex__iff,axiom,
! [A: $tType,M: A,Ms: list @ A,N: A,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R2 ) )
= ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
| ( ( ( size_size @ ( list @ A ) @ Ms )
= ( size_size @ ( list @ A ) @ Ns ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_6242_arg__min__SOME__Min,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( lattic7623131987881927897min_on @ A @ B @ F3 @ S )
= ( fChoice @ A
@ ^ [Y3: A] :
( ( member @ A @ Y3 @ S )
& ( ( F3 @ Y3 )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F3 @ S ) ) ) ) ) ) ) ) ).
% arg_min_SOME_Min
thf(fact_6243_Pow__set_I2_J,axiom,
! [B: $tType,X: B,Xs: list @ B] :
( ( pow2 @ B @ ( set2 @ B @ ( cons @ B @ X @ Xs ) ) )
= ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) @ ( image2 @ ( set @ B ) @ ( set @ B ) @ ( insert2 @ B @ X ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_6244_concat__inth,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys2: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ Ys2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X ) ).
% concat_inth
thf(fact_6245_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I4: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_6246_set__append,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( set2 @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) ) ) ).
% set_append
thf(fact_6247_distinct__append,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( distinct @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( ( distinct @ A @ Xs )
& ( distinct @ A @ Ys2 )
& ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% distinct_append
thf(fact_6248_list__update__append1,axiom,
! [A: $tType,I: nat,Xs: list @ A,Ys2: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys2 ) @ I @ X )
= ( append @ A @ ( list_update @ A @ Xs @ I @ X ) @ Ys2 ) ) ) ).
% list_update_append1
thf(fact_6249_nth__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys2 ) @ N )
= ( nth @ A @ Xs @ N ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys2 ) @ N )
= ( nth @ A @ Ys2 @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_6250_list__update__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A,X: A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys2 ) @ N @ X )
= ( append @ A @ ( list_update @ A @ Xs @ N @ X ) @ Ys2 ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys2 ) @ N @ X )
= ( append @ A @ Xs @ ( list_update @ A @ Ys2 @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_6251_comm__append__is__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys2
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs @ Ys2 )
= ( append @ A @ Ys2 @ Xs ) )
=> ? [N3: nat,Zs2: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N3 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N3 @ Zs2 ) )
= ( append @ A @ Xs @ Ys2 ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_6252_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W2: list @ A,R2: set @ ( product_prod @ A @ A ),V2: list @ A,Z: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W2 ) @ ( lexord @ A @ R2 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W2 ) @ ( size_size @ ( list @ A ) @ U ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V2 ) @ ( append @ A @ W2 @ Z ) ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_sufI
thf(fact_6253_nths__Cons,axiom,
! [A: $tType,X: A,L: list @ A,A4: set @ nat] :
( ( nths @ A @ ( cons @ A @ X @ L ) @ A4 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A4 ) @ ( cons @ A @ X @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( suc @ J3 ) @ A4 ) ) ) ) ) ).
% nths_Cons
thf(fact_6254_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_6255_nth__repl,axiom,
! [A: $tType,M: nat,Xs: list @ A,N: nat,X: A] :
( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( M != N )
=> ( ( nth @ A @ ( append @ A @ ( take @ A @ N @ Xs ) @ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ ( drop @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) @ M )
= ( nth @ A @ Xs @ M ) ) ) ) ) ).
% nth_repl
thf(fact_6256_pos__n__replace,axiom,
! [A: $tType,N: nat,Xs: list @ A,Y: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ ( append @ A @ ( take @ A @ N @ Xs ) @ ( append @ A @ ( cons @ A @ Y @ ( nil @ A ) ) @ ( drop @ A @ ( suc @ N ) @ Xs ) ) ) ) ) ) ).
% pos_n_replace
thf(fact_6257_drop0,axiom,
! [A: $tType] :
( ( drop @ A @ ( zero_zero @ nat ) )
= ( ^ [X2: list @ A] : X2 ) ) ).
% drop0
thf(fact_6258_drop__update__cancel,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ N @ M )
=> ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N @ X ) )
= ( drop @ A @ M @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_6259_drop__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( nil @ A )
= ( drop @ A @ N @ Xs ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_6260_drop__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( drop @ A @ N @ Xs )
= ( nil @ A ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_6261_drop__all,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( drop @ A @ N @ Xs )
= ( nil @ A ) ) ) ).
% drop_all
thf(fact_6262_nth__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,I: nat] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_6263_set__drop__subset,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_drop_subset
thf(fact_6264_drop__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( drop @ A @ ( zero_zero @ nat ) @ Xs )
= Xs ) ).
% drop_0
thf(fact_6265_drop__eq__nths,axiom,
! [A: $tType] :
( ( drop @ A )
= ( ^ [N2: nat,Xs3: list @ A] : ( nths @ A @ Xs3 @ ( collect @ nat @ ( ord_less_eq @ nat @ N2 ) ) ) ) ) ).
% drop_eq_nths
thf(fact_6266_set__drop__subset__set__drop,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ M @ Xs ) ) @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_6267_drop__update__swap,axiom,
! [A: $tType,M: nat,N: nat,Xs: list @ A,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N @ X ) )
= ( list_update @ A @ ( drop @ A @ M @ Xs ) @ ( minus_minus @ nat @ N @ M ) @ X ) ) ) ).
% drop_update_swap
thf(fact_6268_drop__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ Xs ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_6269_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_6270_hd__drop__conv__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hd @ A @ ( drop @ A @ N @ Xs ) )
= ( nth @ A @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_6271_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_6272_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Vs )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I @ Vs ) ) @ ( set2 @ A @ ( drop @ A @ J @ Vs ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_take_disj_set_drop_if_distinct
thf(fact_6273_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_6274_upd__conv__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A,A3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ Xs @ I @ A3 )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ A3 @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_6275_take__hd__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( append @ A @ ( take @ A @ N @ Xs ) @ ( cons @ A @ ( hd @ A @ ( drop @ A @ N @ Xs ) ) @ ( nil @ A ) ) )
= ( take @ A @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_6276_upto_Opelims,axiom,
! [X: int,Xa3: int,Y: list @ int] :
( ( ( upto @ X @ Xa3 )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa3 ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X @ Xa3 )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa3 ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa3 )
=> ( Y
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa3 ) ) ) ) ) ).
% upto.pelims
thf(fact_6277_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_6278_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= ( nil @ int ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil
thf(fact_6279_upto__Nil2,axiom,
! [I: int,J: int] :
( ( ( nil @ int )
= ( upto @ I @ J ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil2
thf(fact_6280_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less @ int @ J @ I )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ).
% upto_empty
thf(fact_6281_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_6282_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(1)
thf(fact_6283_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(4)
thf(fact_6284_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(3)
thf(fact_6285_upto__rec__numeral_I2_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(2)
thf(fact_6286_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_6287_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_6288_upto_Osimps,axiom,
( upto
= ( ^ [I4: int,J3: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I4 @ J3 ) @ ( cons @ int @ I4 @ ( upto @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J3 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_6289_upto_Oelims,axiom,
! [X: int,Xa3: int,Y: list @ int] :
( ( ( upto @ X @ Xa3 )
= Y )
=> ( ( ( ord_less_eq @ int @ X @ Xa3 )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa3 ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa3 )
=> ( Y
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_6290_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_6291_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_6292_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_6293_to__nat__on__def,axiom,
! [A: $tType] :
( ( countable_to_nat_on @ A )
= ( ^ [S7: set @ A] :
( fChoice @ ( A > nat )
@ ^ [F2: A > nat] :
( ( ( finite_finite2 @ A @ S7 )
=> ( bij_betw @ A @ nat @ F2 @ S7 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S7 ) ) ) )
& ( ~ ( finite_finite2 @ A @ S7 )
=> ( bij_betw @ A @ nat @ F2 @ S7 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% to_nat_on_def
thf(fact_6294_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,M: A > ( option @ B ),X: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( map_upds @ A @ B @ M @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ Ys2 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ Xs @ Ys2 ) @ X @ ( some @ B @ ( nth @ B @ Ys2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_6295_map__upds__list__update2__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,M: A > ( option @ B ),Ys2: list @ B,Y: B] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( map_upds @ A @ B @ M @ Xs @ ( list_update @ B @ Ys2 @ I @ Y ) )
= ( map_upds @ A @ B @ M @ Xs @ Ys2 ) ) ) ).
% map_upds_list_update2_drop
thf(fact_6296_to__nat__on__finite,axiom,
! [A: $tType,S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( bij_betw @ A @ nat @ ( countable_to_nat_on @ A @ S ) @ S @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S ) ) ) ) ).
% to_nat_on_finite
thf(fact_6297_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),X: A,Y: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_6298_restrict__map__upds,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,D3: set @ A,M: A > ( option @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ D3 )
=> ( ( restrict_map @ A @ B @ ( map_upds @ A @ B @ M @ Xs @ Ys2 ) @ D3 )
= ( map_upds @ A @ B @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D3 @ ( set2 @ A @ Xs ) ) ) @ Xs @ Ys2 ) ) ) ) ).
% restrict_map_upds
thf(fact_6299_restrict__map__to__empty,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M @ ( bot_bot @ ( set @ A ) ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% restrict_map_to_empty
thf(fact_6300_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,X: A,D3: set @ A,M: A > ( option @ B ),Y: option @ B] :
( ( ( member @ A @ X @ D3 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ Y ) @ D3 )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) )
& ( ~ ( member @ A @ X @ D3 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ Y ) @ D3 )
= ( restrict_map @ A @ B @ M @ D3 ) ) ) ) ).
% restrict_fun_upd
thf(fact_6301_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X: A,D3: set @ A,M: A > ( option @ B ),Y: option @ B] :
( ( member @ A @ X @ D3 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D3 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ) ).
% fun_upd_restrict_conv
thf(fact_6302_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X: A,D3: set @ A,M: A > ( option @ B )] :
( ( ( member @ A @ X @ D3 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D3 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
& ( ~ ( member @ A @ X @ D3 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D3 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ D3 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_6303_restrict__map__insert,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B ),A3: A,A4: set @ A] :
( ( restrict_map @ A @ B @ F3 @ ( insert2 @ A @ A3 @ A4 ) )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ F3 @ A4 ) @ A3 @ ( F3 @ A3 ) ) ) ).
% restrict_map_insert
thf(fact_6304_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),D3: set @ A,X: A,Y: option @ B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D3 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ).
% fun_upd_restrict
thf(fact_6305_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B ),X: A] :
( ( restrict_map @ A @ B @ F3 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ ( none @ B ) ) ) ).
% restrict_complement_singleton_eq
thf(fact_6306_ran__map__upd,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A3: B,B3: A] :
( ( ( M @ A3 )
= ( none @ A ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M @ A3 @ ( some @ A @ B3 ) ) )
= ( insert2 @ A @ B3 @ ( ran @ B @ A @ M ) ) ) ) ).
% ran_map_upd
thf(fact_6307_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys2: list @ B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) @ I )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_6308_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X2: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_6309_set__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [I4: nat] :
( ( Uu3
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I4 ) @ ( nth @ B @ Ys2 @ I4 ) ) )
& ( ord_less @ nat @ I4 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ) ).
% set_zip
thf(fact_6310_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),X: B,Y: A,Z: A] :
( ( ( M @ X )
= ( some @ A @ Y ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M @ ( dom @ B @ A @ M ) )
=> ( ~ ( member @ A @ Z @ ( ran @ B @ A @ M ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M @ X @ ( some @ A @ Z ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Z @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_6311_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,I: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ ( nth @ A @ Xs @ I ) )
= ( some @ B @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_6312_dom__eq__empty__conv,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B )] :
( ( ( dom @ A @ B @ F3 )
= ( bot_bot @ ( set @ A ) ) )
= ( F3
= ( ^ [X2: A] : ( none @ B ) ) ) ) ).
% dom_eq_empty_conv
thf(fact_6313_dom__empty,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B
@ ^ [X2: A] : ( none @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% dom_empty
thf(fact_6314_dom__fun__upd,axiom,
! [B: $tType,A: $tType,Y: option @ B,F3: A > ( option @ B ),X: A] :
( ( ( Y
= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ Y ) )
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F3 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( ( Y
!= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ Y ) )
= ( insert2 @ A @ X @ ( dom @ A @ B @ F3 ) ) ) ) ) ).
% dom_fun_upd
thf(fact_6315_dom__map__upds,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),Xs: list @ A,Ys2: list @ B] :
( ( dom @ A @ B @ ( map_upds @ A @ B @ M @ Xs @ Ys2 ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys2 ) @ Xs ) ) @ ( dom @ A @ B @ M ) ) ) ).
% dom_map_upds
thf(fact_6316_finite__ran,axiom,
! [B: $tType,A: $tType,P6: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ P6 ) )
=> ( finite_finite2 @ B @ ( ran @ A @ B @ P6 ) ) ) ).
% finite_ran
thf(fact_6317_insert__dom,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),X: B,Y: A] :
( ( ( F3 @ X )
= ( some @ A @ Y ) )
=> ( ( insert2 @ B @ X @ ( dom @ B @ A @ F3 ) )
= ( dom @ B @ A @ F3 ) ) ) ).
% insert_dom
thf(fact_6318_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_6319_dom__if,axiom,
! [B: $tType,A: $tType,P: A > $o,F3: A > ( option @ B ),G2: A > ( option @ B )] :
( ( dom @ A @ B
@ ^ [X2: A] : ( if @ ( option @ B ) @ ( P @ X2 ) @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ F3 ) @ ( collect @ A @ P ) )
@ ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ G2 )
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) ) ) ) ).
% dom_if
thf(fact_6320_finite__map__freshness,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ F3 ) )
=> ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ? [X3: A] :
( ( F3 @ X3 )
= ( none @ B ) ) ) ) ).
% finite_map_freshness
thf(fact_6321_dom__minus,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),X: B,A4: set @ B] :
( ( ( F3 @ X )
= ( none @ A ) )
=> ( ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F3 ) @ ( insert2 @ B @ X @ A4 ) )
= ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F3 ) @ A4 ) ) ) ).
% dom_minus
thf(fact_6322_finite__set__of__finite__maps,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ ( A > ( option @ B ) )
@ ( collect @ ( A > ( option @ B ) )
@ ^ [M2: A > ( option @ B )] :
( ( ( dom @ A @ B @ M2 )
= A4 )
& ( ord_less_eq @ ( set @ B ) @ ( ran @ A @ B @ M2 ) @ B2 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_6323_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
@ ^ [X2: A] : ( none @ B ) )
=> ( ! [K2: A,V3: B,M4: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M4 ) )
=> ( ~ ( member @ A @ K2 @ ( dom @ A @ B @ M4 ) )
=> ( ( P @ M4 )
=> ( P @ ( fun_upd @ A @ ( option @ B ) @ M4 @ K2 @ ( some @ B @ V3 ) ) ) ) ) )
=> ( P @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_6324_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B ),X: A] :
( ( ( dom @ A @ B @ F3 )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ? [V6: B] :
( F3
= ( fun_upd @ A @ ( option @ B )
@ ^ [X2: A] : ( none @ B )
@ X
@ ( some @ B @ V6 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_6325_dom__override__on,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B ),G2: A > ( option @ B ),A4: set @ A] :
( ( dom @ A @ B @ ( override_on @ A @ ( option @ B ) @ F3 @ G2 @ A4 ) )
= ( sup_sup @ ( set @ A )
@ ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F3 )
@ ( collect @ A
@ ^ [A5: A] : ( member @ A @ A5 @ ( minus_minus @ ( set @ A ) @ A4 @ ( dom @ A @ B @ G2 ) ) ) ) )
@ ( collect @ A
@ ^ [A5: A] : ( member @ A @ A5 @ ( inf_inf @ ( set @ A ) @ A4 @ ( dom @ A @ B @ G2 ) ) ) ) ) ) ).
% dom_override_on
thf(fact_6326_card__quotient__disjoint,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [X2: A] : ( equiv_quotient @ A @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
@ A4 )
=> ( ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ R2 ) )
= ( finite_card @ A @ A4 ) ) ) ) ).
% card_quotient_disjoint
thf(fact_6327_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F3: A > B,G2: A > B] :
( ( override_on @ A @ B @ F3 @ G2 @ ( bot_bot @ ( set @ A ) ) )
= F3 ) ).
% override_on_emptyset
thf(fact_6328_override__on__insert_H,axiom,
! [B: $tType,A: $tType,F3: A > B,G2: A > B,X: A,X4: set @ A] :
( ( override_on @ A @ B @ F3 @ G2 @ ( insert2 @ A @ X @ X4 ) )
= ( override_on @ A @ B @ ( fun_upd @ A @ B @ F3 @ X @ ( G2 @ X ) ) @ G2 @ X4 ) ) ).
% override_on_insert'
thf(fact_6329_override__on__insert,axiom,
! [B: $tType,A: $tType,F3: A > B,G2: A > B,X: A,X4: set @ A] :
( ( override_on @ A @ B @ F3 @ G2 @ ( insert2 @ A @ X @ X4 ) )
= ( fun_upd @ A @ B @ ( override_on @ A @ B @ F3 @ G2 @ X4 ) @ X @ ( G2 @ X ) ) ) ).
% override_on_insert
thf(fact_6330_quotient__is__empty2,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ( bot_bot @ ( set @ ( set @ A ) ) )
= ( equiv_quotient @ A @ A4 @ R2 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty2
thf(fact_6331_quotient__is__empty,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ( equiv_quotient @ A @ A4 @ R2 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty
thf(fact_6332_quotient__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( bot_bot @ ( set @ A ) ) @ R2 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% quotient_empty
thf(fact_6333_quotient__diff1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,A3: A] :
( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [A5: A] : ( equiv_quotient @ A @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
@ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( equiv_quotient @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ R2 )
= ( minus_minus @ ( set @ ( set @ A ) ) @ ( equiv_quotient @ A @ A4 @ R2 ) @ ( equiv_quotient @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) @ R2 ) ) ) ) ) ).
% quotient_diff1
thf(fact_6334_quotient__def,axiom,
! [A: $tType] :
( ( equiv_quotient @ A )
= ( ^ [A6: set @ A,R5: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ A @ ( set @ ( set @ A ) )
@ ^ [X2: A] : ( insert2 @ ( set @ A ) @ ( image @ A @ A @ R5 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A6 ) ) ) ) ).
% quotient_def
thf(fact_6335_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ).
% sum_list_update
thf(fact_6336_ImageI,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,R2: set @ ( product_prod @ A @ B ),A4: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 )
=> ( ( member @ A @ A3 @ A4 )
=> ( member @ B @ B3 @ ( image @ A @ B @ R2 @ A4 ) ) ) ) ).
% ImageI
thf(fact_6337_Image__empty2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
( ( image @ B @ A @ R @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty2
thf(fact_6338_sum__list_ONil,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
= ( zero_zero @ A ) ) ) ).
% sum_list.Nil
thf(fact_6339_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [Ns: list @ A] :
( ( ( groups8242544230860333062m_list @ A @ Ns )
= ( zero_zero @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ns ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_6340_Image__empty1,axiom,
! [B: $tType,A: $tType,X4: set @ B] :
( ( image @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty1
thf(fact_6341_Image__Id__on,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( image @ A @ A @ ( id_on @ A @ A4 ) @ B2 )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ).
% Image_Id_on
thf(fact_6342_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A ),A3: B] :
( ( member @ A @ B3 @ ( image @ B @ A @ R2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B3 ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_6343_member__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% member_le_sum_list
thf(fact_6344_Image__Int__subset,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),A4: set @ B,B2: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ R @ A4 ) @ ( image @ B @ A @ R @ B2 ) ) ) ).
% Image_Int_subset
thf(fact_6345_Image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ B @ A ),B2: C > ( set @ B ),A4: set @ C] :
( ( image @ B @ A @ R2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B2 @ A4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X2: C] : ( image @ B @ A @ R2 @ ( B2 @ X2 ) )
@ A4 ) ) ) ).
% Image_UN
thf(fact_6346_Image__closed__trancl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ X4 ) @ X4 )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ X4 )
= X4 ) ) ).
% Image_closed_trancl
thf(fact_6347_Image__mono,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),A17: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A17 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R4 @ A17 ) @ ( image @ A @ B @ R2 @ A4 ) ) ) ) ).
% Image_mono
thf(fact_6348_Un__Image,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A ),A4: set @ B] :
( ( image @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ R @ S ) @ A4 )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ R @ A4 ) @ ( image @ B @ A @ S @ A4 ) ) ) ).
% Un_Image
thf(fact_6349_Image__Un,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),A4: set @ B,B2: set @ B] :
( ( image @ B @ A @ R @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ R @ A4 ) @ ( image @ B @ A @ R @ B2 ) ) ) ).
% Image_Un
thf(fact_6350_rev__ImageI,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ A4 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 )
=> ( member @ B @ B3 @ ( image @ A @ B @ R2 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_6351_Image__iff,axiom,
! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A ),A4: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ R2 @ A4 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ B3 ) @ R2 ) ) ) ) ).
% Image_iff
thf(fact_6352_ImageE,axiom,
! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A ),A4: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ R2 @ A4 ) )
=> ~ ! [X3: B] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ B3 ) @ R2 )
=> ~ ( member @ B @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_6353_finite__Image,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),A4: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( finite_finite2 @ B @ ( image @ A @ B @ R @ A4 ) ) ) ).
% finite_Image
thf(fact_6354_quotientE,axiom,
! [A: $tType,X4: set @ A,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ A ) @ X4 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ~ ! [X3: A] :
( ( X4
= ( image @ A @ A @ R2 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ~ ( member @ A @ X3 @ A4 ) ) ) ).
% quotientE
thf(fact_6355_quotientI,axiom,
! [A: $tType,X: A,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ A4 )
=> ( member @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( equiv_quotient @ A @ A4 @ R2 ) ) ) ).
% quotientI
thf(fact_6356_finite__rtrancl__Image,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ A4 ) ) ) ) ).
% finite_rtrancl_Image
thf(fact_6357_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_6358_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ( ( groups8242544230860333062m_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_6359_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_6360_Image__singleton,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A ),A3: B] :
( ( image @ B @ A @ R2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( collect @ A
@ ^ [B5: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B5 ) @ R2 ) ) ) ).
% Image_singleton
thf(fact_6361_Image__INT__subset,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ B @ A ),B2: C > ( set @ B ),A4: set @ C] :
( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B2 @ A4 ) ) )
@ ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X2: C] : ( image @ B @ A @ R2 @ ( B2 @ X2 ) )
@ A4 ) ) ) ).
% Image_INT_subset
thf(fact_6362_Image__eq__UN,axiom,
! [A: $tType,B: $tType] :
( ( image @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ A ),B6: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : ( image @ B @ A @ R5 @ ( insert2 @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) )
@ B6 ) ) ) ) ).
% Image_eq_UN
thf(fact_6363_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_6364_UN__Image,axiom,
! [A: $tType,B: $tType,C: $tType,X4: C > ( set @ ( product_prod @ B @ A ) ),I5: set @ C,S: set @ B] :
( ( image @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ X4 @ I5 ) ) @ S )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [I4: C] : ( image @ B @ A @ ( X4 @ I4 ) @ S )
@ I5 ) ) ) ).
% UN_Image
thf(fact_6365_card__length__sum__list__rec,axiom,
! [M: nat,N6: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N6 ) ) ) )
= ( plus_plus @ nat
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= ( minus_minus @ nat @ M @ ( one_one @ nat ) ) )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N6 ) ) ) )
@ ( 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 ) )
= N6 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_6366_singleton__quotient,axiom,
! [A: $tType,X: A,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
= ( insert2 @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% singleton_quotient
thf(fact_6367_sum__list__sum__nth,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ( ( groups8242544230860333062m_list @ B )
= ( ^ [Xs3: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% sum_list_sum_nth
thf(fact_6368_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X2: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y3: nat,Z6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V6: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y3 @ V6 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) )
@ ( rep_Integ @ X2 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_6369_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list @ A,X4: set @ A,F3: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X4 )
=> ( ( finite_finite2 @ A @ X4 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X2 ) @ ( F3 @ X2 ) )
@ X4 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_6370_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( zero_zero @ A )
@ Xs ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_6371_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F3: A > B] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ B @ ( map @ A @ B @ F3 @ Xs ) @ N )
= ( F3 @ ( nth @ A @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_6372_inj__on__map__eq__map,axiom,
! [B: $tType,A: $tType,F3: A > B,Xs: list @ A,Ys2: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) ) )
=> ( ( ( map @ A @ B @ F3 @ Xs )
= ( map @ A @ B @ F3 @ Ys2 ) )
= ( Xs = Ys2 ) ) ) ).
% inj_on_map_eq_map
thf(fact_6373_map__inj__on,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,Ys2: list @ B] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( map @ B @ A @ F3 @ Ys2 ) )
=> ( ( inj_on @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% map_inj_on
thf(fact_6374_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_6375_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs: list @ A,F3: A > B,G2: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G2 @ Xs ) ) ) ) ) ).
% sum_list_mono
thf(fact_6376_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( strict9044650504122735259up_add @ B ) )
=> ! [Xs: list @ A,F3: A > B,G2: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G2 @ Xs ) ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_6377_map__removeAll__inj__on,axiom,
! [B: $tType,A: $tType,F3: A > B,X: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) )
=> ( ( map @ A @ B @ F3 @ ( removeAll @ A @ X @ Xs ) )
= ( removeAll @ B @ ( F3 @ X ) @ ( map @ A @ B @ F3 @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_6378_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X2: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y3: nat,Z6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V6: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y3 @ V6 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) )
@ ( rep_Integ @ X2 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_6379_less__eq__int_Oabs__eq,axiom,
! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V6: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) )
@ Xa3
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_6380_less__int_Oabs__eq,axiom,
! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V6: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) )
@ Xa3
@ X ) ) ).
% less_int.abs_eq
thf(fact_6381_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_6382_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N2 @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_6383_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_6384_horner__sum__bit__eq__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ A @ A3 ) @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ).
% horner_sum_bit_eq_take_bit
thf(fact_6385_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F3: nat > B,Ns: list @ nat] :
( ! [X3: nat,Y2: nat] :
( ( ord_less_eq @ nat @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F3 @ Ns ) ) ) ) ) ) ).
% sorted_wrt_less_sum_mono_lowerbound
thf(fact_6386_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( hd @ nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_6387_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( upt @ I @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_6388_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_6389_take__upt,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ M ) @ N )
=> ( ( take @ nat @ M @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus @ nat @ I @ M ) ) ) ) ).
% take_upt
thf(fact_6390_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_6391_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
= ( cons @ nat @ ( numeral_numeral @ nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) )
& ( ~ ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
= ( nil @ nat ) ) ) ) ).
% upt_rec_numeral
thf(fact_6392_sum__list__upt,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups8242544230860333062m_list @ nat @ ( upt @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum_list_upt
thf(fact_6393_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ I4 @ N )
@ ( upt @ ( zero_zero @ nat ) @ M ) )
= ( upt @ N @ ( plus_plus @ nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_6394_sorted__map,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X2: B,Y3: B] : ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y3 ) )
@ Xs ) ) ) ).
% sorted_map
thf(fact_6395_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_6396_sorted__replicate,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: nat,X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N @ X ) ) ) ).
% sorted_replicate
thf(fact_6397_sorted__drop,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( drop @ A @ N @ Xs ) ) ) ) ).
% sorted_drop
thf(fact_6398_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A4 ) ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_6399_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M @ N ) ) ).
% sorted_wrt_upt
thf(fact_6400_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_6401_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A4 ) ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_6402_sorted__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A3 @ Xs ) ) ) ) ).
% sorted_remove1
thf(fact_6403_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_6404_sorted__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M @ N ) ) ).
% sorted_upt
thf(fact_6405_sorted__nths,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I5: set @ nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nths @ A @ Xs @ I5 ) ) ) ) ).
% sorted_nths
thf(fact_6406_sorted__take,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( take @ A @ N @ Xs ) ) ) ) ).
% sorted_take
thf(fact_6407_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_6408_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ Xs ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted_insort
thf(fact_6409_sorted2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Zs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% sorted2
thf(fact_6410_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_6411_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ).
% strict_sorted_simps(1)
thf(fact_6412_sorted0,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% sorted0
thf(fact_6413_strict__sorted__equal,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys2 )
=> ( ( ( set2 @ A @ Ys2 )
= ( set2 @ A @ Xs ) )
=> ( Ys2 = Xs ) ) ) ) ) ).
% strict_sorted_equal
thf(fact_6414_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N2: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% atLeast_upt
thf(fact_6415_map__replicate__trivial,axiom,
! [A: $tType,X: A,I: nat] :
( ( map @ nat @ A
@ ^ [I4: nat] : X
@ ( upt @ ( zero_zero @ nat ) @ I ) )
= ( replicate @ A @ I @ X ) ) ).
% map_replicate_trivial
thf(fact_6416_sorted1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% sorted1
thf(fact_6417_sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ Ys2 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ys2 ) )
=> ( ord_less_eq @ A @ X @ X2 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys2 ) ) ) ) ).
% sorted_simps(2)
thf(fact_6418_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X @ Ys2 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ys2 ) )
=> ( ord_less @ A @ X @ X2 ) )
& ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys2 ) ) ) ) ).
% strict_sorted_simps(2)
thf(fact_6419_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_6420_sorted__append,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ Ys2 ) )
= ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys2 )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Ys2 ) )
=> ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ) ).
% sorted_append
thf(fact_6421_sorted__distinct__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys2 )
=> ( ( distinct @ A @ Ys2 )
=> ( ( ( set2 @ A @ Xs )
= ( set2 @ A @ Ys2 ) )
=> ( Xs = Ys2 ) ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_6422_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P3: A > A > $o,Xs3: list @ A] :
! [I4: nat,J3: nat] :
( ( ord_less @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P3 @ ( nth @ A @ Xs3 @ I4 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_6423_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_6424_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_6425_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ).
% sorted_insort_key
thf(fact_6426_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ).
% sorted_map_remove1
thf(fact_6427_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N2: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) ) ) ).
% atMost_upto
thf(fact_6428_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_6429_upt__rec,axiom,
( upt
= ( ^ [I4: nat,J3: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I4 @ J3 ) @ ( cons @ nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_6430_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_6431_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_6432_map__upt__Suc,axiom,
! [A: $tType,F3: nat > A,N: nat] :
( ( map @ nat @ A @ F3 @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( cons @ A @ ( F3 @ ( zero_zero @ nat ) )
@ ( map @ nat @ A
@ ^ [I4: nat] : ( F3 @ ( suc @ I4 ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_6433_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map @ nat @ nat
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_6434_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_6435_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_6436_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_6437_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [X3: list @ A] :
( ( ( set2 @ A @ X3 )
= A4 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X3 )
& ( distinct @ A @ X3 )
& ! [Y5: list @ A] :
( ( ( ( set2 @ A @ Y5 )
= A4 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y5 )
& ( distinct @ A @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_6438_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_6439_nth__map__upt,axiom,
! [A: $tType,I: nat,N: nat,M: nat,F3: nat > A] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N @ M ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F3 @ ( upt @ M @ N ) ) @ I )
= ( F3 @ ( plus_plus @ nat @ M @ I ) ) ) ) ).
% nth_map_upt
thf(fact_6440_insort__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,Xs: list @ A] :
( ( member @ A @ A3 @ ( set2 @ A @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ A3
@ ( remove1 @ A @ A3 @ Xs ) )
= Xs ) ) ) ) ).
% insort_remove1
thf(fact_6441_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X @ Xs ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_6442_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ ( suc @ I4 ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_6443_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ~ ! [L7: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L7 )
=> ( ( ( set2 @ A @ L7 )
= A4 )
=> ( ( size_size @ ( list @ A ) @ L7 )
!= ( finite_card @ A @ A4 ) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_6444_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_6445_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_6446_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_6447_map__upt__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,M: nat,F3: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( minus_minus @ nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( F3 @ ( plus_plus @ nat @ M @ I2 ) ) ) )
=> ( ( map @ nat @ A @ F3 @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_6448_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
& ( P @ X5 ) )
=> ( ( find @ A @ P @ Xs )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_6449_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,Ys2: list @ B] :
( ( inj_on @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys2 ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Ys2 ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F3 @ Ys2 ) )
=> ( ( ( set2 @ B @ Xs )
= ( set2 @ B @ Ys2 ) )
=> ( Xs = Ys2 ) ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_6450_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,L: list @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
& ( ( set2 @ A @ L )
= A4 )
& ( ( size_size @ ( list @ A ) @ L )
= ( finite_card @ A @ A4 ) ) )
= ( ( linord4507533701916653071of_set @ A @ A4 )
= L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_6451_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ A3 ) )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ A3
@ Xs )
= ( append @ A @ Xs @ ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_insort_is_snoc
thf(fact_6452_transpose__rectangle,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( N
= ( zero_zero @ nat ) ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I2 ) )
= N ) )
=> ( ( transpose @ A @ Xs )
= ( map @ nat @ ( list @ A )
@ ^ [I4: nat] :
( map @ nat @ A
@ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J3 ) @ I4 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_6453_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S )
=> ( ( finite_finite2 @ B @ A4 )
=> ~ ! [L7: list @ B] :
( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ L7 ) )
=> ( ( ( set2 @ B @ L7 )
= A4 )
=> ( ( size_size @ ( list @ B ) @ L7 )
!= ( finite_card @ B @ A4 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_6454_sorted__upto,axiom,
! [M: int,N: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M @ N ) ) ).
% sorted_upto
thf(fact_6455_sorted__wrt__upto,axiom,
! [I: int,J: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I @ J ) ) ).
% sorted_wrt_upto
thf(fact_6456_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( folding_insort_key @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ ( set @ A ) )
@ ^ [X2: A] : X2 ) ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_6457_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,A4: set @ B,L: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ L ) )
& ( ( set2 @ B @ L )
= A4 )
& ( ( size_size @ ( list @ B ) @ L )
= ( finite_card @ B @ A4 ) ) )
= ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A4 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_6458_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,X: B,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( remove1 @ B @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A4 ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
thf(fact_6459_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,A4: set @ B,B2: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ S )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A4 )
= ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ B2 ) )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( A4 = B2 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_inject
thf(fact_6460_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( bot_bot @ ( set @ B ) ) )
= ( nil @ B ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_empty
thf(fact_6461_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( set2 @ B @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A4 ) )
= A4 ) ) ) ) ).
% folding_insort_key.set_sorted_key_list_of_set
thf(fact_6462_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S )
=> ( ( size_size @ ( list @ B ) @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A4 ) )
= ( finite_card @ B @ A4 ) ) ) ) ).
% folding_insort_key.length_sorted_key_list_of_set
thf(fact_6463_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S )
=> ( distinct @ A @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A4 ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_6464_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S )
=> ( sorted_wrt @ A @ Less_eq @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A4 ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_6465_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S )
=> ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A4 ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_6466_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A4 )
= ( nil @ B ) )
= ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
thf(fact_6467_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( set2 @ B @ Xs ) @ S )
=> ( ( sorted_wrt @ A @ Less_eq @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( distinct @ B @ Xs )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( set2 @ B @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_6468_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,X: B,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( insert2 @ B @ X @ A4 ) )
= ( insort_key @ A @ B @ Less_eq @ F3 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
thf(fact_6469_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,X: B,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ~ ( member @ B @ X @ A4 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( insert2 @ B @ X @ A4 ) )
= ( insort_key @ A @ B @ Less_eq @ F3 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A4 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert
thf(fact_6470_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_6471_length__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( foldr @ ( list @ A ) @ nat
@ ^ [Xs3: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) )
@ Xs
@ ( zero_zero @ nat ) ) ) ).
% length_transpose
thf(fact_6472_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_6473_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs3 @ ( zero_zero @ A ) ) ) ) ) ).
% sum_list.eq_foldr
thf(fact_6474_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_6475_rev__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rev @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_6476_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_6477_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I4 ) ) @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_6478_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_6479_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J3 ) @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_6480_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F2: B > A,A5: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X2: B,B5: A] : ( plus_plus @ A @ ( F2 @ X2 ) @ ( times_times @ A @ A5 @ B5 ) )
@ Xs3
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_6481_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_6482_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_6483_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G2: ( list @ A ) > A,Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter2 @ A
@ ^ [X2: A] :
( X2
= ( G2 @ Xs ) )
@ Xs ) ) ) ).
% sorted_same
thf(fact_6484_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_6485_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_6486_length__filter__less,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_6487_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,P: B > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ).
% sorted_filter
thf(fact_6488_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,Y: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert2 @ A @ Y @ ( set2 @ A @ Xs ) ) )
=> ( ( filter2 @ A
@ ^ [X2: A] :
( ( F3 @ Y )
= ( F3 @ X2 ) )
@ Xs )
= ( filter2 @ A
@ ( ^ [Y4: A,Z2: A] : Y4 = Z2
@ Y )
@ Xs ) ) ) ).
% inj_on_filter_key_eq
thf(fact_6489_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,G2: ( list @ B ) > A,Xs: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F3
@ ( filter2 @ B
@ ^ [X2: B] :
( ( F3 @ X2 )
= ( G2 @ Xs ) )
@ Xs ) ) ) ) ).
% sorted_map_same
thf(fact_6490_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F3: B > A,P: B > $o,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( F3 @ X2 ) @ ( zero_zero @ A ) )
@ Xs ) ) ) ) ).
% sum_list_map_filter'
thf(fact_6491_sum__list__filter__le__nat,axiom,
! [A: $tType,F3: A > nat,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ ( filter2 @ A @ P @ Xs ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) ) ) ).
% sum_list_filter_le_nat
thf(fact_6492_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B,P: B > $o,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ~ ( P @ X3 )
=> ( ( F3 @ X3 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_6493_filter__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,P: B > $o,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( P @ X )
=> ( ( filter2 @ B @ P @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( linorder_insort_key @ B @ A @ F3 @ X @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ) ).
% filter_insort
thf(fact_6494_set__minus__filter__out,axiom,
! [A: $tType,Xs: list @ A,Y: A] :
( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X2: A] : X2 != Y
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_6495_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys2 ) )
=> ( ( filter2 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( set2 @ A @ Ys2 ) )
@ Zs )
= Ys2 ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_6496_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys2 ) )
=> ( ( filter2 @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ ( set2 @ A @ Ys2 ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_6497_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys2 ) )
=> ( ( filter2 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_6498_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys2 ) )
=> ( ( filter2 @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
@ Zs )
= Ys2 ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_6499_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
( nths @ A @ Xs3
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
& ( P3 @ ( nth @ A @ Xs3 @ I4 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_6500_length__filter__conv__card,axiom,
! [A: $tType,P6: A > $o,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P6 @ Xs ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P6 @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_6501_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [A3: B,Xs: list @ B,F3: B > A] :
( ( member @ B @ A3 @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( ( hd @ B
@ ( filter2 @ B
@ ^ [X2: B] :
( ( F3 @ A3 )
= ( F3 @ X2 ) )
@ Xs ) )
= A3 )
=> ( ( linorder_insort_key @ B @ A @ F3 @ A3 @ ( remove1 @ B @ A3 @ Xs ) )
= Xs ) ) ) ) ) ).
% insort_key_remove1
thf(fact_6502_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Xss: list @ ( list @ B )] :
( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs ) )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [Xs3: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) )
@ Xss
@ ( zero_zero @ nat ) ) )
= ( suc
@ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [X2: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys3: list @ B] :
( Ys3
!= ( nil @ B ) )
@ Xss )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_6503_nth__transpose,axiom,
! [A: $tType,I: nat,Xs: list @ ( list @ A )] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I )
= ( map @ ( list @ A ) @ A
@ ^ [Xs3: list @ A] : ( nth @ A @ Xs3 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_6504_transpose__max__length,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( foldr @ ( list @ A ) @ nat
@ ^ [Xs3: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) )
@ ( transpose @ A @ Xs )
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [X2: list @ A] :
( X2
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_6505_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_6506_transpose__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( transpose @ A @ ( transpose @ A @ Xs ) )
= ( takeWhile @ ( list @ A )
@ ^ [X2: list @ A] :
( X2
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_transpose
thf(fact_6507_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,T2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F3 @ Xs ) ) )
=> ( ( filter2 @ B
@ ^ [X2: B] : ( ord_less @ A @ T2 @ ( F3 @ X2 ) )
@ Xs )
= ( takeWhile @ B
@ ^ [X2: B] : ( ord_less @ A @ T2 @ ( F3 @ X2 ) )
@ Xs ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_6508_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_6509_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_6510_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_6511_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_6512_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( P @ ( nth @ A @ Xs @ I2 ) ) )
=> ( ( 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_6513_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I2 ) ) ) )
=> ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P @ ( nth @ A @ Xs @ N ) ) )
=> ( ( takeWhile @ A @ P @ Xs )
= ( take @ A @ N @ Xs ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_6514_SUP__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F3 ) @ Xs @ ( bot_bot @ A ) ) ) ) ).
% SUP_set_fold
thf(fact_6515_finite__sequence__to__countable__set,axiom,
! [A: $tType,X4: set @ A] :
( ( countable_countable @ A @ X4 )
=> ~ ! [F5: nat > ( set @ A )] :
( ! [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( F5 @ I3 ) @ X4 )
=> ( ! [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) )
=> ( ! [I3: nat] : ( finite_finite2 @ A @ ( F5 @ I3 ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ F5 @ ( top_top @ ( set @ nat ) ) ) )
!= X4 ) ) ) ) ) ).
% finite_sequence_to_countable_set
thf(fact_6516_countable__empty,axiom,
! [A: $tType] : ( countable_countable @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% countable_empty
thf(fact_6517_countable__insert__eq,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( countable_countable @ A @ ( insert2 @ A @ X @ A4 ) )
= ( countable_countable @ A @ A4 ) ) ).
% countable_insert_eq
thf(fact_6518_countable__insert,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ( countable_countable @ A @ A4 )
=> ( countable_countable @ A @ ( insert2 @ A @ A3 @ A4 ) ) ) ).
% countable_insert
thf(fact_6519_countable__Un__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( countable_countable @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ( countable_countable @ A @ A4 )
& ( countable_countable @ A @ B2 ) ) ) ).
% countable_Un_iff
thf(fact_6520_countable__Un,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ( countable_countable @ A @ B2 )
=> ( countable_countable @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% countable_Un
thf(fact_6521_ccSup__insert,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,A3: A] :
( ( countable_countable @ A @ A4 )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% ccSup_insert
thf(fact_6522_ccInf__insert,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,A3: A] :
( ( countable_countable @ A @ A4 )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ).
% ccInf_insert
thf(fact_6523_countable__Diff__eq,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( countable_countable @ A @ A4 ) ) ).
% countable_Diff_eq
thf(fact_6524_ccSUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,F3: B > A,A3: B] :
( ( countable_countable @ B @ A4 )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( sup_sup @ A @ ( F3 @ A3 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ) ).
% ccSUP_insert
thf(fact_6525_ccINF__insert,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,F3: B > A,A3: B] :
( ( countable_countable @ B @ A4 )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( inf_inf @ A @ ( F3 @ A3 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ) ).
% ccINF_insert
thf(fact_6526_countable__image__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,S: set @ B] :
( ( countable_countable @ A @ ( image2 @ B @ A @ F3 @ S ) )
= ( ? [T10: set @ B] :
( ( countable_countable @ B @ T10 )
& ( ord_less_eq @ ( set @ B ) @ T10 @ S )
& ( ( image2 @ B @ A @ F3 @ S )
= ( image2 @ B @ A @ F3 @ T10 ) ) ) ) ) ).
% countable_image_eq
thf(fact_6527_countable__subset__image,axiom,
! [A: $tType,B: $tType,B2: set @ A,F3: B > A,A4: set @ B] :
( ( ( countable_countable @ A @ B2 )
& ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F3 @ A4 ) ) )
= ( ? [A14: set @ B] :
( ( countable_countable @ B @ A14 )
& ( ord_less_eq @ ( set @ B ) @ A14 @ A4 )
& ( B2
= ( image2 @ B @ A @ F3 @ A14 ) ) ) ) ) ).
% countable_subset_image
thf(fact_6528_ex__countable__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,S: set @ B,P: ( set @ A ) > $o] :
( ( ? [T10: set @ A] :
( ( countable_countable @ A @ T10 )
& ( ord_less_eq @ ( set @ A ) @ T10 @ ( image2 @ B @ A @ F3 @ S ) )
& ( P @ T10 ) ) )
= ( ? [T10: set @ B] :
( ( countable_countable @ B @ T10 )
& ( ord_less_eq @ ( set @ B ) @ T10 @ S )
& ( P @ ( image2 @ B @ A @ F3 @ T10 ) ) ) ) ) ).
% ex_countable_subset_image
thf(fact_6529_all__countable__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,S: set @ B,P: ( set @ A ) > $o] :
( ( ! [T10: set @ A] :
( ( ( countable_countable @ A @ T10 )
& ( ord_less_eq @ ( set @ A ) @ T10 @ ( image2 @ B @ A @ F3 @ S ) ) )
=> ( P @ T10 ) ) )
= ( ! [T10: set @ B] :
( ( ( countable_countable @ B @ T10 )
& ( ord_less_eq @ ( set @ B ) @ T10 @ S ) )
=> ( P @ ( image2 @ B @ A @ F3 @ T10 ) ) ) ) ) ).
% all_countable_subset_image
thf(fact_6530_ccInf__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( countable_countable @ A @ B2 )
=> ( ( countable_countable @ A @ A4 )
=> ( ! [B7: A] :
( ( member @ A @ B7 @ B2 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A4 )
& ( ord_less_eq @ A @ X5 @ B7 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ) ) ).
% ccInf_mono
thf(fact_6531_ccInf__lower,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,X: A] :
( ( countable_countable @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ X ) ) ) ) ).
% ccInf_lower
thf(fact_6532_ccInf__lower2,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,U: A,V2: A] :
( ( countable_countable @ A @ A4 )
=> ( ( member @ A @ U @ A4 )
=> ( ( ord_less_eq @ A @ U @ V2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ V2 ) ) ) ) ) ).
% ccInf_lower2
thf(fact_6533_le__ccInf__iff,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B3: A] :
( ( countable_countable @ A @ A4 )
=> ( ( ord_less_eq @ A @ B3 @ ( complete_Inf_Inf @ A @ A4 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ A @ B3 @ X2 ) ) ) ) ) ) ).
% le_ccInf_iff
thf(fact_6534_ccInf__greatest,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,Z: A] :
( ( countable_countable @ A @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ Z @ X3 ) )
=> ( ord_less_eq @ A @ Z @ ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ).
% ccInf_greatest
thf(fact_6535_countable__subset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( countable_countable @ A @ B2 )
=> ( countable_countable @ A @ A4 ) ) ) ).
% countable_subset
thf(fact_6536_ccSup__upper2,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,U: A,V2: A] :
( ( countable_countable @ A @ A4 )
=> ( ( member @ A @ U @ A4 )
=> ( ( ord_less_eq @ A @ V2 @ U )
=> ( ord_less_eq @ A @ V2 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ).
% ccSup_upper2
thf(fact_6537_ccSup__le__iff,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B3: A] :
( ( countable_countable @ A @ A4 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ B3 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ A @ X2 @ B3 ) ) ) ) ) ) ).
% ccSup_le_iff
thf(fact_6538_ccSup__upper,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,X: A] :
( ( countable_countable @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% ccSup_upper
thf(fact_6539_ccSup__least,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,Z: A] :
( ( countable_countable @ A @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ Z ) ) ) ) ).
% ccSup_least
thf(fact_6540_ccSup__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( countable_countable @ A @ B2 )
=> ( ( countable_countable @ A @ A4 )
=> ( ! [A7: A] :
( ( member @ A @ A7 @ A4 )
=> ? [X5: A] :
( ( member @ A @ X5 @ B2 )
& ( ord_less_eq @ A @ A7 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ).
% ccSup_mono
thf(fact_6541_less__ccSup__iff,axiom,
! [A: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [S: set @ A,A3: A] :
( ( countable_countable @ A @ S )
=> ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ S ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S )
& ( ord_less @ A @ A3 @ X2 ) ) ) ) ) ) ).
% less_ccSup_iff
thf(fact_6542_ccInf__less__iff,axiom,
! [A: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [S: set @ A,A3: A] :
( ( countable_countable @ A @ S )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S ) @ A3 )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S )
& ( ord_less @ A @ X2 @ A3 ) ) ) ) ) ) ).
% ccInf_less_iff
thf(fact_6543_to__nat__on__surj,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( countable_countable @ A @ A4 )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ( countable_to_nat_on @ A @ A4 @ X3 )
= N ) ) ) ) ).
% to_nat_on_surj
thf(fact_6544_countable__finite,axiom,
! [A: $tType,S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( countable_countable @ A @ S ) ) ).
% countable_finite
thf(fact_6545_uncountable__infinite,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( countable_countable @ A @ A4 )
=> ~ ( finite_finite2 @ A @ A4 ) ) ).
% uncountable_infinite
thf(fact_6546_countable__Collect__finite,axiom,
! [A: $tType] :
( ( countable @ A )
=> ( countable_countable @ ( set @ A ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ).
% countable_Collect_finite
thf(fact_6547_infinite__countable__subset_H,axiom,
! [A: $tType,X4: set @ A] :
( ~ ( finite_finite2 @ A @ X4 )
=> ? [C7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C7 @ X4 )
& ( countable_countable @ A @ C7 )
& ~ ( finite_finite2 @ A @ C7 ) ) ) ).
% infinite_countable_subset'
thf(fact_6548_countable__Collect__finite__subset,axiom,
! [A: $tType,T4: set @ A] :
( ( countable_countable @ A @ T4 )
=> ( countable_countable @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
& ( ord_less_eq @ ( set @ A ) @ A6 @ T4 ) ) ) ) ) ).
% countable_Collect_finite_subset
thf(fact_6549_ccSup__subset__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( countable_countable @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ).
% ccSup_subset_mono
thf(fact_6550_ccInf__superset__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ) ).
% ccInf_superset_mono
thf(fact_6551_countable__image__eq__inj,axiom,
! [A: $tType,B: $tType,F3: B > A,S: set @ B] :
( ( countable_countable @ A @ ( image2 @ B @ A @ F3 @ S ) )
= ( ? [T10: set @ B] :
( ( countable_countable @ B @ T10 )
& ( ord_less_eq @ ( set @ B ) @ T10 @ S )
& ( ( image2 @ B @ A @ F3 @ S )
= ( image2 @ B @ A @ F3 @ T10 ) )
& ( inj_on @ B @ A @ F3 @ T10 ) ) ) ) ).
% countable_image_eq_inj
thf(fact_6552_ex__countable__subset__image__inj,axiom,
! [A: $tType,B: $tType,F3: B > A,S: set @ B,P: ( set @ A ) > $o] :
( ( ? [T10: set @ A] :
( ( countable_countable @ A @ T10 )
& ( ord_less_eq @ ( set @ A ) @ T10 @ ( image2 @ B @ A @ F3 @ S ) )
& ( P @ T10 ) ) )
= ( ? [T10: set @ B] :
( ( countable_countable @ B @ T10 )
& ( ord_less_eq @ ( set @ B ) @ T10 @ S )
& ( inj_on @ B @ A @ F3 @ T10 )
& ( P @ ( image2 @ B @ A @ F3 @ T10 ) ) ) ) ) ).
% ex_countable_subset_image_inj
thf(fact_6553_all__countable__subset__image__inj,axiom,
! [A: $tType,B: $tType,F3: B > A,S: set @ B,P: ( set @ A ) > $o] :
( ( ! [T10: set @ A] :
( ( ( countable_countable @ A @ T10 )
& ( ord_less_eq @ ( set @ A ) @ T10 @ ( image2 @ B @ A @ F3 @ S ) ) )
=> ( P @ T10 ) ) )
= ( ! [T10: set @ B] :
( ( ( countable_countable @ B @ T10 )
& ( ord_less_eq @ ( set @ B ) @ T10 @ S )
& ( inj_on @ B @ A @ F3 @ T10 ) )
=> ( P @ ( image2 @ B @ A @ F3 @ T10 ) ) ) ) ) ).
% all_countable_subset_image_inj
thf(fact_6554_ccSup__union__distrib,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ( countable_countable @ A @ B2 )
=> ( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ).
% ccSup_union_distrib
thf(fact_6555_ccInf__union__distrib,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ( countable_countable @ A @ B2 )
=> ( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ) ) ).
% ccInf_union_distrib
thf(fact_6556_countable__Image,axiom,
! [B: $tType,A: $tType,Y6: set @ A,X4: set @ ( product_prod @ A @ B )] :
( ! [Y2: A] :
( ( member @ A @ Y2 @ Y6 )
=> ( countable_countable @ B @ ( image @ A @ B @ X4 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( countable_countable @ A @ Y6 )
=> ( countable_countable @ B @ ( image @ A @ B @ X4 @ Y6 ) ) ) ) ).
% countable_Image
thf(fact_6557_uncountable__def,axiom,
! [A: $tType,A4: set @ A] :
( ( ~ ( countable_countable @ A @ A4 ) )
= ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
& ~ ? [F2: nat > A] :
( ( image2 @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) )
= A4 ) ) ) ).
% uncountable_def
thf(fact_6558_union__set__fold,axiom,
! [A: $tType,Xs: list @ A,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 )
= ( fold @ A @ ( set @ A ) @ ( insert2 @ A ) @ Xs @ A4 ) ) ).
% union_set_fold
thf(fact_6559_ccSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,B2: set @ C,F3: B > A,G2: C > A] :
( ( countable_countable @ B @ A4 )
=> ( ( countable_countable @ C @ B2 )
=> ( ! [N3: B] :
( ( member @ B @ N3 @ A4 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B2 )
& ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G2 @ X5 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G2 @ B2 ) ) ) ) ) ) ) ).
% ccSUP_mono
thf(fact_6560_ccSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,F3: B > A,U: A] :
( ( countable_countable @ B @ A4 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ U ) ) ) ) ).
% ccSUP_least
thf(fact_6561_ccSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,I: B,F3: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ I ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ) ).
% ccSUP_upper
thf(fact_6562_ccSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,F3: B > A,U: A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ U )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ X2 ) @ U ) ) ) ) ) ) ).
% ccSUP_le_iff
thf(fact_6563_ccSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,I: B,U: A,F3: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( member @ B @ I @ A4 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ I ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ) ) ).
% ccSUP_upper2
thf(fact_6564_countable__infiniteE_H,axiom,
! [A: $tType,A4: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ~ ! [G9: nat > A] :
~ ( bij_betw @ nat @ A @ G9 @ ( top_top @ ( set @ nat ) ) @ A4 ) ) ) ).
% countable_infiniteE'
thf(fact_6565_less__ccSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [A4: set @ B,A3: A,F3: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ A3 @ ( F3 @ X2 ) ) ) ) ) ) ) ).
% less_ccSUP_iff
thf(fact_6566_ccINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,U: A,F3: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ U @ ( F3 @ I2 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ) ) ).
% ccINF_greatest
thf(fact_6567_le__ccINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,U: A,F3: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ord_less_eq @ A @ U @ ( F3 @ X2 ) ) ) ) ) ) ) ).
% le_ccINF_iff
thf(fact_6568_ccINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,I: B,F3: B > A,U: A] :
( ( countable_countable @ B @ A4 )
=> ( ( member @ B @ I @ A4 )
=> ( ( ord_less_eq @ A @ ( F3 @ I ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ U ) ) ) ) ) ).
% ccINF_lower2
thf(fact_6569_ccINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,I: B,F3: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( F3 @ I ) ) ) ) ) ).
% ccINF_lower
thf(fact_6570_ccINF__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,B2: set @ C,F3: B > A,G2: C > A] :
( ( countable_countable @ B @ A4 )
=> ( ( countable_countable @ C @ B2 )
=> ( ! [M4: C] :
( ( member @ C @ M4 @ B2 )
=> ? [X5: B] :
( ( member @ B @ X5 @ A4 )
& ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G2 @ M4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G2 @ B2 ) ) ) ) ) ) ) ).
% ccINF_mono
thf(fact_6571_ccINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [A4: set @ B,F3: B > A,A3: A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ A3 )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ ( F3 @ X2 ) @ A3 ) ) ) ) ) ) ).
% ccINF_less_iff
thf(fact_6572_ccSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,F3: B > A,G2: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A5: B] : ( sup_sup @ A @ ( F3 @ A5 ) @ ( G2 @ A5 ) )
@ A4 ) ) ) ) ) ).
% ccSUP_sup_distrib
thf(fact_6573_ccINF__sup__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( counta4013691401010221786attice @ A )
=> ! [A4: set @ B,B2: set @ C,F3: B > A,G2: C > A] :
( ( countable_countable @ B @ A4 )
=> ( ( countable_countable @ C @ B2 )
=> ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G2 @ B2 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A5: B] :
( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [B5: C] : ( sup_sup @ A @ ( F3 @ A5 ) @ ( G2 @ B5 ) )
@ B2 ) )
@ A4 ) ) ) ) ) ) ).
% ccINF_sup_distrib2
thf(fact_6574_sup__ccInf,axiom,
! [A: $tType] :
( ( counta4013691401010221786attice @ A )
=> ! [B2: set @ A,A3: A] :
( ( countable_countable @ A @ B2 )
=> ( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ B2 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ A @ A @ ( sup_sup @ A @ A3 ) @ B2 ) ) ) ) ) ).
% sup_ccInf
thf(fact_6575_sup__ccINF,axiom,
! [A: $tType,B: $tType] :
( ( counta4013691401010221786attice @ A )
=> ! [B2: set @ B,A3: A,F3: B > A] :
( ( countable_countable @ B @ B2 )
=> ( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ B2 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( sup_sup @ A @ A3 @ ( F3 @ B5 ) )
@ B2 ) ) ) ) ) ).
% sup_ccINF
thf(fact_6576_ccInf__sup,axiom,
! [A: $tType] :
( ( counta4013691401010221786attice @ A )
=> ! [B2: set @ A,A3: A] :
( ( countable_countable @ A @ B2 )
=> ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B2 ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image2 @ A @ A
@ ^ [B5: A] : ( sup_sup @ A @ B5 @ A3 )
@ B2 ) ) ) ) ) ).
% ccInf_sup
thf(fact_6577_ccINF__sup,axiom,
! [A: $tType,B: $tType] :
( ( counta4013691401010221786attice @ A )
=> ! [B2: set @ B,F3: B > A,A3: A] :
( ( countable_countable @ B @ B2 )
=> ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ B2 ) ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( sup_sup @ A @ ( F3 @ B5 ) @ A3 )
@ B2 ) ) ) ) ) ).
% ccINF_sup
thf(fact_6578_countableE__infinite,axiom,
! [A: $tType,S: set @ A] :
( ( countable_countable @ A @ S )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ! [E: A > nat] :
~ ( bij_betw @ A @ nat @ E @ S @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% countableE_infinite
thf(fact_6579_ccSup__inter__less__eq,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ( countable_countable @ A @ B2 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ).
% ccSup_inter_less_eq
thf(fact_6580_less__eq__ccInf__inter,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ( countable_countable @ A @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ) ).
% less_eq_ccInf_inter
thf(fact_6581_ccSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B2: set @ B,A4: set @ B,F3: B > A,G2: B > A] :
( ( countable_countable @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ B2 ) ) ) ) ) ) ) ).
% ccSUP_subset_mono
thf(fact_6582_ccINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,B2: set @ B,F3: B > A,G2: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ A4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B2 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ B2 ) ) ) ) ) ) ) ).
% ccINF_superset_mono
thf(fact_6583_Sup__set__fold,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Xs: list @ A] :
( ( complete_Sup_Sup @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( sup_sup @ A ) @ Xs @ ( bot_bot @ A ) ) ) ) ).
% Sup_set_fold
thf(fact_6584_ccSUP__union,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,B2: set @ B,M5: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( countable_countable @ B @ B2 )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M5 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M5 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M5 @ B2 ) ) ) ) ) ) ) ).
% ccSUP_union
thf(fact_6585_mono__ccSup,axiom,
! [B: $tType,A: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F3: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( countable_countable @ A @ A4 )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ A4 ) ) @ ( F3 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ).
% mono_ccSup
thf(fact_6586_mono__ccSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F3: A > B,I5: set @ C,A4: C > A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( countable_countable @ C @ I5 )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( F3 @ ( A4 @ X2 ) )
@ I5 ) )
@ ( F3 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A4 @ I5 ) ) ) ) ) ) ) ).
% mono_ccSUP
thf(fact_6587_ccINF__union,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,B2: set @ B,M5: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( countable_countable @ B @ B2 )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M5 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M5 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M5 @ B2 ) ) ) ) ) ) ) ).
% ccINF_union
thf(fact_6588_mono__ccINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F3: A > B,I5: set @ C,A4: C > A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( countable_countable @ C @ I5 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A4 @ I5 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( F3 @ ( A4 @ X2 ) )
@ I5 ) ) ) ) ) ) ).
% mono_ccINF
thf(fact_6589_mono__ccInf,axiom,
! [B: $tType,A: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F3: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( countable_countable @ A @ A4 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ A4 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ A4 ) ) ) ) ) ) ).
% mono_ccInf
thf(fact_6590_countable__as__injective__image,axiom,
! [A: $tType,A4: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ~ ! [F6: nat > A] :
( ( A4
= ( image2 @ nat @ A @ F6 @ ( top_top @ ( set @ nat ) ) ) )
=> ~ ( inj_on @ nat @ A @ F6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% countable_as_injective_image
thf(fact_6591_image__to__nat__on,axiom,
! [A: $tType,A4: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ( ( image2 @ A @ nat @ ( countable_to_nat_on @ A @ A4 ) @ A4 )
= ( top_top @ ( set @ nat ) ) ) ) ) ).
% image_to_nat_on
thf(fact_6592_Sup__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic5882676163264333800up_fin @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( sup_sup @ A ) @ Xs @ X ) ) ) ).
% Sup_fin.set_eq_fold
thf(fact_6593_to__nat__on__infinite,axiom,
! [A: $tType,S: set @ A] :
( ( countable_countable @ A @ S )
=> ( ~ ( finite_finite2 @ A @ S )
=> ( bij_betw @ A @ nat @ ( countable_to_nat_on @ A @ S ) @ S @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% to_nat_on_infinite
thf(fact_6594_comp__fun__idem__on_Ofold__set__fold,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,Xs: list @ A,Y: B] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ S )
=> ( ( finite_fold @ A @ B @ F3 @ Y @ ( set2 @ A @ Xs ) )
= ( fold @ A @ B @ F3 @ Xs @ Y ) ) ) ) ).
% comp_fun_idem_on.fold_set_fold
thf(fact_6595_countable__enum__cases,axiom,
! [A: $tType,S: set @ A] :
( ( countable_countable @ A @ S )
=> ( ( ( finite_finite2 @ A @ S )
=> ! [F6: A > nat] :
~ ( bij_betw @ A @ nat @ F6 @ S @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S ) ) ) )
=> ~ ( ~ ( finite_finite2 @ A @ S )
=> ! [F6: A > nat] :
~ ( bij_betw @ A @ nat @ F6 @ S @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% countable_enum_cases
thf(fact_6596_range__from__nat__into,axiom,
! [A: $tType,A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( countable_countable @ A @ A4 )
=> ( ( image2 @ nat @ A @ ( counta4804993851260445106t_into @ A @ A4 ) @ ( top_top @ ( set @ nat ) ) )
= A4 ) ) ) ).
% range_from_nat_into
thf(fact_6597_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,Xs: list @ A,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ S )
=> ( ( finite_fold @ A @ B @ F3 @ Y @ ( set2 @ A @ Xs ) )
= ( fold @ A @ B @ F3 @ ( remdups @ A @ Xs ) @ Y ) ) ) ) ).
% comp_fun_commute_on.fold_set_fold_remdups
thf(fact_6598_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_6599_from__nat__into__inject,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( countable_countable @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( countable_countable @ A @ B2 )
=> ( ( ( counta4804993851260445106t_into @ A @ A4 )
= ( counta4804993851260445106t_into @ A @ B2 ) )
= ( A4 = B2 ) ) ) ) ) ) ).
% from_nat_into_inject
thf(fact_6600_from__nat__into__inj__infinite,axiom,
! [A: $tType,A4: set @ A,M: nat,N: nat] :
( ( countable_countable @ A @ A4 )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ( ( ( counta4804993851260445106t_into @ A @ A4 @ M )
= ( counta4804993851260445106t_into @ A @ A4 @ N ) )
= ( M = N ) ) ) ) ).
% from_nat_into_inj_infinite
thf(fact_6601_to__nat__on__from__nat__into__infinite,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( countable_countable @ A @ A4 )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ( ( countable_to_nat_on @ A @ A4 @ ( counta4804993851260445106t_into @ A @ A4 @ N ) )
= N ) ) ) ).
% to_nat_on_from_nat_into_infinite
thf(fact_6602_from__nat__into,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( counta4804993851260445106t_into @ A @ A4 @ N ) @ A4 ) ) ).
% from_nat_into
thf(fact_6603_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_6604_inj__on__from__nat__into,axiom,
! [A: $tType] :
( inj_on @ ( set @ A ) @ ( nat > A ) @ ( counta4804993851260445106t_into @ A )
@ ( collect @ ( set @ A )
@ ^ [A6: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
& ( countable_countable @ A @ A6 ) ) ) ) ).
% inj_on_from_nat_into
thf(fact_6605_range__from__nat__into__subset,axiom,
! [A: $tType,A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ ( counta4804993851260445106t_into @ A @ A4 ) @ ( top_top @ ( set @ nat ) ) ) @ A4 ) ) ).
% range_from_nat_into_subset
thf(fact_6606_subset__range__from__nat__into,axiom,
! [A: $tType,A4: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( image2 @ nat @ A @ ( counta4804993851260445106t_into @ A @ A4 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% subset_range_from_nat_into
thf(fact_6607_bij__betw__from__nat__into__finite,axiom,
! [A: $tType,S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( bij_betw @ nat @ A @ ( counta4804993851260445106t_into @ A @ S ) @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S ) ) @ S ) ) ).
% bij_betw_from_nat_into_finite
thf(fact_6608_bij__betw__from__nat__into,axiom,
! [A: $tType,S: set @ A] :
( ( countable_countable @ A @ S )
=> ( ~ ( finite_finite2 @ A @ S )
=> ( bij_betw @ nat @ A @ ( counta4804993851260445106t_into @ A @ S ) @ ( top_top @ ( set @ nat ) ) @ S ) ) ) ).
% bij_betw_from_nat_into
thf(fact_6609_comp__fun__idem__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite673082921795544331dem_on @ A @ B )
= ( ^ [S7: set @ A,F2: A > B > B] :
( ( finite4664212375090638736ute_on @ A @ B @ S7 @ F2 )
& ( finite4980608107308702382axioms @ A @ B @ S7 @ F2 ) ) ) ) ).
% comp_fun_idem_on_def
thf(fact_6610_comp__fun__idem__on_Ointro,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( finite4980608107308702382axioms @ A @ B @ S @ F3 )
=> ( finite673082921795544331dem_on @ A @ B @ S @ F3 ) ) ) ).
% comp_fun_idem_on.intro
thf(fact_6611_comp__fun__idem__on__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite4980608107308702382axioms @ A @ B )
= ( ^ [S7: set @ A,F2: A > B > B] :
! [X2: A] :
( ( member @ A @ X2 @ S7 )
=> ( ( comp @ B @ B @ B @ ( F2 @ X2 ) @ ( F2 @ X2 ) )
= ( F2 @ X2 ) ) ) ) ) ).
% comp_fun_idem_on_axioms_def
thf(fact_6612_comp__fun__idem__on__axioms_Ointro,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( comp @ B @ B @ B @ ( F3 @ X3 ) @ ( F3 @ X3 ) )
= ( F3 @ X3 ) ) )
=> ( finite4980608107308702382axioms @ A @ B @ S @ F3 ) ) ).
% comp_fun_idem_on_axioms.intro
thf(fact_6613_comp__fun__idem__on_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F3 )
=> ( finite4980608107308702382axioms @ A @ B @ S @ F3 ) ) ).
% comp_fun_idem_on.axioms(2)
thf(fact_6614_Field__insert,axiom,
! [A: $tType,A3: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 ) )
= ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) ) ) ).
% Field_insert
thf(fact_6615_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,A4: set @ A,Z: B,Y: B,A3: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A4 @ Y )
=> ( ( member @ A @ A3 @ A4 )
=> ? [Y8: B] :
( ( Y
= ( F3 @ A3 @ Y8 ) )
& ( finite_fold_graph @ A @ B @ F3 @ Z @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ Y8 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE_aux
thf(fact_6616_Field__empty,axiom,
! [A: $tType] :
( ( field2 @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Field_empty
thf(fact_6617_Field__Un,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) )
= ( sup_sup @ ( set @ A ) @ ( field2 @ A @ R2 ) @ ( field2 @ A @ S3 ) ) ) ).
% Field_Un
thf(fact_6618_Field__Union,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( field2 @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ A ) @ ( field2 @ A ) @ R ) ) ) ).
% Field_Union
thf(fact_6619_fold__graph_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold_graph @ A @ B )
= ( ^ [F2: A > B > B,Z6: B,A12: set @ A,A23: B] :
( ( ( A12
= ( bot_bot @ ( set @ A ) ) )
& ( A23 = Z6 ) )
| ? [X2: A,A6: set @ A,Y3: B] :
( ( A12
= ( insert2 @ A @ X2 @ A6 ) )
& ( A23
= ( F2 @ X2 @ Y3 ) )
& ~ ( member @ A @ X2 @ A6 )
& ( finite_fold_graph @ A @ B @ F2 @ Z6 @ A6 @ Y3 ) ) ) ) ) ).
% fold_graph.simps
thf(fact_6620_fold__graph_Ocases,axiom,
! [A: $tType,B: $tType,F3: A > B > B,Z: B,A13: set @ A,A24: B] :
( ( finite_fold_graph @ A @ B @ F3 @ Z @ A13 @ A24 )
=> ( ( ( A13
= ( bot_bot @ ( set @ A ) ) )
=> ( A24 != Z ) )
=> ~ ! [X3: A,A9: set @ A] :
( ( A13
= ( insert2 @ A @ X3 @ A9 ) )
=> ! [Y2: B] :
( ( A24
= ( F3 @ X3 @ Y2 ) )
=> ( ~ ( member @ A @ X3 @ A9 )
=> ~ ( finite_fold_graph @ A @ B @ F3 @ Z @ A9 @ Y2 ) ) ) ) ) ) ).
% fold_graph.cases
thf(fact_6621_finite__imp__fold__graph,axiom,
! [A: $tType,B: $tType,A4: set @ A,F3: A > B > B,Z: B] :
( ( finite_finite2 @ A @ A4 )
=> ? [X_1: B] : ( finite_fold_graph @ A @ B @ F3 @ Z @ A4 @ X_1 ) ) ).
% finite_imp_fold_graph
thf(fact_6622_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_6623_empty__fold__graphE,axiom,
! [A: $tType,B: $tType,F3: A > B > B,Z: B,X: B] :
( ( finite_fold_graph @ A @ B @ F3 @ Z @ ( bot_bot @ ( set @ A ) ) @ X )
=> ( X = Z ) ) ).
% empty_fold_graphE
thf(fact_6624_fold__graph_OemptyI,axiom,
! [A: $tType,B: $tType,F3: A > B > B,Z: B] : ( finite_fold_graph @ A @ B @ F3 @ Z @ ( bot_bot @ ( set @ A ) ) @ Z ) ).
% fold_graph.emptyI
thf(fact_6625_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_6626_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_6627_fold__graph_OinsertI,axiom,
! [A: $tType,B: $tType,X: A,A4: set @ A,F3: A > B > B,Z: B,Y: B] :
( ~ ( member @ A @ X @ A4 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A4 @ Y )
=> ( finite_fold_graph @ A @ B @ F3 @ Z @ ( insert2 @ A @ X @ A4 ) @ ( F3 @ X @ Y ) ) ) ) ).
% fold_graph.insertI
thf(fact_6628_fold__graph__closed__eq,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,F3: A > B > B,G2: A > B > B,Z: B] :
( ! [A7: A,B7: B] :
( ( member @ A @ A7 @ A4 )
=> ( ( member @ B @ B7 @ B2 )
=> ( ( F3 @ A7 @ B7 )
= ( G2 @ A7 @ B7 ) ) ) )
=> ( ! [A7: A,B7: B] :
( ( member @ A @ A7 @ A4 )
=> ( ( member @ B @ B7 @ B2 )
=> ( member @ B @ ( G2 @ A7 @ B7 ) @ B2 ) ) )
=> ( ( member @ B @ Z @ B2 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A4 )
= ( finite_fold_graph @ A @ B @ G2 @ Z @ A4 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_6629_fold__graph__closed__lemma,axiom,
! [A: $tType,B: $tType,G2: A > B > B,Z: B,A4: set @ A,X: B,B2: set @ B,F3: A > B > B] :
( ( finite_fold_graph @ A @ B @ G2 @ Z @ A4 @ X )
=> ( ! [A7: A,B7: B] :
( ( member @ A @ A7 @ A4 )
=> ( ( member @ B @ B7 @ B2 )
=> ( ( F3 @ A7 @ B7 )
= ( G2 @ A7 @ B7 ) ) ) )
=> ( ! [A7: A,B7: B] :
( ( member @ A @ A7 @ A4 )
=> ( ( member @ B @ B7 @ B2 )
=> ( member @ B @ ( G2 @ A7 @ B7 ) @ B2 ) ) )
=> ( ( member @ B @ Z @ B2 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A4 @ X )
& ( member @ B @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_6630_mono__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 )
=> ( ord_less_eq @ ( set @ A ) @ ( field2 @ A @ R2 ) @ ( field2 @ A @ S3 ) ) ) ).
% mono_Field
thf(fact_6631_comp__fun__commute__on_Ofold__graph__finite,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,Z: B,A4: set @ A,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A4 @ Y )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% comp_fun_commute_on.fold_graph_finite
thf(fact_6632_comp__fun__commute__on_Ofold__graph__determ,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,A4: set @ A,Z: B,X: B,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A4 @ X )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A4 @ Y )
=> ( Y = X ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_determ
thf(fact_6633_fold__graph__image,axiom,
! [C: $tType,B: $tType,A: $tType,G2: A > B,A4: set @ A,F3: B > C > C,Z: C] :
( ( inj_on @ A @ B @ G2 @ A4 )
=> ( ( finite_fold_graph @ B @ C @ F3 @ Z @ ( image2 @ A @ B @ G2 @ A4 ) )
= ( finite_fold_graph @ A @ C @ ( comp @ B @ ( C > C ) @ A @ F3 @ G2 ) @ Z @ A4 ) ) ) ).
% fold_graph_image
thf(fact_6634_comp__fun__commute__on_Ofold__graph__insertE,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,X: A,A4: set @ A,Z: B,V2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ ( insert2 @ A @ X @ A4 ) @ V2 )
=> ( ~ ( member @ A @ X @ A4 )
=> ~ ! [Y2: B] :
( ( V2
= ( F3 @ X @ Y2 ) )
=> ~ ( finite_fold_graph @ A @ B @ F3 @ Z @ A4 @ Y2 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE
thf(fact_6635_comp__fun__commute__on_Ofold__equality,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,A4: set @ A,Z: B,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A4 @ Y )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ A4 )
= Y ) ) ) ) ).
% comp_fun_commute_on.fold_equality
thf(fact_6636_Finite__Set_Ofold__def,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold @ A @ B )
= ( ^ [F2: A > B > B,Z6: B,A6: set @ A] : ( if @ B @ ( finite_finite2 @ A @ A6 ) @ ( the @ B @ ( finite_fold_graph @ A @ B @ F2 @ Z6 @ A6 ) ) @ Z6 ) ) ) ).
% Finite_Set.fold_def
thf(fact_6637_comp__fun__commute__on_Ofold__graph__fold,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,A4: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( finite_fold_graph @ A @ B @ F3 @ Z @ A4 @ ( finite_fold @ A @ B @ F3 @ Z @ A4 ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_fold
thf(fact_6638_Field__natLeq__on,axiom,
! [N: nat] :
( ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X2 @ Y3 ) ) ) ) )
= ( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_6639_subset__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A3 ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_6640_preorder__on__empty,axiom,
! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% preorder_on_empty
thf(fact_6641_subset__Image__Image__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ A4 ) @ ( image @ A @ A @ R2 @ B2 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R2 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_6642_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ A ),As: A > B] :
! [I4: A,J3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I4 @ J3 ) @ R5 )
=> ( ord_less_eq @ B @ ( As @ I4 ) @ ( As @ J3 ) ) ) ) ) ) ).
% relChain_def
thf(fact_6643_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% natLess_def
thf(fact_6644_linear__order__on__singleton,axiom,
! [A: $tType,X: A] : ( order_679001287576687338der_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% linear_order_on_singleton
thf(fact_6645_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 ) ) ) )
| ? [A7: A] :
( R2
= ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ A7 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% Total_subset_Id
thf(fact_6646_pair__in__Id__conv,axiom,
! [A: $tType,A3: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( id2 @ A ) )
= ( A3 = B3 ) ) ).
% pair_in_Id_conv
thf(fact_6647_IdI,axiom,
! [A: $tType,A3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id2 @ A ) ) ).
% IdI
thf(fact_6648_Image__Id,axiom,
! [A: $tType,A4: set @ A] :
( ( image @ A @ A @ ( id2 @ A ) @ A4 )
= A4 ) ).
% Image_Id
thf(fact_6649_total__on__diff__Id,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ A4 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
= ( total_on @ A @ A4 @ R2 ) ) ).
% total_on_diff_Id
thf(fact_6650_IdE,axiom,
! [A: $tType,P6: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P6 @ ( id2 @ A ) )
=> ~ ! [X3: A] :
( P6
!= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ).
% IdE
thf(fact_6651_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_6652_Id__def,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [P5: product_prod @ A @ A] :
? [X2: A] :
( P5
= ( product_Pair @ A @ A @ X2 @ X2 ) ) ) ) ).
% Id_def
thf(fact_6653_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_6654_Total__Id__Field,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 ) )
=> ( ( field2 @ A @ R2 )
= ( field2 @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ) ).
% Total_Id_Field
thf(fact_6655_bsqr__def,axiom,
! [A: $tType] :
( ( bNF_Wellorder_bsqr @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) )
@ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o
@ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o )
@ ^ [A12: A,A23: A] :
( product_case_prod @ A @ A @ $o
@ ^ [B14: A,B23: A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A12 @ ( insert2 @ A @ A23 @ ( insert2 @ A @ B14 @ ( insert2 @ A @ B23 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R5 ) )
& ( ( ( A12 = B14 )
& ( A23 = B23 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A12 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B14 @ B23 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B14 @ B23 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B14 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B14 @ B23 ) )
& ( A12 = B14 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A23 @ B23 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_6656_max__ext_Omax__extI,axiom,
! [A: $tType,X4: set @ A,Y6: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X4 )
=> ( ( finite_finite2 @ A @ Y6 )
=> ( ( Y6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Y6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa ) @ R ) ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X4 @ Y6 ) @ ( max_ext @ A @ R ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_6657_max__ext__additive,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,R: set @ ( product_prod @ A @ A ),C2: set @ A,D3: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A4 @ B2 ) @ ( max_ext @ A @ R ) )
=> ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ C2 @ D3 ) @ ( max_ext @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ C2 ) @ ( sup_sup @ ( set @ A ) @ B2 @ D3 ) ) @ ( max_ext @ A @ R ) ) ) ) ).
% max_ext_additive
thf(fact_6658_max__ext_Ocases,axiom,
! [A: $tType,A13: set @ A,A24: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A13 @ A24 ) @ ( max_ext @ A @ R ) )
=> ~ ( ( finite_finite2 @ A @ A13 )
=> ( ( finite_finite2 @ A @ A24 )
=> ( ( A24
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ A13 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ A24 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa2 ) @ R ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_6659_max__ext_Osimps,axiom,
! [A: $tType,A13: set @ A,A24: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A13 @ A24 ) @ ( max_ext @ A @ R ) )
= ( ( finite_finite2 @ A @ A13 )
& ( finite_finite2 @ A @ A24 )
& ( A24
!= ( bot_bot @ ( set @ A ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A13 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A24 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R ) ) ) ) ) ).
% max_ext.simps
thf(fact_6660_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 ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A6 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_6661_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
@ ^ [X8: set @ A,Y7: set @ A] :
( ( finite_finite2 @ A @ X8 )
& ( finite_finite2 @ A @ Y7 )
& ( Y7
!= ( bot_bot @ ( set @ A ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ Y7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R6 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_6662_bex__empty,axiom,
! [A: $tType,P: A > $o] :
~ ? [X5: A] :
( ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
& ( P @ X5 ) ) ).
% bex_empty
thf(fact_6663_finite__Collect__bex,axiom,
! [B: $tType,A: $tType,A4: set @ A,Q: B > A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
? [Y3: A] :
( ( member @ A @ Y3 @ A4 )
& ( Q @ X2 @ Y3 ) ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Y3: B] : ( Q @ Y3 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_6664_bex__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) )
& ( P @ X2 ) ) )
= ( ? [X8: A] : ( P @ X8 ) ) ) ).
% bex_UNIV
thf(fact_6665_Image__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: B > A > $o,A4: set @ B] :
( ( image @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) @ A4 )
= ( collect @ A
@ ^ [Y3: A] :
? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( P @ X2 @ Y3 ) ) ) ) ).
% Image_Collect_case_prod
thf(fact_6666_wf__if__measure,axiom,
! [A: $tType,P: A > $o,F3: A > nat,G2: A > A] :
( ! [X3: A] :
( ( P @ X3 )
=> ( ord_less @ nat @ ( F3 @ ( G2 @ X3 ) ) @ ( F3 @ X3 ) ) )
=> ( wf @ A
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,X2: A] :
( ( P @ X2 )
& ( Y3
= ( G2 @ X2 ) ) ) ) ) ) ) ).
% wf_if_measure
thf(fact_6667_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_6668_wf__less,axiom,
wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ).
% wf_less
thf(fact_6669_wf__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P6: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ P6 @ R2 )
=> ( wf @ A @ P6 ) ) ) ).
% wf_subset
thf(fact_6670_Bex__def,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A6: set @ A,P3: A > $o] :
? [X2: A] :
( ( member @ A @ X2 @ A6 )
& ( P3 @ X2 ) ) ) ) ).
% Bex_def
thf(fact_6671_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_6672_Image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B ),S8: set @ A] :
( collect @ B
@ ^ [Y3: B] :
? [X2: A] :
( ( member @ A @ X2 @ S8 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R5 ) ) ) ) ) ).
% Image_def
thf(fact_6673_image__def,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ A @ B )
= ( ^ [F2: A > B,A6: set @ A] :
( collect @ B
@ ^ [Y3: B] :
? [X2: A] :
( ( member @ A @ X2 @ A6 )
& ( Y3
= ( F2 @ X2 ) ) ) ) ) ) ).
% image_def
thf(fact_6674_wf__bounded__measure,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > nat,F3: A > nat] :
( ! [A7: A,B7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B7 @ A7 ) @ R2 )
=> ( ( ord_less_eq @ nat @ ( Ub @ B7 ) @ ( Ub @ A7 ) )
& ( ord_less_eq @ nat @ ( F3 @ B7 ) @ ( Ub @ A7 ) )
& ( ord_less @ nat @ ( F3 @ A7 ) @ ( F3 @ B7 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wf_bounded_measure
thf(fact_6675_wfE__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( wf @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( image @ A @ A @ R @ A4 ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% wfE_pf
thf(fact_6676_wfI__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A9 @ ( image @ A @ A @ R @ A9 ) )
=> ( A9
= ( bot_bot @ ( set @ A ) ) ) )
=> ( wf @ A @ R ) ) ).
% wfI_pf
thf(fact_6677_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 )
=> ( ! [X3: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ ( transitive_trancl @ A @ R2 ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) )
=> ( ( P @ K )
=> ? [X3: B] :
( ( P @ X3 )
& ! [Y5: B] :
( ( P @ Y5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M @ X3 ) @ ( M @ Y5 ) ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_6678_Bex__fold,axiom,
! [A: $tType,A4: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( P @ X2 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K3: A,S8: $o] :
( S8
| ( P @ K3 ) )
@ $false
@ A4 ) ) ) ).
% Bex_fold
thf(fact_6679_nths__nths,axiom,
! [A: $tType,Xs: list @ A,A4: set @ nat,B2: set @ nat] :
( ( nths @ A @ ( nths @ A @ Xs @ A4 ) @ B2 )
= ( nths @ A @ Xs
@ ( collect @ nat
@ ^ [I4: nat] :
( ( member @ nat @ I4 @ A4 )
& ( member @ nat
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [I9: nat] :
( ( member @ nat @ I9 @ A4 )
& ( ord_less @ nat @ I9 @ I4 ) ) ) )
@ B2 ) ) ) ) ) ).
% nths_nths
thf(fact_6680_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 ) ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A6 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R5 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_6681_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F3: A > ( set @ B )] :
( ! [A7: A,B7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B7 @ A7 ) @ R2 )
=> ( ( finite_finite2 @ B @ ( Ub @ A7 ) )
& ( ord_less_eq @ ( set @ B ) @ ( Ub @ B7 ) @ ( Ub @ A7 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F3 @ B7 ) @ ( Ub @ A7 ) )
& ( ord_less @ ( set @ B ) @ ( F3 @ A7 ) @ ( F3 @ B7 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wf_bounded_set
thf(fact_6682_finite__subset__wf,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( wf @ ( set @ A )
@ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X8: set @ A,Y7: set @ A] :
( ( ord_less @ ( set @ A ) @ X8 @ Y7 )
& ( ord_less_eq @ ( set @ A ) @ Y7 @ A4 ) ) ) ) ) ) ).
% finite_subset_wf
thf(fact_6683_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 )] :
? [X8: set @ A,Y7: set @ A] :
( ( Uu3
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X8 @ Y7 ) )
& ( X8
!= ( bot_bot @ ( set @ A ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ Y7 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ X8 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R5 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_6684_max__extp_Omax__extI,axiom,
! [A: $tType,X4: set @ A,Y6: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ X4 )
=> ( ( finite_finite2 @ A @ Y6 )
=> ( ( Y6
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Y6 )
& ( R @ X3 @ Xa ) ) )
=> ( max_extp @ A @ R @ X4 @ Y6 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_6685_max__extp_Ocases,axiom,
! [A: $tType,R: A > A > $o,A13: set @ A,A24: set @ A] :
( ( max_extp @ A @ R @ A13 @ A24 )
=> ~ ( ( finite_finite2 @ A @ A13 )
=> ( ( finite_finite2 @ A @ A24 )
=> ( ( A24
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ A13 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ A24 )
& ( R @ X5 @ Xa2 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_6686_max__extp_Osimps,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R6: A > A > $o,A12: set @ A,A23: set @ A] :
( ( finite_finite2 @ A @ A12 )
& ( finite_finite2 @ A @ A23 )
& ( A23
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A12 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A23 )
& ( R6 @ X2 @ Y3 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_6687_cauchy__def,axiom,
( cauchy
= ( ^ [X8: nat > rat] :
! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ K3 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) ) @ R5 ) ) ) ) ) ) ).
% cauchy_def
thf(fact_6688_cauchyI,axiom,
! [X4: nat > rat] :
( ! [R3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R3 )
=> ? [K10: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ K10 @ M4 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ K10 @ N3 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X4 @ M4 ) @ ( X4 @ N3 ) ) ) @ R3 ) ) ) )
=> ( cauchy @ X4 ) ) ).
% cauchyI
thf(fact_6689_cauchy__imp__bounded,axiom,
! [X4: nat > rat] :
( ( cauchy @ X4 )
=> ? [B7: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B7 )
& ! [N4: nat] : ( ord_less @ rat @ ( abs_abs @ rat @ ( X4 @ N4 ) ) @ B7 ) ) ) ).
% cauchy_imp_bounded
thf(fact_6690_cauchyD,axiom,
! [X4: nat > rat,R2: rat] :
( ( cauchy @ X4 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ? [K2: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ K2 @ M3 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ K2 @ N4 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X4 @ M3 ) @ ( X4 @ N4 ) ) ) @ R2 ) ) ) ) ) ).
% cauchyD
thf(fact_6691_le__Real,axiom,
! [X4: nat > rat,Y6: nat > rat] :
( ( cauchy @ X4 )
=> ( ( cauchy @ Y6 )
=> ( ( ord_less_eq @ real @ ( real2 @ X4 ) @ ( real2 @ Y6 ) )
= ( ! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less_eq @ rat @ ( X4 @ N2 ) @ ( plus_plus @ rat @ ( Y6 @ N2 ) @ R5 ) ) ) ) ) ) ) ) ).
% le_Real
thf(fact_6692_cauchy__not__vanishes,axiom,
! [X4: nat > rat] :
( ( cauchy @ X4 )
=> ( ~ ( vanishes @ X4 )
=> ? [B7: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B7 )
& ? [K2: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ K2 @ N4 )
=> ( ord_less @ rat @ B7 @ ( abs_abs @ rat @ ( X4 @ N4 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes
thf(fact_6693_vanishes__mult__bounded,axiom,
! [X4: nat > rat,Y6: nat > rat] :
( ? [A10: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ A10 )
& ! [N3: nat] : ( ord_less @ rat @ ( abs_abs @ rat @ ( X4 @ N3 ) ) @ A10 ) )
=> ( ( vanishes @ Y6 )
=> ( vanishes
@ ^ [N2: nat] : ( times_times @ rat @ ( X4 @ N2 ) @ ( Y6 @ N2 ) ) ) ) ) ).
% vanishes_mult_bounded
thf(fact_6694_vanishesD,axiom,
! [X4: nat > rat,R2: rat] :
( ( vanishes @ X4 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ? [K2: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ K2 @ N4 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X4 @ N4 ) ) @ R2 ) ) ) ) ).
% vanishesD
thf(fact_6695_vanishesI,axiom,
! [X4: nat > rat] :
( ! [R3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R3 )
=> ? [K10: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K10 @ N3 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X4 @ N3 ) ) @ R3 ) ) )
=> ( vanishes @ X4 ) ) ).
% vanishesI
thf(fact_6696_vanishes__def,axiom,
( vanishes
= ( ^ [X8: nat > rat] :
! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X8 @ N2 ) ) @ R5 ) ) ) ) ) ).
% vanishes_def
thf(fact_6697_cauchy__not__vanishes__cases,axiom,
! [X4: nat > rat] :
( ( cauchy @ X4 )
=> ( ~ ( vanishes @ X4 )
=> ? [B7: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B7 )
& ? [K2: nat] :
( ! [N4: nat] :
( ( ord_less_eq @ nat @ K2 @ N4 )
=> ( ord_less @ rat @ B7 @ ( uminus_uminus @ rat @ ( X4 @ N4 ) ) ) )
| ! [N4: nat] :
( ( ord_less_eq @ nat @ K2 @ N4 )
=> ( ord_less @ rat @ B7 @ ( X4 @ N4 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes_cases
thf(fact_6698_not__positive__Real,axiom,
! [X4: nat > rat] :
( ( cauchy @ X4 )
=> ( ( ~ ( positive2 @ ( real2 @ X4 ) ) )
= ( ! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less_eq @ rat @ ( X4 @ N2 ) @ R5 ) ) ) ) ) ) ).
% not_positive_Real
thf(fact_6699_positive__Real,axiom,
! [X4: nat > rat] :
( ( cauchy @ X4 )
=> ( ( positive2 @ ( real2 @ X4 ) )
= ( ? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less @ rat @ R5 @ ( X4 @ N2 ) ) ) ) ) ) ) ).
% positive_Real
thf(fact_6700_less__real__def,axiom,
( ( ord_less @ real )
= ( ^ [X2: real,Y3: real] : ( positive2 @ ( minus_minus @ real @ Y3 @ X2 ) ) ) ) ).
% less_real_def
thf(fact_6701_Real_Opositive_Orep__eq,axiom,
( positive2
= ( ^ [X2: real] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less @ rat @ R5 @ ( rep_real @ X2 @ N2 ) ) ) ) ) ) ).
% Real.positive.rep_eq
thf(fact_6702_finite__def,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( complete_lattice_lfp @ ( ( set @ A ) > $o )
@ ^ [P5: ( set @ A ) > $o,X2: set @ A] :
( ( X2
= ( bot_bot @ ( set @ A ) ) )
| ? [A6: set @ A,A5: A] :
( ( X2
= ( insert2 @ A @ A5 @ A6 ) )
& ( P5 @ A6 ) ) ) ) ) ).
% finite_def
thf(fact_6703_lfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F4: A > A,X: A] :
( ( order_mono @ A @ A @ F4 )
=> ( ( ( F4 @ X )
= X )
=> ( ! [Z3: A] :
( ( ( F4 @ Z3 )
= Z3 )
=> ( ord_less_eq @ A @ X @ Z3 ) )
=> ( ( complete_lattice_lfp @ A @ F4 )
= X ) ) ) ) ) ).
% lfp_eqI
thf(fact_6704_lfp__lfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A > A] :
( ! [X3: A,Y2: A,W: A,Z3: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ W @ Z3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 @ W ) @ ( F3 @ Y2 @ Z3 ) ) ) )
=> ( ( complete_lattice_lfp @ A
@ ^ [X2: A] : ( complete_lattice_lfp @ A @ ( F3 @ X2 ) ) )
= ( complete_lattice_lfp @ A
@ ^ [X2: A] : ( F3 @ X2 @ X2 ) ) ) ) ) ).
% lfp_lfp
thf(fact_6705_lfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,G2: A > A] :
( ! [Z10: A] : ( ord_less_eq @ A @ ( F3 @ Z10 ) @ ( G2 @ Z10 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F3 ) @ ( complete_lattice_lfp @ A @ G2 ) ) ) ) ).
% lfp_mono
thf(fact_6706_lfp__lowerbound,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,A4: A] :
( ( ord_less_eq @ A @ ( F3 @ A4 ) @ A4 )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F3 ) @ A4 ) ) ) ).
% lfp_lowerbound
thf(fact_6707_lfp__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,A4: A] :
( ! [U4: A] :
( ( ord_less_eq @ A @ ( F3 @ U4 ) @ U4 )
=> ( ord_less_eq @ A @ A4 @ U4 ) )
=> ( ord_less_eq @ A @ A4 @ ( complete_lattice_lfp @ A @ F3 ) ) ) ) ).
% lfp_greatest
thf(fact_6708_lfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_lfp @ A )
= ( ^ [F2: A > A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [U2: A] : ( ord_less_eq @ A @ ( F2 @ U2 ) @ U2 ) ) ) ) ) ) ).
% lfp_def
thf(fact_6709_def__lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: A,F3: A > A,P: A] :
( ( A4
= ( complete_lattice_lfp @ A @ F3 ) )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ ( inf_inf @ A @ A4 @ P ) ) @ P )
=> ( ord_less_eq @ A @ A4 @ P ) ) ) ) ) ).
% def_lfp_induct
thf(fact_6710_lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,P: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ ( inf_inf @ A @ ( complete_lattice_lfp @ A @ F3 ) @ P ) ) @ P )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F3 ) @ P ) ) ) ) ).
% lfp_induct
thf(fact_6711_lfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,P: A > $o] :
( ( order_mono @ A @ A @ F3 )
=> ( ! [S2: A] :
( ( P @ S2 )
=> ( ( ord_less_eq @ A @ S2 @ ( complete_lattice_lfp @ A @ F3 ) )
=> ( P @ ( F3 @ S2 ) ) ) )
=> ( ! [M9: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ M9 )
=> ( P @ X5 ) )
=> ( P @ ( complete_Sup_Sup @ A @ M9 ) ) )
=> ( P @ ( complete_lattice_lfp @ A @ F3 ) ) ) ) ) ) ).
% lfp_ordinal_induct
thf(fact_6712_lfp__funpow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,N: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( complete_lattice_lfp @ A @ ( compow @ ( A > A ) @ ( suc @ N ) @ F3 ) )
= ( complete_lattice_lfp @ A @ F3 ) ) ) ) ).
% lfp_funpow
thf(fact_6713_lfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,K: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F3 @ ( bot_bot @ A ) )
= ( compow @ ( A > A ) @ K @ F3 @ ( bot_bot @ A ) ) )
=> ( ( complete_lattice_lfp @ A @ F3 )
= ( compow @ ( A > A ) @ K @ F3 @ ( bot_bot @ A ) ) ) ) ) ) ).
% lfp_Kleene_iter
thf(fact_6714_iteratesp__def,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F2: A > A] :
( complete_lattice_lfp @ ( A > $o )
@ ^ [P5: A > $o,X2: A] :
( ? [Y3: A] :
( ( X2
= ( F2 @ Y3 ) )
& ( P5 @ Y3 ) )
| ? [M8: set @ A] :
( ( X2
= ( complete_Sup_Sup @ A @ M8 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ M8 )
=> ( P5 @ Y3 ) ) ) ) ) ) ) ) ).
% iteratesp_def
thf(fact_6715_lfp__transfer__bounded,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [P: A > $o,F3: A > A,Alpha: A > B,G2: B > B] :
( ( P @ ( bot_bot @ A ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( P @ ( F3 @ X3 ) ) )
=> ( ! [M9: nat > A] :
( ! [I3: nat] : ( P @ ( M9 @ I3 ) )
=> ( P @ ( complete_Sup_Sup @ A @ ( image2 @ nat @ A @ M9 @ ( top_top @ ( set @ nat ) ) ) ) ) )
=> ( ! [M9: nat > A] :
( ( order_mono @ nat @ A @ M9 )
=> ( ! [I3: nat] : ( P @ ( M9 @ I3 ) )
=> ( ( Alpha @ ( complete_Sup_Sup @ A @ ( image2 @ nat @ A @ M9 @ ( top_top @ ( set @ nat ) ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ nat @ B
@ ^ [I4: nat] : ( Alpha @ ( M9 @ I4 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) )
=> ( ( order_sup_continuous @ A @ A @ F3 )
=> ( ( order_sup_continuous @ B @ B @ G2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ ( complete_lattice_lfp @ A @ F3 ) )
=> ( ( Alpha @ ( F3 @ X3 ) )
= ( G2 @ ( Alpha @ X3 ) ) ) ) )
=> ( ! [X3: B] : ( ord_less_eq @ B @ ( Alpha @ ( bot_bot @ A ) ) @ ( G2 @ X3 ) )
=> ( ( Alpha @ ( complete_lattice_lfp @ A @ F3 ) )
= ( complete_lattice_lfp @ B @ G2 ) ) ) ) ) ) ) ) ) ) ) ).
% lfp_transfer_bounded
thf(fact_6716_sup__continuous__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F3: A > B,G2: A > B] :
( ( order_sup_continuous @ A @ B @ F3 )
=> ( ( order_sup_continuous @ A @ B @ G2 )
=> ( order_sup_continuous @ A @ B
@ ^ [X2: A] : ( sup_sup @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% sup_continuous_sup
thf(fact_6717_lfp__induct2,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,F3: ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F3 ) )
=> ( ( order_mono @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ ( product_prod @ A @ B ) ) @ F3 )
=> ( ! [A7: A,B7: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B7 ) @ ( F3 @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) ) ) )
=> ( P @ A7 @ B7 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% lfp_induct2
thf(fact_6718_lfp__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [Alpha: A > B,F3: A > A,G2: B > B] :
( ( order_sup_continuous @ A @ B @ Alpha )
=> ( ( order_sup_continuous @ A @ A @ F3 )
=> ( ( order_sup_continuous @ B @ B @ G2 )
=> ( ! [X3: B] : ( ord_less_eq @ B @ ( Alpha @ ( bot_bot @ A ) ) @ ( G2 @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less_eq @ A @ X3 @ ( complete_lattice_lfp @ A @ F3 ) )
=> ( ( Alpha @ ( F3 @ X3 ) )
= ( G2 @ ( Alpha @ X3 ) ) ) )
=> ( ( Alpha @ ( complete_lattice_lfp @ A @ F3 ) )
= ( complete_lattice_lfp @ B @ G2 ) ) ) ) ) ) ) ) ).
% lfp_transfer
thf(fact_6719_cclfp__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( counta3822494911875563373attice @ B )
& ( counta3822494911875563373attice @ A ) )
=> ! [Alpha: A > B,F3: A > A,G2: B > B] :
( ( order_sup_continuous @ A @ B @ Alpha )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ( ( Alpha @ ( bot_bot @ A ) )
= ( bot_bot @ B ) )
=> ( ! [X3: A] :
( ( Alpha @ ( F3 @ X3 ) )
= ( G2 @ ( Alpha @ X3 ) ) )
=> ( ( Alpha @ ( order_532582986084564980_cclfp @ A @ F3 ) )
= ( order_532582986084564980_cclfp @ B @ G2 ) ) ) ) ) ) ) ).
% cclfp_transfer
thf(fact_6720_iteratesp_Osimps,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F2: A > A,A5: A] :
( ? [X2: A] :
( ( A5
= ( F2 @ X2 ) )
& ( comple7512665784863727008ratesp @ A @ F2 @ X2 ) )
| ? [M8: set @ A] :
( ( A5
= ( complete_Sup_Sup @ A @ M8 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
& ! [X2: A] :
( ( member @ A @ X2 @ M8 )
=> ( comple7512665784863727008ratesp @ A @ F2 @ X2 ) ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_6721_iteratesp_Ocases,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F3: A > A,A3: A] :
( ( comple7512665784863727008ratesp @ A @ F3 @ A3 )
=> ( ! [X3: A] :
( ( A3
= ( F3 @ X3 ) )
=> ~ ( comple7512665784863727008ratesp @ A @ F3 @ X3 ) )
=> ~ ! [M9: set @ A] :
( ( A3
= ( complete_Sup_Sup @ A @ M9 ) )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ M9 )
=> ( comple7512665784863727008ratesp @ A @ F3 @ X5 ) ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_6722_iteratesp_OSup,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [M5: set @ A,F3: A > A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ M5 )
=> ( comple7512665784863727008ratesp @ A @ F3 @ X3 ) )
=> ( comple7512665784863727008ratesp @ A @ F3 @ ( complete_Sup_Sup @ A @ M5 ) ) ) ) ) ).
% iteratesp.Sup
thf(fact_6723_sup__continuous__lfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F4: A > A] :
( ( order_sup_continuous @ A @ A @ F4 )
=> ( ( complete_lattice_lfp @ A @ F4 )
= ( complete_Sup_Sup @ A
@ ( image2 @ nat @ A
@ ^ [I4: nat] : ( compow @ ( A > A ) @ I4 @ F4 @ ( bot_bot @ A ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% sup_continuous_lfp
thf(fact_6724_ord__class_Olexordp__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P5: ( list @ A ) > ( list @ A ) > $o,X17: list @ A,X25: list @ A] :
( ? [Y3: A,Ys3: list @ A] :
( ( X17
= ( nil @ A ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) ) )
| ? [X2: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( X17
= ( cons @ A @ X2 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ord_less @ A @ X2 @ Y3 ) )
| ? [X2: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( X17
= ( cons @ A @ X2 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( ord_less @ A @ X2 @ Y3 )
& ~ ( ord_less @ A @ Y3 @ X2 )
& ( P5 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% ord_class.lexordp_def
thf(fact_6725_butlast__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( butlast @ A @ ( take @ A @ N @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% butlast_take
thf(fact_6726_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( ( ord_less @ A @ X @ Y )
| ( ~ ( ord_less @ A @ Y @ X )
& ( ord_lexordp @ A @ Xs @ Ys2 ) ) ) ) ) ).
% lexordp_simps(3)
thf(fact_6727_lexordp__append__leftD,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A,Us: list @ A,Vs: list @ A] :
( ( ord_lexordp @ A @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) )
=> ( ! [A7: A] :
~ ( ord_less @ A @ A7 @ A7 )
=> ( ord_lexordp @ A @ Us @ Vs ) ) ) ) ).
% lexordp_append_leftD
thf(fact_6728_lexordp__irreflexive,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 )
=> ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% lexordp_irreflexive
thf(fact_6729_lexordp_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ).
% lexordp.Cons
thf(fact_6730_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ~ ( ord_less @ A @ Y @ X )
=> ( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_6731_lexordp__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ! [Y2: A,Ys4: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y2 @ Ys4 ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: A,Ys4: list @ A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys4 ) ) )
=> ( ! [X3: A,Xs2: list @ A,Ys4: list @ A] :
( ( ord_lexordp @ A @ Xs2 @ Ys4 )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ X3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ) ).
% lexordp_induct
thf(fact_6732_lexordp__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ( ( Xs
= ( nil @ A ) )
=> ! [Y2: A,Ys5: list @ A] :
( Ys2
!= ( cons @ A @ Y2 @ Ys5 ) ) )
=> ( ! [X3: A] :
( ? [Xs4: list @ A] :
( Xs
= ( cons @ A @ X3 @ Xs4 ) )
=> ! [Y2: A] :
( ? [Ys5: list @ A] :
( Ys2
= ( cons @ A @ Y2 @ Ys5 ) )
=> ~ ( ord_less @ A @ X3 @ Y2 ) ) )
=> ~ ! [X3: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Xs4 ) )
=> ! [Ys5: list @ A] :
( ( Ys2
= ( cons @ A @ X3 @ Ys5 ) )
=> ~ ( ord_lexordp @ A @ Xs4 @ Ys5 ) ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_6733_lexordp_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Y3: A,Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) ) )
| ? [X2: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ord_less @ A @ X2 @ Y3 ) )
| ? [X2: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( ord_less @ A @ X2 @ Y3 )
& ~ ( ord_less @ A @ Y3 @ X2 )
& ( ord_lexordp @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% lexordp.simps
thf(fact_6734_lexordp_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A13: list @ A,A24: list @ A] :
( ( ord_lexordp @ A @ A13 @ A24 )
=> ( ( ( A13
= ( nil @ A ) )
=> ! [Y2: A,Ys4: list @ A] :
( A24
!= ( cons @ A @ Y2 @ Ys4 ) ) )
=> ( ! [X3: A] :
( ? [Xs2: list @ A] :
( A13
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys4: list @ A] :
( A24
= ( cons @ A @ Y2 @ Ys4 ) )
=> ~ ( ord_less @ A @ X3 @ Y2 ) ) )
=> ~ ! [X3: A,Y2: A,Xs2: list @ A] :
( ( A13
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys4: list @ A] :
( ( A24
= ( cons @ A @ Y2 @ Ys4 ) )
=> ( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ~ ( ord_less @ A @ Y2 @ X3 )
=> ~ ( ord_lexordp @ A @ Xs2 @ Ys4 ) ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_6735_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Us: list @ A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp @ A @ ( append @ A @ Us @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ).
% lexordp_append_left_rightI
thf(fact_6736_lexordp__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ? [X2: A,Vs2: list @ A] :
( Ys3
= ( append @ A @ Xs3 @ ( cons @ A @ X2 @ Vs2 ) ) )
| ? [Us2: list @ A,A5: A,B5: A,Vs2: list @ A,Ws: list @ A] :
( ( ord_less @ A @ A5 @ B5 )
& ( Xs3
= ( append @ A @ Us2 @ ( cons @ A @ A5 @ Vs2 ) ) )
& ( Ys3
= ( append @ A @ Us2 @ ( cons @ A @ B5 @ Ws ) ) ) ) ) ) ) ) ).
% lexordp_iff
thf(fact_6737_sorted__butlast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( butlast @ A @ Xs ) ) ) ) ) ).
% sorted_butlast
thf(fact_6738_nth__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) ) )
=> ( ( nth @ A @ ( butlast @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_6739_take__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ N @ ( butlast @ A @ Xs ) )
= ( take @ A @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_6740_lexordp__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% lexordp_conv_lexord
thf(fact_6741_finite__refines__card__le,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ R ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( equiv_equiv @ A @ A4 @ R )
=> ( ( equiv_equiv @ A @ A4 @ S )
=> ( ord_less_eq @ nat @ ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ S ) ) @ ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ R ) ) ) ) ) ) ) ).
% finite_refines_card_le
thf(fact_6742_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% semiring_bit_operations_class.even_mask_iff
thf(fact_6743_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% mask_nat_positive_iff
thf(fact_6744_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_6745_mask__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( ( bit_se2239418461657761734s_mask @ A @ N )
= ( zero_zero @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% mask_eq_0_iff
thf(fact_6746_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_6747_in__quotient__imp__subset,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X4: set @ A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ ( set @ A ) @ X4 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ( ord_less_eq @ ( set @ A ) @ X4 @ A4 ) ) ) ).
% in_quotient_imp_subset
thf(fact_6748_in__quotient__imp__non__empty,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X4: set @ A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ ( set @ A ) @ X4 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ( X4
!= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_6749_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% less_eq_mask
thf(fact_6750_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N ) ) ).
% mask_nonnegative_int
thf(fact_6751_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( zero_zero @ int ) ) ).
% not_mask_negative_int
thf(fact_6752_equiv__class__self,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ A @ A3 @ A4 )
=> ( member @ A @ A3 @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_self
thf(fact_6753_quotient__disj,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X4: set @ A,Y6: set @ A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ ( set @ A ) @ X4 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ( ( X4 = Y6 )
| ( ( inf_inf @ ( set @ A ) @ X4 @ Y6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% quotient_disj
thf(fact_6754_less__mask,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ) ).
% less_mask
thf(fact_6755_finite__refines__finite,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ R ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( equiv_equiv @ A @ A4 @ R )
=> ( ( equiv_equiv @ A @ A4 @ S )
=> ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ S ) ) ) ) ) ) ).
% finite_refines_finite
thf(fact_6756_equiv__class__eq__iff,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
= ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( member @ A @ X @ A4 )
& ( member @ A @ Y @ A4 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_6757_eq__equiv__class__iff,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_6758_equiv__class__eq,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 )
=> ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_eq
thf(fact_6759_eq__equiv__class,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A,A4: set @ A] :
( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ A @ B3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_6760_eq__equiv__class__iff2,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ( equiv_quotient @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
= ( equiv_quotient @ A @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) @ R2 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_6761_refines__equiv__class__eq2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A4: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( equiv_equiv @ A @ A4 @ R )
=> ( ( equiv_equiv @ A @ A4 @ S )
=> ( ( image @ A @ A @ S @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_6762_refines__equiv__class__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A4: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( equiv_equiv @ A @ A4 @ R )
=> ( ( equiv_equiv @ A @ A4 @ S )
=> ( ( image @ A @ A @ R @ ( image @ A @ A @ S @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_6763_equiv__imp__dvd__card,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ! [X9: set @ A] :
( ( member @ ( set @ A ) @ X9 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ X9 ) ) )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% equiv_imp_dvd_card
thf(fact_6764_refines__equiv__image__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( equiv_equiv @ A @ A4 @ R )
=> ( ( equiv_equiv @ A @ A4 @ S )
=> ( ( image2 @ ( set @ A ) @ ( set @ A ) @ ( image @ A @ A @ S ) @ ( equiv_quotient @ A @ A4 @ R ) )
= ( equiv_quotient @ A @ A4 @ S ) ) ) ) ) ).
% refines_equiv_image_eq
thf(fact_6765_subset__equiv__class,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),B3: A,A3: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ A @ B3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 ) ) ) ) ).
% subset_equiv_class
thf(fact_6766_equiv__class__subset,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_subset
thf(fact_6767_equiv__class__nondisjoint,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,A3: A,B3: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_6768_in__quotient__imp__in__rel,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X4: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ ( set @ A ) @ X4 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ X4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_6769_mask__nat__less__exp,axiom,
! [N: nat] : ( ord_less @ nat @ ( bit_se2239418461657761734s_mask @ nat @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% mask_nat_less_exp
thf(fact_6770_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A18: set @ A,R12: set @ ( product_prod @ A @ A ),A26: set @ B,R23: set @ ( product_prod @ B @ B ),F3: A > B > ( set @ C ),A13: A,A24: B] :
( ( equiv_equiv @ A @ A18 @ R12 )
=> ( ( equiv_equiv @ B @ A26 @ R23 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R12 @ R23 @ F3 )
=> ( ( member @ A @ A13 @ A18 )
=> ( ( member @ B @ A24 @ A26 )
=> ( ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [X17: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F3 @ X17 ) @ ( image @ B @ B @ R23 @ ( insert2 @ B @ A24 @ ( bot_bot @ ( set @ B ) ) ) ) ) )
@ ( image @ A @ A @ R12 @ ( insert2 @ A @ A13 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F3 @ A13 @ A24 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_6771_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > ( set @ B ),A3: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R2 @ F3 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F3 @ A3 ) ) ) ) ) ).
% UN_equiv_class
thf(fact_6772_congruent2__implies__congruent__UN,axiom,
! [B: $tType,C: $tType,A: $tType,A18: set @ A,R12: set @ ( product_prod @ A @ A ),A26: set @ B,R23: set @ ( product_prod @ B @ B ),F3: A > B > ( set @ C ),A3: B] :
( ( equiv_equiv @ A @ A18 @ R12 )
=> ( ( equiv_equiv @ B @ A26 @ R23 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R12 @ R23 @ F3 )
=> ( ( member @ B @ A3 @ A26 )
=> ( equiv_congruent @ A @ ( set @ C ) @ R12
@ ^ [X17: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F3 @ X17 ) @ ( image @ B @ B @ R23 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ) ).
% congruent2_implies_congruent_UN
thf(fact_6773_proj__iff,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 )
=> ( ( ( equiv_proj @ A @ A @ R2 @ X )
= ( equiv_proj @ A @ A @ R2 @ Y ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_6774_independentD,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A,T2: set @ A,U: A > real,V2: A] :
( ~ ( real_V358717886546972837endent @ A @ S3 )
=> ( ( finite_finite2 @ A @ T2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S3 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V6: A] : ( real_V8093663219630862766scaleR @ A @ ( U @ V6 ) @ V6 )
@ T2 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ V2 @ T2 )
=> ( ( U @ V2 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% independentD
thf(fact_6775_independent__empty,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ~ ( real_V358717886546972837endent @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% independent_empty
thf(fact_6776_dependent__single,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V358717886546972837endent @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% dependent_single
thf(fact_6777_dependent__mono,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( real_V358717886546972837endent @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( real_V358717886546972837endent @ A @ A4 ) ) ) ) ).
% dependent_mono
thf(fact_6778_independent__mono,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ~ ( real_V358717886546972837endent @ A @ B2 ) ) ) ) ).
% independent_mono
thf(fact_6779_independent__Union__directed,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [C2: set @ ( set @ A )] :
( ! [C5: set @ A,D6: set @ A] :
( ( member @ ( set @ A ) @ C5 @ C2 )
=> ( ( member @ ( set @ A ) @ D6 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ C5 @ D6 )
| ( ord_less_eq @ ( set @ A ) @ D6 @ C5 ) ) ) )
=> ( ! [C5: set @ A] :
( ( member @ ( set @ A ) @ C5 @ C2 )
=> ~ ( real_V358717886546972837endent @ A @ C5 ) )
=> ~ ( real_V358717886546972837endent @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) ) ) ) ) ).
% independent_Union_directed
thf(fact_6780_dependent__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ A4 )
=> ( real_V358717886546972837endent @ A @ A4 ) ) ) ).
% dependent_zero
thf(fact_6781_unique__representation,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,F3: A > real,G2: A > real] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ! [V3: A] :
( ( ( F3 @ V3 )
!= ( zero_zero @ real ) )
=> ( member @ A @ V3 @ Basis ) )
=> ( ! [V3: A] :
( ( ( G2 @ V3 )
!= ( zero_zero @ real ) )
=> ( member @ A @ V3 @ Basis ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [V6: A] :
( ( F3 @ V6 )
!= ( zero_zero @ real ) ) ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [V6: A] :
( ( G2 @ V6 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V6: A] : ( real_V8093663219630862766scaleR @ A @ ( F3 @ V6 ) @ V6 )
@ ( collect @ A
@ ^ [V6: A] :
( ( F3 @ V6 )
!= ( zero_zero @ real ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V6: A] : ( real_V8093663219630862766scaleR @ A @ ( G2 @ V6 ) @ V6 )
@ ( collect @ A
@ ^ [V6: A] :
( ( G2 @ V6 )
!= ( zero_zero @ real ) ) ) ) )
=> ( F3 = G2 ) ) ) ) ) ) ) ) ).
% unique_representation
thf(fact_6782_dependent__finite,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ( real_V358717886546972837endent @ A @ S )
= ( ? [U2: A > real] :
( ? [X2: A] :
( ( member @ A @ X2 @ S )
& ( ( U2 @ X2 )
!= ( zero_zero @ real ) ) )
& ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V6: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V6 ) @ V6 )
@ S )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% dependent_finite
thf(fact_6783_independent__if__scalars__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [F6: A > real,X3: A] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [Y3: A] : ( real_V8093663219630862766scaleR @ A @ ( F6 @ Y3 ) @ Y3 )
@ A4 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( F6 @ X3 )
= ( zero_zero @ real ) ) ) )
=> ~ ( real_V358717886546972837endent @ A @ A4 ) ) ) ) ).
% independent_if_scalars_zero
thf(fact_6784_independentD__unique,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A,X4: A > real,Y6: A > real] :
( ~ ( real_V358717886546972837endent @ A @ B2 )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( X4 @ X2 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( X4 @ X2 )
!= ( zero_zero @ real ) ) )
@ B2 )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( Y6 @ X2 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( Y6 @ X2 )
!= ( zero_zero @ real ) ) )
@ B2 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( X4 @ X2 ) @ X2 )
@ ( collect @ A
@ ^ [X2: A] :
( ( X4 @ X2 )
!= ( zero_zero @ real ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( Y6 @ X2 ) @ X2 )
@ ( collect @ A
@ ^ [X2: A] :
( ( Y6 @ X2 )
!= ( zero_zero @ real ) ) ) ) )
=> ( X4 = Y6 ) ) ) ) ) ) ) ) ).
% independentD_unique
thf(fact_6785_proj__def,axiom,
! [A: $tType,B: $tType] :
( ( equiv_proj @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ A ),X2: B] : ( image @ B @ A @ R5 @ ( insert2 @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% proj_def
thf(fact_6786_independent__explicit__finite__subsets,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ A4 ) )
= ( ! [S7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S7 @ A4 )
=> ( ( finite_finite2 @ A @ S7 )
=> ! [U2: A > real] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V6: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V6 ) @ V6 )
@ S7 )
= ( zero_zero @ A ) )
=> ! [X2: A] :
( ( member @ A @ X2 @ S7 )
=> ( ( U2 @ X2 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% independent_explicit_finite_subsets
thf(fact_6787_independent__explicit__module,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ S3 ) )
= ( ! [T3: set @ A,U2: A > real,V6: A] :
( ( finite_finite2 @ A @ T3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T3 @ S3 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [W3: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ W3 ) @ W3 )
@ T3 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ V6 @ T3 )
=> ( ( U2 @ V6 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% independent_explicit_module
thf(fact_6788_dependent__explicit,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [S8: set @ A] :
? [T3: set @ A] :
( ( finite_finite2 @ A @ T3 )
& ( ord_less_eq @ ( set @ A ) @ T3 @ S8 )
& ? [U2: A > real] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V6: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V6 ) @ V6 )
@ T3 )
= ( zero_zero @ A ) )
& ? [X2: A] :
( ( member @ A @ X2 @ T3 )
& ( ( U2 @ X2 )
!= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% dependent_explicit
thf(fact_6789_independentD__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A,X4: A > real,X: A] :
( ~ ( real_V358717886546972837endent @ A @ B2 )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( X4 @ X2 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( X4 @ X2 )
!= ( zero_zero @ real ) ) )
@ B2 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( X4 @ X2 ) @ X2 )
@ ( collect @ A
@ ^ [X2: A] :
( ( X4 @ X2 )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
=> ( ( X4 @ X )
= ( zero_zero @ real ) ) ) ) ) ) ) ).
% independentD_alt
thf(fact_6790_independent__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ B2 ) )
= ( ! [X8: A > real] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( X8 @ X2 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( X8 @ X2 )
!= ( zero_zero @ real ) ) )
@ B2 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( X8 @ X2 ) @ X2 )
@ ( collect @ A
@ ^ [X2: A] :
( ( X8 @ X2 )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
=> ! [X2: A] :
( ( X8 @ X2 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% independent_alt
thf(fact_6791_dependent__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [B6: set @ A] :
? [X8: A > real] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( X8 @ X2 )
!= ( zero_zero @ real ) ) ) )
& ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( X8 @ X2 )
!= ( zero_zero @ real ) ) )
@ B6 )
& ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( X8 @ X2 ) @ X2 )
@ ( collect @ A
@ ^ [X2: A] :
( ( X8 @ X2 )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
& ? [X2: A] :
( ( X8 @ X2 )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% dependent_alt
thf(fact_6792_isUCont__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F3: A > B] :
( ( topolo6026614971017936543ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S8 )
& ! [X2: A,Y3: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y3 ) @ S8 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) @ R5 ) ) ) ) ) ) ) ).
% isUCont_def
thf(fact_6793_possible__bit__def,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se6407376104438227557le_bit @ A )
= ( ^ [Tyrep: itself @ A,N2: nat] :
( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% possible_bit_def
thf(fact_6794_possible__bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Ty: itself @ A] : ( bit_se6407376104438227557le_bit @ A @ Ty @ ( zero_zero @ nat ) ) ) ).
% possible_bit_0
thf(fact_6795_possible__bit__less__imp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Tyrep2: itself @ A,I: nat,J: nat] :
( ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ I )
=> ( ( ord_less_eq @ nat @ J @ I )
=> ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ J ) ) ) ) ).
% possible_bit_less_imp
thf(fact_6796_uniformly__continuous__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ( ( topolo6026614971017936543ous_on @ A @ B )
= ( ^ [S8: set @ A,F2: A > B] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X2: A] :
( ( member @ A @ X2 @ S8 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ S8 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y3 @ X2 ) @ D5 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) @ E3 ) ) ) ) ) ) ) ) ) ).
% uniformly_continuous_on_def
thf(fact_6797_drop__bit__exp__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% drop_bit_exp_eq
thf(fact_6798_bit__minus__2__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% bit_minus_2_iff
thf(fact_6799_CHAR__eq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) ) ) ).
% CHAR_eq_0
thf(fact_6800_of__nat__CHAR,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_CHAR
thf(fact_6801_bit__mask__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M ) ) ) ) ).
% bit_mask_iff
thf(fact_6802_CHAR__eqI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C3: nat] :
( ( ( semiring_1_of_nat @ A @ C3 )
= ( zero_zero @ A ) )
=> ( ! [X3: nat] :
( ( ( semiring_1_of_nat @ A @ X3 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ nat @ C3 @ X3 ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C3 ) ) ) ) ).
% CHAR_eqI
thf(fact_6803_of__nat__eq__0__iff__char__dvd,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ A @ N )
= ( zero_zero @ A ) )
= ( dvd_dvd @ nat @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) @ N ) ) ) ).
% of_nat_eq_0_iff_char_dvd
thf(fact_6804_CHAR__eq0__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
= ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( semiring_1_of_nat @ A @ N2 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_eq0_iff
thf(fact_6805_CHAR__eq__posI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( ( ( semiring_1_of_nat @ A @ C3 )
= ( zero_zero @ A ) )
=> ( ! [X3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
=> ( ( ord_less @ nat @ X3 @ C3 )
=> ( ( semiring_1_of_nat @ A @ X3 )
!= ( zero_zero @ A ) ) ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C3 ) ) ) ) ) ).
% CHAR_eq_posI
thf(fact_6806_CHAR__pos__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ( semiring_1_of_nat @ A @ N2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_pos_iff
thf(fact_6807_bit__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4730199178511100633sh_bit @ A @ M @ A3 ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_push_bit_iff
thf(fact_6808_fold__possible__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
= ( ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% fold_possible_bit
thf(fact_6809_bit__minus__exp__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% bit_minus_exp_iff
thf(fact_6810_bit__mask__sub__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M ) ) ) ) ).
% bit_mask_sub_iff
thf(fact_6811_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
& ( N
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% bit_double_iff
thf(fact_6812_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P5: rat,Q6: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B5: int,D5: int] : ( ord_less_eq @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C6 @ B5 ) )
@ ( quotient_of @ Q6 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_eq_code
thf(fact_6813_of__real__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( real_Vector_of_real @ complex @ ( sqrt @ X ) )
= ( csqrt @ ( real_Vector_of_real @ complex @ X ) ) ) ) ).
% of_real_sqrt
thf(fact_6814_quotient__of__denom__pos,axiom,
! [R2: rat,P6: int,Q5: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair @ int @ int @ P6 @ Q5 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q5 ) ) ).
% quotient_of_denom_pos
thf(fact_6815_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P5: rat,Q6: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B5: int,D5: int] : ( ord_less @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C6 @ B5 ) )
@ ( quotient_of @ Q6 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_code
thf(fact_6816_numeral__xor__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% numeral_xor_num
thf(fact_6817_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K: num] :
( ( ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
= ( insert2 @ 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_6818_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_6819_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( ( suc @ N )
= ( numeral_numeral @ nat @ K ) )
= ( N
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_6820_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
= ( suc @ N ) )
= ( ( pred_numeral @ K )
= N ) ) ).
% eq_numeral_Suc
thf(fact_6821_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% pred_numeral_inc
thf(fact_6822_less__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( ord_less @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% less_numeral_Suc
thf(fact_6823_less__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% less_Suc_numeral
thf(fact_6824_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_6825_le__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% le_numeral_Suc
thf(fact_6826_le__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less_eq @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% le_Suc_numeral
thf(fact_6827_diff__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( minus_minus @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% diff_numeral_Suc
thf(fact_6828_diff__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( minus_minus @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% diff_Suc_numeral
thf(fact_6829_max__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_max @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_max @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% max_Suc_numeral
thf(fact_6830_max__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_max @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% max_numeral_Suc
thf(fact_6831_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral @ nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_6832_min__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_min @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_min @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% min_numeral_Suc
thf(fact_6833_min__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_min @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_min @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% min_Suc_numeral
thf(fact_6834_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_6835_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_6836_lessThan__nat__numeral,axiom,
! [K: num] :
( ( set_ord_lessThan @ nat @ ( numeral_numeral @ nat @ K ) )
= ( insert2 @ nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan @ nat @ ( pred_numeral @ K ) ) ) ) ).
% lessThan_nat_numeral
thf(fact_6837_atMost__nat__numeral,axiom,
! [K: num] :
( ( set_ord_atMost @ nat @ ( numeral_numeral @ nat @ K ) )
= ( insert2 @ nat @ ( numeral_numeral @ nat @ K ) @ ( set_ord_atMost @ nat @ ( pred_numeral @ K ) ) ) ) ).
% atMost_nat_numeral
thf(fact_6838_xor__num__eq__None__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num] :
( ( ( bit_un2480387367778600638or_num @ M @ N )
= ( none @ num ) )
= ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% xor_num_eq_None_iff
thf(fact_6839_rec__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F3: nat > A > A,V2: num,N: nat] :
( ( rec_nat @ A @ A3 @ F3 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N ) )
= ( F3 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N ) @ ( rec_nat @ A @ A3 @ F3 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N ) ) ) ) ).
% rec_nat_add_eq_if
thf(fact_6840_case__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F3: nat > A,V2: num,N: nat] :
( ( case_nat @ A @ A3 @ F3 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N ) )
= ( F3 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N ) ) ) ).
% case_nat_add_eq_if
thf(fact_6841_old_Onat_Osimps_I7_J,axiom,
! [T: $tType,F16: T,F25: nat > T > T,Nat: nat] :
( ( rec_nat @ T @ F16 @ F25 @ ( suc @ Nat ) )
= ( F25 @ Nat @ ( rec_nat @ T @ F16 @ F25 @ Nat ) ) ) ).
% old.nat.simps(7)
thf(fact_6842_old_Onat_Osimps_I6_J,axiom,
! [T: $tType,F16: T,F25: nat > T > T] :
( ( rec_nat @ T @ F16 @ F25 @ ( zero_zero @ nat ) )
= F16 ) ).
% old.nat.simps(6)
thf(fact_6843_case__nat__numeral,axiom,
! [A: $tType,A3: A,F3: nat > A,V2: num] :
( ( case_nat @ A @ A3 @ F3 @ ( numeral_numeral @ nat @ V2 ) )
= ( F3 @ ( pred_numeral @ V2 ) ) ) ).
% case_nat_numeral
thf(fact_6844_rec__nat__numeral,axiom,
! [A: $tType,A3: A,F3: nat > A > A,V2: num] :
( ( rec_nat @ A @ A3 @ F3 @ ( numeral_numeral @ nat @ V2 ) )
= ( F3 @ ( pred_numeral @ V2 ) @ ( rec_nat @ A @ A3 @ F3 @ ( pred_numeral @ V2 ) ) ) ) ).
% rec_nat_numeral
thf(fact_6845_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_6846_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_6847_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F16: A,F25: nat > A,X23: nat] :
( ( case_nat @ A @ F16 @ F25 @ ( suc @ X23 ) )
= ( F25 @ X23 ) ) ).
% old.nat.simps(5)
thf(fact_6848_old_Onat_Osimps_I4_J,axiom,
! [A: $tType,F16: A,F25: nat > A] :
( ( case_nat @ A @ F16 @ F25 @ ( zero_zero @ nat ) )
= F16 ) ).
% old.nat.simps(4)
thf(fact_6849_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F16: A,F25: nat > A,Nat: nat] :
( ( H @ ( case_nat @ A @ F16 @ F25 @ Nat ) )
= ( case_nat @ B @ ( H @ F16 )
@ ^ [X2: nat] : ( H @ ( F25 @ X2 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_6850_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
= ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_6851_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M6: nat] : ( suc @ ( ord_max @ nat @ M6 @ N ) )
@ M ) ) ).
% max_Suc2
thf(fact_6852_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M6: nat] : ( suc @ ( ord_max @ nat @ N @ M6 ) )
@ M ) ) ).
% max_Suc1
thf(fact_6853_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [K3: nat] : K3
@ ( minus_minus @ nat @ M @ N ) ) ) ).
% diff_Suc
thf(fact_6854_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M6: nat] : ( suc @ ( ord_min @ nat @ M6 @ N ) )
@ M ) ) ).
% min_Suc2
thf(fact_6855_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M6: nat] : ( suc @ ( ord_min @ nat @ N @ M6 ) )
@ M ) ) ).
% min_Suc1
thf(fact_6856_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X2: A,F2: nat > A,N2: nat] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ X2
@ ( F2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_6857_old_Orec__nat__def,axiom,
! [T: $tType] :
( ( rec_nat @ T )
= ( ^ [F18: T,F26: nat > T > T,X2: nat] : ( the @ T @ ( rec_set_nat @ T @ F18 @ F26 @ X2 ) ) ) ) ).
% old.rec_nat_def
thf(fact_6858_rec__nat__0__imp,axiom,
! [A: $tType,F3: nat > A,F16: A,F25: nat > A > A] :
( ( F3
= ( rec_nat @ A @ F16 @ F25 ) )
=> ( ( F3 @ ( zero_zero @ nat ) )
= F16 ) ) ).
% rec_nat_0_imp
thf(fact_6859_subset__Collect__iff,axiom,
! [A: $tType,B2: set @ A,A4: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( P @ X2 ) ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_6860_subset__CollectI,axiom,
! [A: $tType,B2: set @ A,A4: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( Q @ X2 ) ) )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_6861_nat_Osplit__sels_I2_J,axiom,
! [A: $tType,P: A > $o,F16: A,F25: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F16 @ F25 @ Nat ) )
= ( ~ ( ( ( Nat
= ( zero_zero @ nat ) )
& ~ ( P @ F16 ) )
| ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
& ~ ( P @ ( F25 @ ( pred @ Nat ) ) ) ) ) ) ) ).
% nat.split_sels(2)
thf(fact_6862_nat_Osplit__sels_I1_J,axiom,
! [A: $tType,P: A > $o,F16: A,F25: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F16 @ F25 @ Nat ) )
= ( ( ( Nat
= ( zero_zero @ nat ) )
=> ( P @ F16 ) )
& ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
=> ( P @ ( F25 @ ( pred @ Nat ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_6863_pred__def,axiom,
( pred
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X25: nat] : X25 ) ) ).
% pred_def
thf(fact_6864_relImage__proj,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Gr4221423524335903396lImage @ A @ ( set @ A ) @ R @ ( equiv_proj @ A @ A @ R ) ) @ ( id_on @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ R ) ) ) ) ).
% relImage_proj
thf(fact_6865_relInvImage__Id__on,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A,B2: set @ B] :
( ! [A1: A,A22: A] :
( ( ( F3 @ A1 )
= ( F3 @ A22 ) )
= ( A1 = A22 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Gr7122648621184425601vImage @ A @ B @ A4 @ ( id_on @ B @ B2 ) @ F3 ) @ ( id2 @ A ) ) ) ).
% relInvImage_Id_on
thf(fact_6866_relImage__mono,axiom,
! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ A @ A ),F3: A > B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Gr4221423524335903396lImage @ A @ B @ R1 @ F3 ) @ ( bNF_Gr4221423524335903396lImage @ A @ B @ R22 @ F3 ) ) ) ).
% relImage_mono
thf(fact_6867_relInvImage__mono,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ A @ A ),A4: set @ B,F3: B > A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Gr7122648621184425601vImage @ B @ A @ A4 @ R1 @ F3 ) @ ( bNF_Gr7122648621184425601vImage @ B @ A @ A4 @ R22 @ F3 ) ) ) ).
% relInvImage_mono
thf(fact_6868_relInvImage__UNIV__relImage,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F3: A > B] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Gr7122648621184425601vImage @ A @ B @ ( top_top @ ( set @ A ) ) @ ( bNF_Gr4221423524335903396lImage @ A @ B @ R @ F3 ) @ F3 ) ) ).
% relInvImage_UNIV_relImage
thf(fact_6869_ran__map__add,axiom,
! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M1 ) @ ( dom @ A @ B @ M22 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ran @ A @ B @ ( map_add @ A @ B @ M1 @ M22 ) )
= ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ M1 ) @ ( ran @ A @ B @ M22 ) ) ) ) ).
% ran_map_add
thf(fact_6870_set__rec,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( rec_list @ ( set @ A ) @ A @ ( bot_bot @ ( set @ A ) )
@ ^ [X2: A,Uu3: list @ A] : ( insert2 @ A @ X2 ) ) ) ).
% set_rec
thf(fact_6871_dom__map__add,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),N: A > ( option @ B )] :
( ( dom @ A @ B @ ( map_add @ A @ B @ M @ N ) )
= ( sup_sup @ ( set @ A ) @ ( dom @ A @ B @ N ) @ ( dom @ A @ B @ M ) ) ) ).
% dom_map_add
thf(fact_6872_map__add__comm,axiom,
! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M1 ) @ ( dom @ A @ B @ M22 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ M1 @ M22 )
= ( map_add @ A @ B @ M22 @ M1 ) ) ) ).
% map_add_comm
thf(fact_6873_graph__map__add,axiom,
! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M1 ) @ ( dom @ A @ B @ M22 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( graph @ A @ B @ ( map_add @ A @ B @ M1 @ M22 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( graph @ A @ B @ M1 ) @ ( graph @ A @ B @ M22 ) ) ) ) ).
% graph_map_add
thf(fact_6874_numeral__and__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ) ).
% numeral_and_num
thf(fact_6875_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_6876_and__num__eq__None__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num] :
( ( ( bit_un7362597486090784418nd_num @ M @ N )
= ( none @ num ) )
= ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% and_num_eq_None_iff
thf(fact_6877_connected__closedD,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,A4: set @ A,B2: set @ A] :
( ( topolo1966860045006549960nected @ A @ S3 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ S3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ( ( topolo7761053866217962861closed @ A @ A4 )
=> ( ( topolo7761053866217962861closed @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ S3 )
= ( bot_bot @ ( set @ A ) ) )
| ( ( inf_inf @ ( set @ A ) @ B2 @ S3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% connected_closedD
thf(fact_6878_connected__closed,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1966860045006549960nected @ A )
= ( ^ [S8: set @ A] :
~ ? [A6: set @ A,B6: set @ A] :
( ( topolo7761053866217962861closed @ A @ A6 )
& ( topolo7761053866217962861closed @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ S8 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) )
& ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ S8 )
= ( bot_bot @ ( set @ A ) ) )
& ( ( inf_inf @ ( set @ A ) @ A6 @ S8 )
!= ( bot_bot @ ( set @ A ) ) )
& ( ( inf_inf @ ( set @ A ) @ B6 @ S8 )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% connected_closed
thf(fact_6879_connected__contains__Ioo,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A4: set @ A,A3: A,B3: A] :
( ( topolo1966860045006549960nected @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( member @ A @ B3 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) @ A4 ) ) ) ) ) ).
% connected_contains_Ioo
thf(fact_6880_connectedD__interval,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [U3: set @ A,X: A,Y: A,Z: A] :
( ( topolo1966860045006549960nected @ A @ U3 )
=> ( ( member @ A @ X @ U3 )
=> ( ( member @ A @ Y @ U3 )
=> ( ( ord_less_eq @ A @ X @ Z )
=> ( ( ord_less_eq @ A @ Z @ Y )
=> ( member @ A @ Z @ U3 ) ) ) ) ) ) ) ).
% connectedD_interval
thf(fact_6881_connectedI__interval,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [U3: set @ A] :
( ! [X3: A,Y2: A,Z3: A] :
( ( member @ A @ X3 @ U3 )
=> ( ( member @ A @ Y2 @ U3 )
=> ( ( ord_less_eq @ A @ X3 @ Z3 )
=> ( ( ord_less_eq @ A @ Z3 @ Y2 )
=> ( member @ A @ Z3 @ U3 ) ) ) ) )
=> ( topolo1966860045006549960nected @ A @ U3 ) ) ) ).
% connectedI_interval
thf(fact_6882_connected__iff__interval,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ( ( topolo1966860045006549960nected @ A )
= ( ^ [U5: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ U5 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ U5 )
=> ! [Z6: A] :
( ( ord_less_eq @ A @ X2 @ Z6 )
=> ( ( ord_less_eq @ A @ Z6 @ Y3 )
=> ( member @ A @ Z6 @ U5 ) ) ) ) ) ) ) ) ).
% connected_iff_interval
thf(fact_6883_connected__contains__Icc,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A4: set @ A,A3: A,B3: A] :
( ( topolo1966860045006549960nected @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( member @ A @ B3 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ A4 ) ) ) ) ) ).
% connected_contains_Icc
thf(fact_6884_connected__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo1966860045006549960nected @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% connected_empty
thf(fact_6885_connected__sing,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X: A] : ( topolo1966860045006549960nected @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% connected_sing
thf(fact_6886_connected__Un,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,T2: set @ A] :
( ( topolo1966860045006549960nected @ A @ S3 )
=> ( ( topolo1966860045006549960nected @ A @ T2 )
=> ( ( ( inf_inf @ ( set @ A ) @ S3 @ T2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( topolo1966860045006549960nected @ A @ ( sup_sup @ ( set @ A ) @ S3 @ T2 ) ) ) ) ) ) ).
% connected_Un
thf(fact_6887_connected__Union,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ ( set @ A )] :
( ! [S4: set @ A] :
( ( member @ ( set @ A ) @ S4 @ S )
=> ( topolo1966860045006549960nected @ A @ S4 ) )
=> ( ( ( complete_Inf_Inf @ ( set @ A ) @ S )
!= ( bot_bot @ ( set @ A ) ) )
=> ( topolo1966860045006549960nected @ A @ ( complete_Sup_Sup @ ( set @ A ) @ S ) ) ) ) ) ).
% connected_Union
thf(fact_6888_not__in__connected__cases,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S: set @ A,X: A] :
( ( topolo1966860045006549960nected @ A @ S )
=> ( ~ ( member @ A @ X @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( condit941137186595557371_above @ A @ S )
=> ~ ! [Y5: A] :
( ( member @ A @ Y5 @ S )
=> ( ord_less_eq @ A @ Y5 @ X ) ) )
=> ~ ( ( condit1013018076250108175_below @ A @ S )
=> ~ ! [Y5: A] :
( ( member @ A @ Y5 @ S )
=> ( ord_less_eq @ A @ X @ Y5 ) ) ) ) ) ) ) ) ).
% not_in_connected_cases
thf(fact_6889_connected__diff__open__from__closed,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,T2: set @ A,U: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ T2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ U )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( topolo7761053866217962861closed @ A @ T2 )
=> ( ( topolo1966860045006549960nected @ A @ U )
=> ( ( topolo1966860045006549960nected @ A @ ( minus_minus @ ( set @ A ) @ T2 @ S3 ) )
=> ( topolo1966860045006549960nected @ A @ ( minus_minus @ ( set @ A ) @ U @ S3 ) ) ) ) ) ) ) ) ) ).
% connected_diff_open_from_closed
thf(fact_6890_connectedD,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A4: set @ A,U3: set @ A,V: set @ A] :
( ( topolo1966860045006549960nected @ A @ A4 )
=> ( ( topolo1002775350975398744n_open @ A @ U3 )
=> ( ( topolo1002775350975398744n_open @ A @ V )
=> ( ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ U3 @ V ) @ A4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ U3 @ V ) )
=> ( ( ( inf_inf @ ( set @ A ) @ U3 @ A4 )
= ( bot_bot @ ( set @ A ) ) )
| ( ( inf_inf @ ( set @ A ) @ V @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% connectedD
thf(fact_6891_connectedI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [U3: set @ A] :
( ! [A9: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A9 )
=> ! [B4: set @ A] :
( ( topolo1002775350975398744n_open @ A @ B4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A9 @ U3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ A ) @ B4 @ U3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A9 @ B4 ) @ U3 )
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( ord_less_eq @ ( set @ A ) @ U3 @ ( sup_sup @ ( set @ A ) @ A9 @ B4 ) ) ) ) ) ) )
=> ( topolo1966860045006549960nected @ A @ U3 ) ) ) ).
% connectedI
thf(fact_6892_connected__def,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1966860045006549960nected @ A )
= ( ^ [S7: set @ A] :
~ ? [A6: set @ A,B6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A6 )
& ( topolo1002775350975398744n_open @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ S7 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) )
& ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ S7 )
= ( bot_bot @ ( set @ A ) ) )
& ( ( inf_inf @ ( set @ A ) @ A6 @ S7 )
!= ( bot_bot @ ( set @ A ) ) )
& ( ( inf_inf @ ( set @ A ) @ B6 @ S7 )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% connected_def
thf(fact_6893_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
& ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N2 ) @ ( nth @ B @ Ys2 @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_6894_disjnt__equiv__class,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( disjnt @ A @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_6895_disjnt__self__iff__empty,axiom,
! [A: $tType,S: set @ A] :
( ( disjnt @ A @ S @ S )
= ( S
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjnt_self_iff_empty
thf(fact_6896_disjnt__insert1,axiom,
! [A: $tType,A3: A,X4: set @ A,Y6: set @ A] :
( ( disjnt @ A @ ( insert2 @ A @ A3 @ X4 ) @ Y6 )
= ( ~ ( member @ A @ A3 @ Y6 )
& ( disjnt @ A @ X4 @ Y6 ) ) ) ).
% disjnt_insert1
thf(fact_6897_disjnt__insert2,axiom,
! [A: $tType,Y6: set @ A,A3: A,X4: set @ A] :
( ( disjnt @ A @ Y6 @ ( insert2 @ A @ A3 @ X4 ) )
= ( ~ ( member @ A @ A3 @ Y6 )
& ( disjnt @ A @ Y6 @ X4 ) ) ) ).
% disjnt_insert2
thf(fact_6898_disjnt__Un2,axiom,
! [A: $tType,C2: set @ A,A4: set @ A,B2: set @ A] :
( ( disjnt @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ( disjnt @ A @ C2 @ A4 )
& ( disjnt @ A @ C2 @ B2 ) ) ) ).
% disjnt_Un2
thf(fact_6899_disjnt__Un1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( disjnt @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C2 )
= ( ( disjnt @ A @ A4 @ C2 )
& ( disjnt @ A @ B2 @ C2 ) ) ) ).
% disjnt_Un1
thf(fact_6900_disjnt__sym,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( disjnt @ A @ A4 @ B2 )
=> ( disjnt @ A @ B2 @ A4 ) ) ).
% disjnt_sym
thf(fact_6901_disjnt__iff,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X2: A] :
~ ( ( member @ A @ X2 @ A6 )
& ( member @ A @ X2 @ B6 ) ) ) ) ).
% disjnt_iff
thf(fact_6902_disjnt__insert,axiom,
! [A: $tType,X: A,N6: set @ A,M5: set @ A] :
( ~ ( member @ A @ X @ N6 )
=> ( ( disjnt @ A @ M5 @ N6 )
=> ( disjnt @ A @ ( insert2 @ A @ X @ M5 ) @ N6 ) ) ) ).
% disjnt_insert
thf(fact_6903_disjnt__empty1,axiom,
! [A: $tType,A4: set @ A] : ( disjnt @ A @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% disjnt_empty1
thf(fact_6904_disjnt__empty2,axiom,
! [A: $tType,A4: set @ A] : ( disjnt @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% disjnt_empty2
thf(fact_6905_disjnt__def,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ B6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjnt_def
thf(fact_6906_disjnt__subset1,axiom,
! [A: $tType,X4: set @ A,Y6: set @ A,Z7: set @ A] :
( ( disjnt @ A @ X4 @ Y6 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z7 @ X4 )
=> ( disjnt @ A @ Z7 @ Y6 ) ) ) ).
% disjnt_subset1
thf(fact_6907_disjnt__subset2,axiom,
! [A: $tType,X4: set @ A,Y6: set @ A,Z7: set @ A] :
( ( disjnt @ A @ X4 @ Y6 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z7 @ Y6 )
=> ( disjnt @ A @ X4 @ Z7 ) ) ) ).
% disjnt_subset2
thf(fact_6908_listrel__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( listrel @ A @ B @ R2 ) @ ( listrel @ A @ B @ S3 ) ) ) ).
% listrel_mono
thf(fact_6909_disjnt__ge__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y6: set @ A,X4: set @ A] :
( ( finite_finite2 @ A @ Y6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ Y6 ) @ X3 ) )
=> ( disjnt @ A @ X4 @ Y6 ) ) ) ) ).
% disjnt_ge_max
thf(fact_6910_card__Un__disjnt,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( disjnt @ A @ A4 @ B2 )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ) ).
% card_Un_disjnt
thf(fact_6911_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_6912_listrel__Cons,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),X: B,Xs: list @ B] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert2 @ ( list @ B ) @ ( cons @ B @ X @ Xs ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( set_Cons @ A @ ( image @ B @ A @ R2 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) @ ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert2 @ ( list @ B ) @ Xs @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) ) ) ) ).
% listrel_Cons
thf(fact_6913_sum__card__image,axiom,
! [B: $tType,A: $tType,A4: set @ A,F3: A > ( set @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( pairwise @ A
@ ^ [S8: A,T3: A] : ( disjnt @ B @ ( F3 @ S8 ) @ ( F3 @ T3 ) )
@ A4 )
=> ( ( groups7311177749621191930dd_sum @ ( set @ B ) @ nat @ ( finite_card @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ A4 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [A5: A] : ( finite_card @ B @ ( F3 @ A5 ) )
@ A4 ) ) ) ) ).
% sum_card_image
thf(fact_6914_pairwise__mono,axiom,
! [A: $tType,P: A > A > $o,A4: set @ A,Q: A > A > $o,B2: set @ A] :
( ( pairwise @ A @ P @ A4 )
=> ( ! [X3: A,Y2: A] :
( ( P @ X3 @ Y2 )
=> ( Q @ X3 @ Y2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( pairwise @ A @ Q @ B2 ) ) ) ) ).
% pairwise_mono
thf(fact_6915_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S: set @ A,T4: set @ A] :
( ( pairwise @ A @ P @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ T4 @ S )
=> ( pairwise @ A @ P @ T4 ) ) ) ).
% pairwise_subset
thf(fact_6916_pairwise__imageI,axiom,
! [B: $tType,A: $tType,A4: set @ A,F3: A > B,P: B > B > $o] :
( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( ( X3 != Y2 )
=> ( ( ( F3 @ X3 )
!= ( F3 @ Y2 ) )
=> ( P @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) ) ) ) )
=> ( pairwise @ B @ P @ ( image2 @ A @ B @ F3 @ A4 ) ) ) ).
% pairwise_imageI
thf(fact_6917_pairwise__image,axiom,
! [A: $tType,B: $tType,R2: A > A > $o,F3: B > A,S3: set @ B] :
( ( pairwise @ A @ R2 @ ( image2 @ B @ A @ F3 @ S3 ) )
= ( pairwise @ B
@ ^ [X2: B,Y3: B] :
( ( ( F3 @ X2 )
!= ( F3 @ Y3 ) )
=> ( R2 @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
@ S3 ) ) ).
% pairwise_image
thf(fact_6918_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_6919_pairwise__insert,axiom,
! [A: $tType,R2: A > A > $o,X: A,S3: set @ A] :
( ( pairwise @ A @ R2 @ ( insert2 @ A @ X @ S3 ) )
= ( ! [Y3: A] :
( ( ( member @ A @ Y3 @ S3 )
& ( Y3 != X ) )
=> ( ( R2 @ X @ Y3 )
& ( R2 @ Y3 @ X ) ) )
& ( pairwise @ A @ R2 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_6920_pairwiseD,axiom,
! [A: $tType,R: A > A > $o,S: set @ A,X: A,Y: A] :
( ( pairwise @ A @ R @ S )
=> ( ( member @ A @ X @ S )
=> ( ( member @ A @ Y @ S )
=> ( ( X != Y )
=> ( R @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_6921_pairwiseI,axiom,
! [A: $tType,S: set @ A,R: A > A > $o] :
( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ S )
=> ( ( member @ A @ Y2 @ S )
=> ( ( X3 != Y2 )
=> ( R @ X3 @ Y2 ) ) ) )
=> ( pairwise @ A @ R @ S ) ) ).
% pairwiseI
thf(fact_6922_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R6: A > A > $o,S7: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S7 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ S7 )
=> ( ( X2 != Y3 )
=> ( R6 @ X2 @ Y3 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_6923_pairwise__trivial,axiom,
! [A: $tType,I5: set @ A] :
( pairwise @ A
@ ^ [I4: A,J3: A] : J3 != I4
@ I5 ) ).
% pairwise_trivial
thf(fact_6924_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A4: A] : ( pairwise @ A @ P @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_6925_pairwise__alt,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R6: A > A > $o,S7: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S7 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ ( minus_minus @ ( set @ A ) @ S7 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( R6 @ X2 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_6926_disjoint__image__subset,axiom,
! [A: $tType,A20: set @ ( set @ A ),F3: ( set @ A ) > ( set @ A )] :
( ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ A20 )
=> ( ! [X9: set @ A] :
( ( member @ ( set @ A ) @ X9 @ A20 )
=> ( ord_less_eq @ ( set @ A ) @ ( F3 @ X9 ) @ X9 ) )
=> ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ F3 @ A20 ) ) ) ) ).
% disjoint_image_subset
thf(fact_6927_card__Union__disjoint,axiom,
! [A: $tType,C2: set @ ( set @ A )] :
( ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ C2 )
=> ( ! [A9: set @ A] :
( ( member @ ( set @ A ) @ A9 @ C2 )
=> ( finite_finite2 @ A @ A9 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
= ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ C2 ) ) ) ) ).
% card_Union_disjoint
thf(fact_6928_infinite__infinite__partition,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ~ ! [C7: nat > ( set @ A )] :
( ( pairwise @ nat
@ ^ [I4: nat,J3: nat] : ( disjnt @ A @ ( C7 @ I4 ) @ ( C7 @ J3 ) )
@ ( top_top @ ( set @ nat ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ C7 @ ( top_top @ ( set @ nat ) ) ) ) @ A4 )
=> ~ ! [I3: nat] :
~ ( finite_finite2 @ A @ ( C7 @ I3 ) ) ) ) ) ).
% infinite_infinite_partition
thf(fact_6929_listrel1__subset__listrel,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R4 )
=> ( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R2 ) @ ( listrel @ A @ A @ R4 ) ) ) ) ).
% listrel1_subset_listrel
thf(fact_6930_lists__empty,axiom,
! [A: $tType] :
( ( lists @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% lists_empty
thf(fact_6931_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_6932_refl__on__Int,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),B2: set @ A,S3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A4 @ R2 )
=> ( ( refl_on @ A @ B2 @ S3 )
=> ( refl_on @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ).
% refl_on_Int
thf(fact_6933_refl__onD2,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A4 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ A @ Y @ A4 ) ) ) ).
% refl_onD2
thf(fact_6934_refl__onD1,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A4 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ A @ X @ A4 ) ) ) ).
% refl_onD1
thf(fact_6935_refl__onD,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( refl_on @ A @ A4 @ R2 )
=> ( ( member @ A @ A3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_6936_refl__on__Un,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),B2: set @ A,S3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A4 @ R2 )
=> ( ( refl_on @ A @ B2 @ S3 )
=> ( refl_on @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ).
% refl_on_Un
thf(fact_6937_refl__Id,axiom,
! [A: $tType] : ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ ( id2 @ A ) ) ).
% refl_Id
thf(fact_6938_refl__on__Id__on,axiom,
! [A: $tType,A4: set @ A] : ( refl_on @ A @ A4 @ ( id_on @ A @ A4 ) ) ).
% refl_on_Id_on
thf(fact_6939_lists__eq__set,axiom,
! [A: $tType] :
( ( lists @ A )
= ( ^ [A6: set @ A] :
( collect @ ( list @ A )
@ ^ [Xs3: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 ) ) ) ) ).
% lists_eq_set
thf(fact_6940_refl__on__def_H,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A6: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ! [X2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Z6: A] :
( ( member @ A @ Y3 @ A6 )
& ( member @ A @ Z6 @ A6 ) )
@ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R5 ) ) ) ) ) ).
% refl_on_def'
thf(fact_6941_lists__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( lists @ A @ A4 ) @ ( lists @ A @ B2 ) ) ) ).
% lists_mono
thf(fact_6942_Collect__finite__eq__lists,axiom,
! [A: $tType] :
( ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) )
= ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( lists @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% Collect_finite_eq_lists
thf(fact_6943_Collect__finite__subset__eq__lists,axiom,
! [A: $tType,T4: set @ A] :
( ( collect @ ( set @ A )
@ ^ [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
& ( ord_less_eq @ ( set @ A ) @ A6 @ T4 ) ) )
= ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( lists @ A @ T4 ) ) ) ).
% Collect_finite_subset_eq_lists
thf(fact_6944_refl__on__UNION,axiom,
! [B: $tType,A: $tType,S: set @ A,A4: A > ( set @ B ),R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( refl_on @ B @ ( A4 @ X3 ) @ ( R2 @ X3 ) ) )
=> ( refl_on @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ S ) ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S ) ) ) ) ).
% refl_on_UNION
thf(fact_6945_refl__on__INTER,axiom,
! [B: $tType,A: $tType,S: set @ A,A4: A > ( set @ B ),R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( refl_on @ B @ ( A4 @ X3 ) @ ( R2 @ X3 ) ) )
=> ( refl_on @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ S ) ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S ) ) ) ) ).
% refl_on_INTER
thf(fact_6946_refl__on__singleton,axiom,
! [A: $tType,X: A] : ( refl_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% refl_on_singleton
thf(fact_6947_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( antisym @ A @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A3 = B3 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_6948_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( last @ nat @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ ( one_one @ nat ) ) ) ) ).
% last_upt
thf(fact_6949_last__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% last_replicate
thf(fact_6950_last__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( last @ A @ ( drop @ A @ N @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_drop
thf(fact_6951_antisym__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 )
=> ( ( antisym @ A @ S3 )
=> ( antisym @ A @ R2 ) ) ) ).
% antisym_subset
thf(fact_6952_antisym__Id__on,axiom,
! [A: $tType,A4: set @ A] : ( antisym @ A @ ( id_on @ A @ A4 ) ) ).
% antisym_Id_on
thf(fact_6953_antisym__Id,axiom,
! [A: $tType] : ( antisym @ A @ ( id2 @ A ) ) ).
% antisym_Id
thf(fact_6954_antisym__def,axiom,
! [A: $tType] :
( ( antisym @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X2: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R5 )
=> ( X2 = Y3 ) ) ) ) ) ).
% antisym_def
thf(fact_6955_antisymI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ R2 )
=> ( X3 = Y2 ) ) )
=> ( antisym @ A @ R2 ) ) ).
% antisymI
thf(fact_6956_antisymD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( antisym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A3 ) @ R2 )
=> ( A3 = B3 ) ) ) ) ).
% antisymD
thf(fact_6957_antisym__empty,axiom,
! [A: $tType] : ( antisym @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% antisym_empty
thf(fact_6958_antisym__singleton,axiom,
! [A: $tType,X: product_prod @ A @ A] : ( antisym @ A @ ( insert2 @ ( product_prod @ A @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% antisym_singleton
thf(fact_6959_subset__subseqs,axiom,
! [A: $tType,X4: set @ A,Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X4 @ ( set2 @ A @ Xs ) )
=> ( member @ ( set @ A ) @ X4 @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_6960_bit__not__iff__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ N )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% bit_not_iff_eq
thf(fact_6961_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_6962_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_6963_bit_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_right
thf(fact_6964_bit_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_left
thf(fact_6965_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_6966_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_6967_take__bit__not__mask__eq__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_not_mask_eq_0
thf(fact_6968_bit_Ocompl__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ X @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ X @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( bit_ri4277139882892585799ns_not @ A @ X )
= Y ) ) ) ) ).
% bit.compl_unique
thf(fact_6969_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_6970_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_6971_abstract__boolean__algebra__sym__diff_Oaxioms_I1_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 ) ) ).
% abstract_boolean_algebra_sym_diff.axioms(1)
thf(fact_6972_abstract__boolean__algebra_Ocompl__one,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Compl @ One2 )
= Zero ) ) ).
% abstract_boolean_algebra.compl_one
thf(fact_6973_abstract__boolean__algebra_Ocompl__zero,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Compl @ Zero )
= One2 ) ) ).
% abstract_boolean_algebra.compl_zero
thf(fact_6974_abstract__boolean__algebra_Ocompl__unique,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( ( Conj @ X @ Y )
= Zero )
=> ( ( ( Disj @ X @ Y )
= One2 )
=> ( ( Compl @ X )
= Y ) ) ) ) ).
% abstract_boolean_algebra.compl_unique
thf(fact_6975_abstract__boolean__algebra_Odouble__compl,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Compl @ ( Compl @ X ) )
= X ) ) ).
% abstract_boolean_algebra.double_compl
thf(fact_6976_abstract__boolean__algebra_Odisj__one__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Disj @ One2 @ X )
= One2 ) ) ).
% abstract_boolean_algebra.disj_one_left
thf(fact_6977_abstract__boolean__algebra_Oconj__one__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Conj @ X @ One2 )
= X ) ) ).
% abstract_boolean_algebra.conj_one_right
thf(fact_6978_abstract__boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Conj @ Zero @ X )
= Zero ) ) ).
% abstract_boolean_algebra.conj_zero_left
thf(fact_6979_abstract__boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Compl @ ( Conj @ X @ Y ) )
= ( Disj @ ( Compl @ X ) @ ( Compl @ Y ) ) ) ) ).
% abstract_boolean_algebra.de_Morgan_conj
thf(fact_6980_abstract__boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Compl @ ( Disj @ X @ Y ) )
= ( Conj @ ( Compl @ X ) @ ( Compl @ Y ) ) ) ) ).
% abstract_boolean_algebra.de_Morgan_disj
thf(fact_6981_abstract__boolean__algebra_Odisj__one__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Disj @ X @ One2 )
= One2 ) ) ).
% abstract_boolean_algebra.disj_one_right
thf(fact_6982_abstract__boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Conj @ X @ Zero )
= Zero ) ) ).
% abstract_boolean_algebra.conj_zero_right
thf(fact_6983_abstract__boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Disj @ X @ Zero )
= X ) ) ).
% abstract_boolean_algebra.disj_zero_right
thf(fact_6984_abstract__boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Conj @ ( Compl @ X ) @ X )
= Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_left
thf(fact_6985_abstract__boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Disj @ ( Compl @ X ) @ X )
= One2 ) ) ).
% abstract_boolean_algebra.disj_cancel_left
thf(fact_6986_abstract__boolean__algebra__sym__diff_Oxor__def,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ X @ Y )
= ( Disj @ ( Conj @ X @ ( Compl @ Y ) ) @ ( Conj @ ( Compl @ X ) @ Y ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def
thf(fact_6987_abstract__boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,A3: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( ( Conj @ A3 @ X )
= Zero )
=> ( ( ( Disj @ A3 @ X )
= One2 )
=> ( ( ( Conj @ A3 @ Y )
= Zero )
=> ( ( ( Disj @ A3 @ Y )
= One2 )
=> ( X = Y ) ) ) ) ) ) ).
% abstract_boolean_algebra.complement_unique
thf(fact_6988_abstract__boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Conj @ X @ ( Compl @ X ) )
= Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_right
thf(fact_6989_abstract__boolean__algebra_Oconj__disj__distrib,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y: A,Z: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Conj @ X @ ( Disj @ Y @ Z ) )
= ( Disj @ ( Conj @ X @ Y ) @ ( Conj @ X @ Z ) ) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib
thf(fact_6990_abstract__boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Disj @ X @ ( Compl @ X ) )
= One2 ) ) ).
% abstract_boolean_algebra.disj_cancel_right
thf(fact_6991_abstract__boolean__algebra_Odisj__conj__distrib,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y: A,Z: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Disj @ X @ ( Conj @ Y @ Z ) )
= ( Conj @ ( Disj @ X @ Y ) @ ( Disj @ X @ Z ) ) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib
thf(fact_6992_abstract__boolean__algebra__sym__diff_Oxor__def2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ X @ Y )
= ( Conj @ ( Disj @ X @ Y ) @ ( Disj @ ( Compl @ X ) @ ( Compl @ Y ) ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def2
thf(fact_6993_abstract__boolean__algebra__sym__diff_Oxor__self,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ X @ X )
= Zero ) ) ).
% abstract_boolean_algebra_sym_diff.xor_self
thf(fact_6994_abstract__boolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( ( Compl @ X )
= ( Compl @ Y ) )
= ( X = Y ) ) ) ).
% abstract_boolean_algebra.compl_eq_compl_iff
thf(fact_6995_abstract__boolean__algebra_Oconj__disj__distrib2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Y: A,Z: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Conj @ ( Disj @ Y @ Z ) @ X )
= ( Disj @ ( Conj @ Y @ X ) @ ( Conj @ Z @ X ) ) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib2
thf(fact_6996_abstract__boolean__algebra_Odisj__conj__distrib2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Y: A,Z: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Disj @ ( Conj @ Y @ Z ) @ X )
= ( Conj @ ( Disj @ Y @ X ) @ ( Disj @ Z @ X ) ) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib2
thf(fact_6997_abstract__boolean__algebra__sym__diff_Oxor__one__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ One2 @ X )
= ( Compl @ X ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_left
thf(fact_6998_abstract__boolean__algebra__sym__diff_Oxor__left__self,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ X @ ( Xor @ X @ Y ) )
= Y ) ) ).
% abstract_boolean_algebra_sym_diff.xor_left_self
thf(fact_6999_abstract__boolean__algebra__sym__diff_Oxor__one__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ X @ One2 )
= ( Compl @ X ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_right
thf(fact_7000_abstract__boolean__algebra__sym__diff_Oxor__compl__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ ( Compl @ X ) @ Y )
= ( Compl @ ( Xor @ X @ Y ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_left
thf(fact_7001_abstract__boolean__algebra__sym__diff_Oxor__cancel__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ ( Compl @ X ) @ X )
= One2 ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_left
thf(fact_7002_abstract__boolean__algebra__sym__diff_Oxor__compl__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ X @ ( Compl @ Y ) )
= ( Compl @ ( Xor @ X @ Y ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_right
thf(fact_7003_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y: A,Z: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Conj @ X @ ( Xor @ Y @ Z ) )
= ( Xor @ ( Conj @ X @ Y ) @ ( Conj @ X @ Z ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib
thf(fact_7004_abstract__boolean__algebra__sym__diff_Oxor__cancel__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ X @ ( Compl @ X ) )
= One2 ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_right
thf(fact_7005_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,Y: A,Z: A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Conj @ ( Xor @ Y @ Z ) @ X )
= ( Xor @ ( Conj @ Y @ X ) @ ( Conj @ Z @ X ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib2
thf(fact_7006_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_7007_abstract__boolean__algebra__sym__diff_Ointro,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor )
=> ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor ) ) ) ).
% abstract_boolean_algebra_sym_diff.intro
thf(fact_7008_abstract__boolean__algebra__sym__diff__def,axiom,
! [A: $tType] :
( ( boolea3799213064322606851m_diff @ A )
= ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One: A,Xor2: A > A > A] :
( ( boolea2506097494486148201lgebra @ A @ Conj2 @ Disj2 @ Compl2 @ Zero2 @ One )
& ( boolea5476839437570043046axioms @ A @ Conj2 @ Disj2 @ Compl2 @ Xor2 ) ) ) ) ).
% abstract_boolean_algebra_sym_diff_def
thf(fact_7009_abstract__boolean__algebra__sym__diff__axioms__def,axiom,
! [A: $tType] :
( ( boolea5476839437570043046axioms @ A )
= ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Xor2: A > A > A] :
! [X2: A,Y3: A] :
( ( Xor2 @ X2 @ Y3 )
= ( Disj2 @ ( Conj2 @ X2 @ ( Compl2 @ Y3 ) ) @ ( Conj2 @ ( Compl2 @ X2 ) @ Y3 ) ) ) ) ) ).
% abstract_boolean_algebra_sym_diff_axioms_def
thf(fact_7010_abstract__boolean__algebra__sym__diff__axioms_Ointro,axiom,
! [A: $tType,Xor: A > A > A,Disj: A > A > A,Conj: A > A > A,Compl: A > A] :
( ! [X3: A,Y2: A] :
( ( Xor @ X3 @ Y2 )
= ( Disj @ ( Conj @ X3 @ ( Compl @ Y2 ) ) @ ( Conj @ ( Compl @ X3 ) @ Y2 ) ) )
=> ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor ) ) ).
% abstract_boolean_algebra_sym_diff_axioms.intro
thf(fact_7011_abstract__boolean__algebra__sym__diff_Oaxioms_I2_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor ) ) ).
% abstract_boolean_algebra_sym_diff.axioms(2)
thf(fact_7012_Real_Opositive_Oabs__eq,axiom,
! [X: nat > rat] :
( ( realrel @ X @ X )
=> ( ( positive2 @ ( real2 @ X ) )
= ( ? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less @ rat @ R5 @ ( X @ N2 ) ) ) ) ) ) ) ).
% Real.positive.abs_eq
thf(fact_7013_sqr_Osimps_I3_J,axiom,
! [N: num] :
( ( sqr @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ ( plus_plus @ num @ ( sqr @ N ) @ N ) ) ) ) ).
% sqr.simps(3)
thf(fact_7014_sqr__conv__mult,axiom,
( sqr
= ( ^ [X2: num] : ( times_times @ num @ X2 @ X2 ) ) ) ).
% sqr_conv_mult
thf(fact_7015_sqr_Osimps_I1_J,axiom,
( ( sqr @ one2 )
= one2 ) ).
% sqr.simps(1)
thf(fact_7016_sqr_Osimps_I2_J,axiom,
! [N: num] :
( ( sqr @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% sqr.simps(2)
thf(fact_7017_pow_Osimps_I3_J,axiom,
! [X: num,Y: num] :
( ( pow @ X @ ( bit1 @ Y ) )
= ( times_times @ num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).
% pow.simps(3)
thf(fact_7018_Real_Opositive_Orsp,axiom,
( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ $o @ $o @ realrel
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2
@ ^ [X8: nat > rat] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less @ rat @ R5 @ ( X8 @ N2 ) ) ) )
@ ^ [X8: nat > rat] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less @ rat @ R5 @ ( X8 @ N2 ) ) ) ) ) ).
% Real.positive.rsp
thf(fact_7019_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one2 )
= X ) ).
% pow.simps(1)
thf(fact_7020_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( semiring_numeral @ B )
& ( monoid_add @ A )
& ( semiring_numeral @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R @ ( bNF_rel_fun @ A @ B @ A @ B @ R @ R ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y4: num,Z2: num] : Y4 = Z2
@ R
@ ( numeral_numeral @ A )
@ ( numeral_numeral @ B ) ) ) ) ) ) ).
% transfer_rule_numeral
thf(fact_7021_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( ring_1 @ B )
& ( ring_1 @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R @ ( bNF_rel_fun @ A @ B @ A @ B @ R @ R ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ A @ B @ R @ R @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
=> ( bNF_rel_fun @ int @ int @ A @ B
@ ^ [Y4: int,Z2: int] : Y4 = Z2
@ R
@ ( ring_1_of_int @ A )
@ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% transfer_rule_of_int
thf(fact_7022_pow_Osimps_I2_J,axiom,
! [X: num,Y: num] :
( ( pow @ X @ ( bit0 @ Y ) )
= ( sqr @ ( pow @ X @ Y ) ) ) ).
% pow.simps(2)
thf(fact_7023_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( semiring_1 @ B )
& ( semiring_1 @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R @ ( bNF_rel_fun @ A @ B @ A @ B @ R @ R ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y4: nat,Z2: nat] : Y4 = Z2
@ R
@ ( semiring_1_of_nat @ A )
@ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% transfer_rule_of_nat
thf(fact_7024_fun_Orel__mono,axiom,
! [D: $tType,B: $tType,A: $tType,R: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra )
=> ( ord_less_eq @ ( ( D > A ) > ( D > B ) > $o )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y4: D,Z2: D] : Y4 = Z2
@ R )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y4: D,Z2: D] : Y4 = Z2
@ Ra ) ) ) ).
% fun.rel_mono
thf(fact_7025_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( ( zero_neq_one @ B )
& ( zero_neq_one @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2
@ R
@ ( zero_neq_one_of_bool @ A )
@ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% transfer_rule_of_bool
thf(fact_7026_less__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y4: nat,Z2: nat] : Y4 = Z2
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y4: nat,Z2: nat] : Y4 = Z2
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( ord_less @ nat )
@ ( ord_less @ nat ) ) ).
% less_natural.rsp
thf(fact_7027_less__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [Y4: int,Z2: int] : Y4 = Z2
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [Y4: int,Z2: int] : Y4 = Z2
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( ord_less @ int )
@ ( ord_less @ int ) ) ).
% less_integer.rsp
thf(fact_7028_less__eq__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [Y4: int,Z2: int] : Y4 = Z2
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [Y4: int,Z2: int] : Y4 = Z2
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( ord_less_eq @ int )
@ ( ord_less_eq @ int ) ) ).
% less_eq_integer.rsp
thf(fact_7029_less__eq__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y4: nat,Z2: nat] : Y4 = Z2
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y4: nat,Z2: nat] : Y4 = Z2
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( ord_less_eq @ nat )
@ ( ord_less_eq @ nat ) ) ).
% less_eq_natural.rsp
thf(fact_7030_fun__mono,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,C2: A > B > $o,A4: A > B > $o,B2: C > D > $o,D3: C > D > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ C2 @ A4 )
=> ( ( ord_less_eq @ ( C > D > $o ) @ B2 @ D3 )
=> ( ord_less_eq @ ( ( A > C ) > ( B > D ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 ) @ ( bNF_rel_fun @ A @ B @ C @ D @ C2 @ D3 ) ) ) ) ).
% fun_mono
thf(fact_7031_Real_Opositive_Otransfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ $o @ $o @ pcr_real
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2
@ ^ [X8: nat > rat] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less @ rat @ R5 @ ( X8 @ N2 ) ) ) )
@ positive2 ) ).
% Real.positive.transfer
thf(fact_7032_less__eq__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V6: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( ord_less_eq @ int ) ) ).
% less_eq_int.transfer
thf(fact_7033_less__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V6: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( ord_less @ int ) ) ).
% less_int.transfer
thf(fact_7034_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% zero_int.transfer
thf(fact_7035_int__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
@ ^ [Y4: nat,Z2: nat] : Y4 = Z2
@ pcr_int
@ ^ [N2: nat] : ( product_Pair @ nat @ nat @ N2 @ ( zero_zero @ nat ) )
@ ( semiring_1_of_nat @ int ) ) ).
% int_transfer
thf(fact_7036_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% one_int.transfer
thf(fact_7037_mono__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ D )
& ( order @ C )
& ( order @ A ) )
=> ! [A4: A > B > $o,B2: C > D > $o] :
( ( bi_total @ A @ B @ A4 )
=> ( ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A4
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( ord_less_eq @ A )
@ ( ord_less_eq @ B ) )
=> ( ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B2
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( ord_less_eq @ C )
@ ( ord_less_eq @ D ) )
=> ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 )
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2
@ ( order_mono @ A @ C )
@ ( order_mono @ B @ D ) ) ) ) ) ) ).
% mono_transfer
thf(fact_7038_arg__min__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F3: A > B,P: A > $o,A3: A] :
( ( inj_on @ A @ B @ F3 @ ( collect @ A @ P ) )
=> ( ( P @ A3 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ A3 ) @ ( F3 @ Y2 ) ) )
=> ( ( lattices_ord_arg_min @ A @ B @ F3 @ P )
= A3 ) ) ) ) ) ).
% arg_min_inj_eq
thf(fact_7039_arg__min__equality,axiom,
! [A: $tType,C: $tType] :
( ( order @ A )
=> ! [P: C > $o,K: C,F3: C > A] :
( ( P @ K )
=> ( ! [X3: C] :
( ( P @ X3 )
=> ( ord_less_eq @ A @ ( F3 @ K ) @ ( F3 @ X3 ) ) )
=> ( ( F3 @ ( lattices_ord_arg_min @ C @ A @ F3 @ P ) )
= ( F3 @ K ) ) ) ) ) ).
% arg_min_equality
thf(fact_7040_arg__min__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ( ( P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) @ ( M @ Y5 ) ) ) ) ) ).
% arg_min_nat_lemma
thf(fact_7041_arg__min__nat__le,axiom,
! [A: $tType,P: A > $o,X: A,M: A > nat] :
( ( P @ X )
=> ( ord_less_eq @ nat @ ( M @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) @ ( M @ X ) ) ) ).
% arg_min_nat_le
thf(fact_7042_arg__minI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [P: A > $o,X: A,F3: A > B,Q: A > $o] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ~ ( ord_less @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ~ ( ord_less @ B @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( lattices_ord_arg_min @ A @ B @ F3 @ P ) ) ) ) ) ) ).
% arg_minI
thf(fact_7043_arg__min__on__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic7623131987881927897min_on @ B @ A )
= ( ^ [F2: B > A,S7: set @ B] :
( lattices_ord_arg_min @ B @ A @ F2
@ ^ [X2: B] : ( member @ B @ X2 @ S7 ) ) ) ) ) ).
% arg_min_on_def
thf(fact_7044_arg__min__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattices_ord_arg_min @ B @ A )
= ( ^ [F2: B > A,P3: B > $o] : ( fChoice @ B @ ( lattic501386751177426532rg_min @ B @ A @ F2 @ P3 ) ) ) ) ) ).
% arg_min_def
thf(fact_7045_in__measures_I2_J,axiom,
! [A: $tType,X: A,Y: A,F3: A > nat,Fs: list @ ( A > nat )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) )
= ( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X )
= ( F3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_7046_arg__min__natI,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ( P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) ) ).
% arg_min_natI
thf(fact_7047_is__arg__min__arg__min__nat,axiom,
! [A: $tType,P: A > $o,X: A,M: A > nat] :
( ( P @ X )
=> ( lattic501386751177426532rg_min @ A @ nat @ M @ P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) ) ).
% is_arg_min_arg_min_nat
thf(fact_7048_is__arg__min__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic501386751177426532rg_min @ B @ A )
= ( ^ [F2: B > A,P3: B > $o,X2: B] :
( ( P3 @ X2 )
& ~ ? [Y3: B] :
( ( P3 @ Y3 )
& ( ord_less @ A @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% is_arg_min_def
thf(fact_7049_is__arg__min__antimono,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F3: A > B,P: A > $o,X: A,Y: A] :
( ( lattic501386751177426532rg_min @ A @ B @ F3 @ P @ X )
=> ( ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X ) )
=> ( ( P @ Y )
=> ( lattic501386751177426532rg_min @ A @ B @ F3 @ P @ Y ) ) ) ) ) ).
% is_arg_min_antimono
thf(fact_7050_is__arg__min__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ( ( lattic501386751177426532rg_min @ A @ B )
= ( ^ [F2: A > B,P3: A > $o,X2: A] :
( ( P3 @ X2 )
& ! [Y3: A] :
( ( P3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ) ) ) ).
% is_arg_min_linorder
thf(fact_7051_measures__less,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) ) ) ).
% measures_less
thf(fact_7052_measures__lesseq,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less_eq @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_7053_ex__is__arg__min__if__finite,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_1: A] :
( lattic501386751177426532rg_min @ A @ B @ F3
@ ^ [X2: A] : ( member @ A @ X2 @ S )
@ X_1 ) ) ) ) ).
% ex_is_arg_min_if_finite
thf(fact_7054_Partial__order__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A3 = B3 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_7055_pred__nat__trancl__eq__le,axiom,
! [M: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N ) @ ( transitive_rtrancl @ nat @ pred_nat ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% pred_nat_trancl_eq_le
thf(fact_7056_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_7057_less__eq,axiom,
! [M: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N ) @ ( transitive_trancl @ nat @ pred_nat ) )
= ( ord_less @ nat @ M @ N ) ) ).
% less_eq
thf(fact_7058_chains__extend,axiom,
! [A: $tType,C3: set @ ( set @ A ),S: set @ ( set @ A ),Z: set @ A] :
( ( member @ ( set @ ( set @ A ) ) @ C3 @ ( chains2 @ A @ S ) )
=> ( ( member @ ( set @ A ) @ Z @ S )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ X3 @ Z ) )
=> ( member @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert2 @ ( set @ A ) @ Z @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C3 ) @ ( chains2 @ A @ S ) ) ) ) ) ).
% chains_extend
thf(fact_7059_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A,A3: B] :
( ( inj_on @ A @ B @ F3 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ A4 )
@ ( insert2 @ A
@ ( the @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ( F3 @ X2 )
= A3 ) ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_on_vimage_singleton
thf(fact_7060_vimageI,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: B,B3: A,B2: set @ A] :
( ( ( F3 @ A3 )
= B3 )
=> ( ( member @ A @ B3 @ B2 )
=> ( member @ B @ A3 @ ( vimage @ B @ A @ F3 @ B2 ) ) ) ) ).
% vimageI
thf(fact_7061_vimage__eq,axiom,
! [A: $tType,B: $tType,A3: A,F3: A > B,B2: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F3 @ B2 ) )
= ( member @ B @ ( F3 @ A3 ) @ B2 ) ) ).
% vimage_eq
thf(fact_7062_vimage__ident,axiom,
! [A: $tType,Y6: set @ A] :
( ( vimage @ A @ A
@ ^ [X2: A] : X2
@ Y6 )
= Y6 ) ).
% vimage_ident
thf(fact_7063_vimage__Collect__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,P: B > $o] :
( ( vimage @ A @ B @ F3 @ ( collect @ B @ P ) )
= ( collect @ A
@ ^ [Y3: A] : ( P @ ( F3 @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_7064_vimage__UNIV,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( vimage @ A @ B @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% vimage_UNIV
thf(fact_7065_vimage__empty,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( vimage @ A @ B @ F3 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% vimage_empty
thf(fact_7066_vimage__Int,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ B,B2: set @ B] :
( ( vimage @ A @ B @ F3 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A4 ) @ ( vimage @ A @ B @ F3 @ B2 ) ) ) ).
% vimage_Int
thf(fact_7067_vimage__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ B,B2: set @ B] :
( ( vimage @ A @ B @ F3 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A4 ) @ ( vimage @ A @ B @ F3 @ B2 ) ) ) ).
% vimage_Un
thf(fact_7068_filtercomap__principal,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ B] :
( ( filtercomap @ A @ B @ F3 @ ( principal @ B @ A4 ) )
= ( principal @ A @ ( vimage @ A @ B @ F3 @ A4 ) ) ) ).
% filtercomap_principal
thf(fact_7069_vimage__const,axiom,
! [B: $tType,A: $tType,C3: B,A4: set @ B] :
( ( ( member @ B @ C3 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : C3
@ A4 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ C3 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : C3
@ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% vimage_const
thf(fact_7070_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ A] :
( ( image2 @ B @ A @ F3 @ ( vimage @ B @ A @ F3 @ A4 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% image_vimage_eq
thf(fact_7071_vimage__if,axiom,
! [B: $tType,A: $tType,C3: B,A4: set @ B,D2: B,B2: set @ A] :
( ( ( member @ B @ C3 @ A4 )
=> ( ( ( member @ B @ D2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B2 ) @ C3 @ D2 )
@ A4 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ D2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B2 ) @ C3 @ D2 )
@ A4 )
= B2 ) ) ) )
& ( ~ ( member @ B @ C3 @ A4 )
=> ( ( ( member @ B @ D2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B2 ) @ C3 @ D2 )
@ A4 )
= ( uminus_uminus @ ( set @ A ) @ B2 ) ) )
& ( ~ ( member @ B @ D2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B2 ) @ C3 @ D2 )
@ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% vimage_if
thf(fact_7072_finite__vimageI,axiom,
! [B: $tType,A: $tType,F4: set @ A,H: B > A] :
( ( finite_finite2 @ A @ F4 )
=> ( ( inj_on @ B @ A @ H @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ H @ F4 ) ) ) ) ).
% finite_vimageI
thf(fact_7073_finite__vimage__Suc__iff,axiom,
! [F4: set @ nat] :
( ( finite_finite2 @ nat @ ( vimage @ nat @ nat @ suc @ F4 ) )
= ( finite_finite2 @ nat @ F4 ) ) ).
% finite_vimage_Suc_iff
thf(fact_7074_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B,G2: A > B,Y: set @ B] :
( ! [W: A] :
( ( member @ A @ W @ S )
=> ( ( F3 @ W )
= ( G2 @ W ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ Y ) @ S )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G2 @ Y ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_7075_vimage__Compl,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ B] :
( ( vimage @ A @ B @ F3 @ ( uminus_uminus @ ( set @ B ) @ A4 ) )
= ( uminus_uminus @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A4 ) ) ) ).
% vimage_Compl
thf(fact_7076_vimageD,axiom,
! [A: $tType,B: $tType,A3: A,F3: A > B,A4: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F3 @ A4 ) )
=> ( member @ B @ ( F3 @ A3 ) @ A4 ) ) ).
% vimageD
thf(fact_7077_vimageE,axiom,
! [A: $tType,B: $tType,A3: A,F3: A > B,B2: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F3 @ B2 ) )
=> ( member @ B @ ( F3 @ A3 ) @ B2 ) ) ).
% vimageE
thf(fact_7078_vimageI2,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: B,A4: set @ A] :
( ( member @ A @ ( F3 @ A3 ) @ A4 )
=> ( member @ B @ A3 @ ( vimage @ B @ A @ F3 @ A4 ) ) ) ).
% vimageI2
thf(fact_7079_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F3: A > B,Q: A > $o] :
( ! [X3: A] :
( ( P @ ( F3 @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage @ A @ B @ F3 @ ( collect @ B @ P ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_7080_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F2: A > B,B6: set @ B] :
( collect @ A
@ ^ [X2: A] : ( member @ B @ ( F2 @ X2 ) @ B6 ) ) ) ) ).
% vimage_def
thf(fact_7081_vimage__Diff,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ B,B2: set @ B] :
( ( vimage @ A @ B @ F3 @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A4 ) @ ( vimage @ A @ B @ F3 @ B2 ) ) ) ).
% vimage_Diff
thf(fact_7082_finite__vimage__iff,axiom,
! [A: $tType,B: $tType,H: A > B,F4: set @ B] :
( ( bij_betw @ A @ B @ H @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( finite_finite2 @ A @ ( vimage @ A @ B @ H @ F4 ) )
= ( finite_finite2 @ B @ F4 ) ) ) ).
% finite_vimage_iff
thf(fact_7083_finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F4: set @ A,H: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ F4 )
=> ( ( inj_on @ B @ A @ H @ A4 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H @ F4 ) @ A4 ) ) ) ) ).
% finite_vimage_IntI
thf(fact_7084_vimage__Suc__insert__Suc,axiom,
! [N: nat,A4: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert2 @ nat @ ( suc @ N ) @ A4 ) )
= ( insert2 @ nat @ N @ ( vimage @ nat @ nat @ suc @ A4 ) ) ) ).
% vimage_Suc_insert_Suc
thf(fact_7085_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A3: A,F3: A > B,B3: B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F3 @ ( insert2 @ B @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( ( F3 @ A3 )
= B3 ) ) ).
% vimage_singleton_eq
thf(fact_7086_vimage__insert,axiom,
! [A: $tType,B: $tType,F3: A > B,A3: B,B2: set @ B] :
( ( vimage @ A @ B @ F3 @ ( insert2 @ B @ A3 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( vimage @ A @ B @ F3 @ B2 ) ) ) ).
% vimage_insert
thf(fact_7087_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F3: B > A,A4: set @ A] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( vimage @ B @ A @ F3 @ A4 )
= ( bot_bot @ ( set @ B ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% surj_vimage_empty
thf(fact_7088_vimage__image__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A] :
( ( vimage @ A @ B @ F3 @ ( image2 @ A @ B @ F3 @ A4 ) )
= ( collect @ A
@ ^ [Y3: A] :
? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ( F3 @ X2 )
= ( F3 @ Y3 ) ) ) ) ) ).
% vimage_image_eq
thf(fact_7089_finite__vimageD,axiom,
! [A: $tType,B: $tType,H: A > B,F4: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ H @ F4 ) )
=> ( ( ( image2 @ A @ B @ H @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B @ F4 ) ) ) ).
% finite_vimageD
thf(fact_7090_vimage__Suc__insert__0,axiom,
! [A4: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert2 @ nat @ ( zero_zero @ nat ) @ A4 ) )
= ( vimage @ nat @ nat @ suc @ A4 ) ) ).
% vimage_Suc_insert_0
thf(fact_7091_vimage__subsetD,axiom,
! [A: $tType,B: $tType,F3: B > A,B2: set @ A,A4: set @ B] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ B2 ) @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_7092_image__subset__iff__subset__vimage,axiom,
! [B: $tType,A: $tType,F3: B > A,A4: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A4 ) @ B2 )
= ( ord_less_eq @ ( set @ B ) @ A4 @ ( vimage @ B @ A @ F3 @ B2 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_7093_image__vimage__subset,axiom,
! [B: $tType,A: $tType,F3: B > A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( vimage @ B @ A @ F3 @ A4 ) ) @ A4 ) ).
% image_vimage_subset
thf(fact_7094_chainsD2,axiom,
! [A: $tType,C3: set @ ( set @ A ),S: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ C3 @ ( chains2 @ A @ S ) )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ C3 @ S ) ) ).
% chainsD2
thf(fact_7095_vimage__mono,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ A,F3: B > A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ A4 ) @ ( vimage @ B @ A @ F3 @ B2 ) ) ) ).
% vimage_mono
thf(fact_7096_subset__vimage__iff,axiom,
! [B: $tType,A: $tType,A4: set @ A,F3: A > B,B2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( vimage @ A @ B @ F3 @ B2 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ B @ ( F3 @ X2 ) @ B2 ) ) ) ) ).
% subset_vimage_iff
thf(fact_7097_Zorn__Lemma2,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ! [X3: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X3 @ ( chains2 @ A @ A4 ) )
=> ? [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A4 )
& ! [Xb: set @ A] :
( ( member @ ( set @ A ) @ Xb @ X3 )
=> ( ord_less_eq @ ( set @ A ) @ Xb @ Xa ) ) ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_7098_chainsD,axiom,
! [A: $tType,C3: set @ ( set @ A ),S: set @ ( set @ A ),X: set @ A,Y: set @ A] :
( ( member @ ( set @ ( set @ A ) ) @ C3 @ ( chains2 @ A @ S ) )
=> ( ( member @ ( set @ A ) @ X @ C3 )
=> ( ( member @ ( set @ A ) @ Y @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X @ Y )
| ( ord_less_eq @ ( set @ A ) @ Y @ X ) ) ) ) ) ).
% chainsD
thf(fact_7099_Zorn__Lemma,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ! [X3: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X3 @ ( chains2 @ A @ A4 ) )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ X3 ) @ A4 ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% Zorn_Lemma
thf(fact_7100_continuous__imp__open__vimage,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [S3: set @ A,F3: A > B,B2: set @ B] :
( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( topolo1002775350975398744n_open @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ B2 ) @ S3 )
=> ( topolo1002775350975398744n_open @ A @ ( vimage @ A @ B @ F3 @ B2 ) ) ) ) ) ) ) ).
% continuous_imp_open_vimage
thf(fact_7101_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A )
=> ! [P: ( A > B ) > A > B > $o] :
( ! [R3: B,F6: A > B,G9: A > B,X3: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X3 )
=> ( ( F6 @ Y5 )
= ( G9 @ Y5 ) ) )
=> ( ( P @ F6 @ X3 @ R3 )
= ( P @ G9 @ X3 @ R3 ) ) )
=> ( ! [X3: A,F6: A > B] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X3 )
=> ( P @ F6 @ Y5 @ ( F6 @ Y5 ) ) )
=> ? [X_12: B] : ( P @ F6 @ X3 @ X_12 ) )
=> ? [F6: A > B] :
! [X5: A] : ( P @ F6 @ X5 @ ( F6 @ X5 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_7102_finite__vimageD_H,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ F3 @ A4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ ( image2 @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) )
=> ( finite_finite2 @ B @ A4 ) ) ) ).
% finite_vimageD'
thf(fact_7103_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F3: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ~ ( finite_finite2 @ B @ A4 )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( image2 @ B @ A @ F3 @ A4 ) )
& ~ ( finite_finite2 @ B @ ( vimage @ B @ A @ F3 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_7104_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F3: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ~ ( finite_finite2 @ B @ A4 )
=> ~ ! [Y2: A] :
( ( member @ A @ Y2 @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ F3 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_7105_vimage__subsetI,axiom,
! [B: $tType,A: $tType,F3: A > B,B2: set @ B,A4: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( image2 @ A @ B @ F3 @ A4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ B2 ) @ A4 ) ) ) ).
% vimage_subsetI
thf(fact_7106_finite__finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F4: set @ A,H: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ F4 )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ F4 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H @ F4 ) @ A4 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_7107_countable__vimage,axiom,
! [B: $tType,A: $tType,B2: set @ A,F3: B > A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ( countable_countable @ B @ ( vimage @ B @ A @ F3 @ B2 ) )
=> ( countable_countable @ A @ B2 ) ) ) ).
% countable_vimage
thf(fact_7108_vimage__subset__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,B2: set @ B,A4: set @ A] :
( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ B2 ) @ A4 )
= ( ord_less_eq @ ( set @ B ) @ B2 @ ( image2 @ A @ B @ F3 @ A4 ) ) ) ) ).
% vimage_subset_eq
thf(fact_7109_vimage__eq__UN,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F2: A > B,B6: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : ( vimage @ A @ B @ F2 @ ( insert2 @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) )
@ B6 ) ) ) ) ).
% vimage_eq_UN
thf(fact_7110_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ~ ( finite_finite2 @ B @ A4 )
=> ~ ! [Y2: A] :
( ( member @ A @ Y2 @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_7111_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( ~ ( finite_finite2 @ B @ A4 )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( image2 @ B @ A @ F3 @ A4 ) )
& ~ ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_7112_card__vimage__inj,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: set @ B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ ( image2 @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( finite_card @ A @ ( vimage @ A @ B @ F3 @ A4 ) )
= ( finite_card @ B @ A4 ) ) ) ) ).
% card_vimage_inj
thf(fact_7113_card__vimage__inj__on__le,axiom,
! [A: $tType,B: $tType,F3: A > B,D3: set @ A,A4: set @ B] :
( ( inj_on @ A @ B @ F3 @ D3 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A4 ) @ D3 ) ) @ ( finite_card @ B @ A4 ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_7114_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
@ ( insert2 @ A
@ ( the @ A
@ ^ [X2: A] :
( ( F3 @ X2 )
= A3 ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_vimage_singleton
thf(fact_7115_chains__def,axiom,
! [A: $tType] :
( ( chains2 @ A )
= ( ^ [A6: set @ ( set @ A )] :
( collect @ ( set @ ( set @ A ) )
@ ^ [C4: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C4 @ A6 )
& ( chain_subset @ A @ C4 ) ) ) ) ) ).
% chains_def
thf(fact_7116_image__split__eq__Sigma,axiom,
! [C: $tType,B: $tType,A: $tType,F3: C > A,G2: C > B,A4: set @ C] :
( ( image2 @ C @ ( product_prod @ A @ B )
@ ^ [X2: C] : ( product_Pair @ A @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ A4 )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F3 @ A4 )
@ ^ [X2: A] : ( image2 @ C @ B @ G2 @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F3 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_7117_Field__square,axiom,
! [A: $tType,X: set @ A] :
( ( field2 @ A
@ ( product_Sigma @ A @ A @ X
@ ^ [Uu3: A] : X ) )
= X ) ).
% Field_square
thf(fact_7118_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B2: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_7119_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_7120_Times__empty,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_7121_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C2: set @ A,A4: set @ B,B2: set @ B] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ C2
@ ^ [Uu3: A] : A4 )
@ ( product_Sigma @ A @ B @ C2
@ ^ [Uu3: A] : B2 ) )
= ( ( C2
= ( bot_bot @ ( set @ A ) ) )
| ( disjnt @ B @ A4 @ B2 ) ) ) ).
% disjnt_Times1_iff
thf(fact_7122_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A4: set @ A,C2: set @ B,B2: set @ A] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : C2 )
@ ( product_Sigma @ A @ B @ B2
@ ^ [Uu3: A] : C2 ) )
= ( ( C2
= ( bot_bot @ ( set @ B ) ) )
| ( disjnt @ A @ A4 @ B2 ) ) ) ).
% disjnt_Times2_iff
thf(fact_7123_finite__SigmaI,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [A7: A] :
( ( member @ A @ A7 @ A4 )
=> ( finite_finite2 @ B @ ( B2 @ A7 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ B2 ) ) ) ) ).
% finite_SigmaI
thf(fact_7124_connected__Times__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [S: set @ A,T4: set @ B] :
( ( topolo1966860045006549960nected @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ S
@ ^ [Uu3: A] : T4 ) )
= ( ( S
= ( bot_bot @ ( set @ A ) ) )
| ( T4
= ( bot_bot @ ( set @ B ) ) )
| ( ( topolo1966860045006549960nected @ A @ S )
& ( topolo1966860045006549960nected @ B @ T4 ) ) ) ) ) ).
% connected_Times_eq
thf(fact_7125_insert__Times__insert,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,B3: B,B2: set @ B] :
( ( product_Sigma @ A @ B @ ( insert2 @ A @ A3 @ A4 )
@ ^ [Uu3: A] : ( insert2 @ B @ B3 @ B2 ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 )
@ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : ( insert2 @ B @ B3 @ B2 ) )
@ ( product_Sigma @ A @ B @ ( insert2 @ A @ A3 @ A4 )
@ ^ [Uu3: A] : B2 ) ) ) ) ).
% insert_Times_insert
thf(fact_7126_card__SigmaI,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( finite_finite2 @ B @ ( B2 @ X3 ) ) )
=> ( ( finite_card @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ B2 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [A5: A] : ( finite_card @ B @ ( B2 @ A5 ) )
@ A4 ) ) ) ) ).
% card_SigmaI
thf(fact_7127_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X: B,A4: set @ B,F3: B > ( set @ A )] :
( ( ( member @ B @ X @ A4 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A4 @ F3 ) )
= ( F3 @ X ) ) )
& ( ~ ( member @ B @ X @ A4 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A4 @ F3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pair_vimage_Sigma
thf(fact_7128_trancl__subset__Sigma,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 )
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) ) ) ).
% trancl_subset_Sigma
thf(fact_7129_listrel__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel @ A @ A @ R2 )
@ ( product_Sigma @ ( list @ A ) @ ( list @ A ) @ ( lists @ A @ A4 )
@ ^ [Uu3: list @ A] : ( lists @ A @ A4 ) ) ) ) ).
% listrel_subset
thf(fact_7130_Id__on__subset__Times,axiom,
! [A: $tType,A4: set @ A] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( id_on @ A @ A4 )
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) ) ).
% Id_on_subset_Times
thf(fact_7131_Restr__subset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) ) ) ) ).
% Restr_subset
thf(fact_7132_Sigma__mono,axiom,
! [B: $tType,A: $tType,A4: set @ A,C2: set @ A,B2: A > ( set @ B ),D3: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( B2 @ X3 ) @ ( D3 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A4 @ B2 ) @ ( product_Sigma @ A @ B @ C2 @ D3 ) ) ) ) ).
% Sigma_mono
thf(fact_7133_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C2: set @ A,A4: set @ B,B2: set @ B] :
( ( member @ A @ X @ C2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu3: B] : C2 )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu3: B] : C2 ) )
= ( ord_less_eq @ ( set @ B ) @ A4 @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_7134_equiv__type,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) ) ) ).
% equiv_type
thf(fact_7135_times__eq__iff,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B,C2: set @ A,D3: set @ B] :
( ( ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 )
= ( product_Sigma @ A @ B @ C2
@ ^ [Uu3: A] : D3 ) )
= ( ( ( A4 = C2 )
& ( B2 = D3 ) )
| ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C2
= ( bot_bot @ ( set @ A ) ) )
| ( D3
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_7136_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I5: set @ A,X4: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I5 @ X4 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ I5 )
=> ( ( X4 @ X2 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_7137_Times__Un__distrib1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ A,C2: set @ B] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A4 @ B2 )
@ ^ [Uu3: A] : C2 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : C2 )
@ ( product_Sigma @ A @ B @ B2
@ ^ [Uu3: A] : C2 ) ) ) ).
% Times_Un_distrib1
thf(fact_7138_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A4: A > ( set @ B ),B2: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I4: A] : ( sup_sup @ ( set @ B ) @ ( A4 @ I4 ) @ ( B2 @ I4 ) ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A4 ) @ ( product_Sigma @ A @ B @ I5 @ B2 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_7139_Sigma__Un__distrib1,axiom,
! [B: $tType,A: $tType,I5: set @ A,J5: set @ A,C2: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I5 @ J5 ) @ C2 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ C2 ) @ ( product_Sigma @ A @ B @ J5 @ C2 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_7140_finite__cartesian__product,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) ) ) ) ).
% finite_cartesian_product
thf(fact_7141_infinite__cartesian__product,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ~ ( finite_finite2 @ B @ B2 )
=> ~ ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_7142_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A4: set @ A,C2: A > ( set @ B ),B2: set @ A] :
( ( disjnt @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ C2 ) @ ( product_Sigma @ A @ B @ B2 @ C2 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
=> ( ( C2 @ X2 )
= ( bot_bot @ ( set @ B ) ) ) )
| ( disjnt @ A @ A4 @ B2 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_7143_times__subset__iff,axiom,
! [A: $tType,B: $tType,A4: set @ A,C2: set @ B,B2: set @ A,D3: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : C2 )
@ ( product_Sigma @ A @ B @ B2
@ ^ [Uu3: A] : D3 ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( C2
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
& ( ord_less_eq @ ( set @ B ) @ C2 @ D3 ) ) ) ) ).
% times_subset_iff
thf(fact_7144_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A3: A,B3: A,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
=> ( ( A3 = B3 )
| ( member @ A @ A3 @ A4 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_7145_Field__Restr__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ord_less_eq @ ( set @ A )
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) ) )
@ A4 ) ).
% Field_Restr_subset
thf(fact_7146_wfI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : B2 ) )
=> ( ! [X3: A,P8: A > $o] :
( ! [Xa: A] :
( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa ) @ R2 )
=> ( P8 @ Y2 ) )
=> ( P8 @ Xa ) )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( member @ A @ X3 @ B2 )
=> ( P8 @ X3 ) ) ) )
=> ( wf @ A @ R2 ) ) ) ).
% wfI
thf(fact_7147_Ex__inj__on__UNION__Sigma,axiom,
! [A: $tType,B: $tType,A4: B > ( set @ A ),I5: set @ B] :
? [F6: A > ( product_prod @ B @ A )] :
( ( inj_on @ A @ ( product_prod @ B @ A ) @ F6 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I5 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ A @ ( product_prod @ B @ A ) @ F6 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I5 ) ) ) @ ( product_Sigma @ B @ A @ I5 @ A4 ) ) ) ).
% Ex_inj_on_UNION_Sigma
thf(fact_7148_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) )
| ( ( finite_finite2 @ A @ A4 )
& ( finite_finite2 @ B @ B2 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_7149_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( finite_finite2 @ B @ B2 ) ) ) ).
% finite_cartesian_productD2
thf(fact_7150_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% finite_cartesian_productD1
thf(fact_7151_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ( B2 @ X2 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ! [A7: A] :
( ( member @ A @ A7 @ A4 )
=> ( finite_finite2 @ B @ ( B2 @ A7 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ B2 ) ) ) ) ).
% finite_SigmaI2
thf(fact_7152_chain__subset__def,axiom,
! [A: $tType] :
( ( chain_subset @ A )
= ( ^ [C4: set @ ( set @ A )] :
! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C4 )
=> ! [Y3: set @ A] :
( ( member @ ( set @ A ) @ Y3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X2 @ Y3 )
| ( ord_less_eq @ ( set @ A ) @ Y3 @ X2 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_7153_Image__subset,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),A4: set @ A,B2: set @ B,C2: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R2 @ C2 ) @ B2 ) ) ).
% Image_subset
thf(fact_7154_trancl__subset__Field2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 )
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R2 )
@ ^ [Uu3: A] : ( field2 @ A @ R2 ) ) ) ).
% trancl_subset_Field2
thf(fact_7155_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 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R5 ) ) ) ) ) ).
% refl_on_def
thf(fact_7156_refl__onI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R2 ) )
=> ( refl_on @ A @ A4 @ R2 ) ) ) ).
% refl_onI
thf(fact_7157_finite__equiv__class,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
=> ( ( member @ ( set @ A ) @ X4 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ( finite_finite2 @ A @ X4 ) ) ) ) ).
% finite_equiv_class
thf(fact_7158_open__prod__intro,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S: set @ ( product_prod @ A @ B )] :
( ! [X3: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X3 @ S )
=> ? [A21: set @ A,B9: set @ B] :
( ( topolo1002775350975398744n_open @ A @ A21 )
& ( topolo1002775350975398744n_open @ B @ B9 )
& ( member @ ( product_prod @ A @ B ) @ X3
@ ( product_Sigma @ A @ B @ A21
@ ^ [Uu3: A] : B9 ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A21
@ ^ [Uu3: A] : B9 )
@ S ) ) )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ B ) @ S ) ) ) ).
% open_prod_intro
thf(fact_7159_open__prod__elim,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S: set @ ( product_prod @ A @ B ),X: product_prod @ A @ B] :
( ( topolo1002775350975398744n_open @ ( product_prod @ A @ B ) @ S )
=> ( ( member @ ( product_prod @ A @ B ) @ X @ S )
=> ~ ! [A9: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A9 )
=> ! [B4: set @ B] :
( ( topolo1002775350975398744n_open @ B @ B4 )
=> ( ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A9
@ ^ [Uu3: A] : B4 ) )
=> ~ ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A9
@ ^ [Uu3: A] : B4 )
@ S ) ) ) ) ) ) ) ).
% open_prod_elim
thf(fact_7160_open__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ( ( topolo1002775350975398744n_open @ ( product_prod @ A @ B ) )
= ( ^ [S7: set @ ( product_prod @ A @ B )] :
! [X2: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X2 @ S7 )
=> ? [A6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A6 )
& ? [B6: set @ B] :
( ( topolo1002775350975398744n_open @ B @ B6 )
& ( member @ ( product_prod @ A @ B ) @ X2
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 )
@ S7 ) ) ) ) ) ) ) ).
% open_prod_def
thf(fact_7161_principal__prod__principal,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ A4 ) @ ( principal @ B @ B2 ) )
= ( principal @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) ) ) ).
% principal_prod_principal
thf(fact_7162_Sigma__Image,axiom,
! [A: $tType,B: $tType,A4: set @ B,B2: B > ( set @ A ),X4: set @ B] :
( ( image @ B @ A @ ( product_Sigma @ B @ A @ A4 @ B2 ) @ X4 )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ ( inf_inf @ ( set @ B ) @ X4 @ A4 ) ) ) ) ).
% Sigma_Image
thf(fact_7163_Refl__Field__Restr2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) ) )
= A4 ) ) ) ).
% Refl_Field_Restr2
thf(fact_7164_card__cartesian__product__singleton,axiom,
! [A: $tType,B: $tType,X: A,A4: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu3: A] : A4 ) )
= ( finite_card @ B @ A4 ) ) ).
% card_cartesian_product_singleton
thf(fact_7165_finite__quotient,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
=> ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ R2 ) ) ) ) ).
% finite_quotient
thf(fact_7166_sum_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B2: B > ( set @ C ),G2: B > C > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( finite_finite2 @ C @ ( B2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G2 @ X2 ) @ ( B2 @ X2 ) )
@ A4 )
= ( groups7311177749621191930dd_sum @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G2 ) @ ( product_Sigma @ B @ C @ A4 @ B2 ) ) ) ) ) ) ).
% sum.Sigma
thf(fact_7167_prod_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B2: B > ( set @ C ),G2: B > C > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( finite_finite2 @ C @ ( B2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G2 @ X2 ) @ ( B2 @ X2 ) )
@ A4 )
= ( groups7121269368397514597t_prod @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G2 ) @ ( product_Sigma @ B @ C @ A4 @ B2 ) ) ) ) ) ) ).
% prod.Sigma
thf(fact_7168_relImage__relInvImage,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F3: B > A,A4: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F3 @ A4 )
@ ^ [Uu3: A] : ( image2 @ B @ A @ F3 @ A4 ) ) )
=> ( ( bNF_Gr4221423524335903396lImage @ B @ A @ ( bNF_Gr7122648621184425601vImage @ B @ A @ A4 @ R @ F3 ) @ F3 )
= R ) ) ).
% relImage_relInvImage
thf(fact_7169_pairs__le__eq__Sigma,axiom,
! [M: nat] :
( ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ 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_7170_product__fold,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X2: A,Z6: set @ ( product_prod @ A @ B )] :
( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y3: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) )
@ Z6
@ B2 )
@ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
@ A4 ) ) ) ) ).
% product_fold
thf(fact_7171_Gr__incl,axiom,
! [A: $tType,B: $tType,A4: set @ A,F3: A > B,B2: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A4 @ F3 )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
= ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ B2 ) ) ).
% Gr_incl
thf(fact_7172_init__seg__of__def,axiom,
! [A: $tType] :
( ( init_seg_of @ A )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ $o
@ ^ [R5: set @ ( product_prod @ A @ A ),S8: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ S8 )
& ! [A5: A,B5: A,C6: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ S8 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C6 ) @ R5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ R5 ) ) ) ) ) ) ).
% init_seg_of_def
thf(fact_7173_comp__fun__commute__Image__fold,axiom,
! [B: $tType,A: $tType,S: set @ A] :
( finite6289374366891150609ommute @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X2: A,Y3: B,A6: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X2 @ S ) @ ( insert2 @ B @ Y3 @ A6 ) @ A6 ) ) ) ).
% comp_fun_commute_Image_fold
thf(fact_7174_comp__fun__commute__relcomp__fold,axiom,
! [A: $tType,B: $tType,C: $tType,S: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S )
=> ( 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 ) ) )
@ ^ [X2: C,Y3: 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,A14: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y3 = W3 ) @ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X2 @ Z6 ) @ A14 ) @ A14 ) )
@ A6
@ S ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_7175_Image__fold,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( image @ A @ B @ R @ S )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X2: A,Y3: B,A6: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X2 @ S ) @ ( insert2 @ B @ Y3 @ A6 ) @ A6 ) )
@ ( bot_bot @ ( set @ B ) )
@ R ) ) ) ).
% Image_fold
thf(fact_7176_lists__length__Suc__eq,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= ( suc @ N ) ) ) )
= ( image2 @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs3: list @ A,N2: A] : ( cons @ A @ N2 @ Xs3 ) )
@ ( product_Sigma @ ( list @ A ) @ A
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) )
@ ^ [Uu3: list @ A] : A4 ) ) ) ).
% lists_length_Suc_eq
thf(fact_7177_Restr__natLeq,axiom,
! [N: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat
@ ( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) )
@ ^ [Uu3: nat] :
( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X2 @ Y3 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_7178_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) )
=> ( ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) )
=> ( Phi @ A3 @ B3 ) )
=> ( ( ( A3 = B3 )
=> ( Phi @ A3 @ B3 ) )
=> ( Phi @ A3 @ B3 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_7179_well__order__induct__imp,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ! [X3: A] :
( ! [Y5: A] :
( ( ( Y5 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X3 ) @ R2 ) )
=> ( ( member @ A @ Y5 @ ( field2 @ A @ R2 ) )
=> ( P @ Y5 ) ) )
=> ( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
=> ( P @ X3 ) ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( P @ A3 ) ) ) ) ).
% well_order_induct_imp
thf(fact_7180_wo__rel_Omax2__among,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B3 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_7181_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less_eq @ nat ) ) ) ) ).
% natLeq_def
thf(fact_7182_wo__rel_Ocases__Total,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 )
=> ( Phi @ A3 @ B3 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A3 ) @ R2 )
=> ( Phi @ A3 @ B3 ) )
=> ( Phi @ A3 @ B3 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_7183_natLeq__on__wo__rel,axiom,
! [N: nat] :
( bNF_Wellorder_wo_rel @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X2 @ Y3 ) ) ) ) ) ).
% natLeq_on_wo_rel
thf(fact_7184_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B3 ) ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B3 ) ) @ R2 )
& ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B3 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_7185_Restr__natLeq2,axiom,
! [N: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat @ ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
@ ^ [Uu3: nat] : ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X2 @ Y3 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_7186_wo__rel_OWell__order__isMinim__exists,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_1: A] : ( bNF_We4791949203932849705sMinim @ A @ R2 @ B2 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_7187_natLeq__underS__less,axiom,
! [N: nat] :
( ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
= ( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) ).
% natLeq_underS_less
thf(fact_7188_underS__Field2,axiom,
! [A: $tType,A3: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( field2 @ A @ R2 ) ) ) ).
% underS_Field2
thf(fact_7189_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [A: $tType,I: A,R: set @ ( product_prod @ A @ A ),J: A] :
( ( member @ A @ I @ ( order_underS @ A @ R @ J ) )
=> ( member @ A @ I @ ( field2 @ A @ R ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_7190_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_7191_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_7192_underS__empty,axiom,
! [A: $tType,A3: A,R2: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( order_underS @ A @ R2 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% underS_empty
thf(fact_7193_Order__Relation_OunderS__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] : ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( field2 @ A @ R2 ) ) ).
% Order_Relation.underS_Field
thf(fact_7194_underS__Field3,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( ( field2 @ A @ R2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( field2 @ A @ R2 ) ) ) ).
% underS_Field3
thf(fact_7195_underS__incl__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( order_underS @ A @ R2 @ B3 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_7196_wo__rel_Ominim__isMinim,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( bNF_We4791949203932849705sMinim @ A @ R2 @ B2 @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B2 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_7197_Refl__under__underS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( order_under @ A @ R2 @ A3 )
= ( sup_sup @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Refl_under_underS
thf(fact_7198_underS__subset__under,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] : ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( order_under @ A @ R2 @ A3 ) ) ).
% underS_subset_under
thf(fact_7199_under__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] : ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R2 @ A3 ) @ ( field2 @ A @ R2 ) ) ).
% under_Field
thf(fact_7200_wo__rel_Ominim__least,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ B2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B2 ) @ B3 ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_7201_wo__rel_Oequals__minim,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ A3 @ B2 )
=> ( ! [B7: A] :
( ( member @ A @ B7 @ B2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B7 ) @ R2 ) )
=> ( A3
= ( bNF_We6954850376910717587_minim @ A @ R2 @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_7202_wo__rel_Ominim__inField,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B2 ) @ ( field2 @ A @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_7203_wo__rel_Ominim__in,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B2 ) @ B2 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_7204_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S )
=> ( ( relcomp @ A @ B @ C @ R @ S )
= ( 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 ) ) )
@ ^ [X2: A,Y3: 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,A14: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y3 = W3 ) @ ( insert2 @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X2 @ Z6 ) @ A14 ) @ A14 ) )
@ A6
@ S ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R ) ) ) ) ).
% relcomp_fold
thf(fact_7205_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_7206_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_7207_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_7208_relcomp__distrib,axiom,
! [A: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B ),T4: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ R @ ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ S @ T4 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( relcomp @ A @ C @ B @ R @ S ) @ ( relcomp @ A @ C @ B @ R @ T4 ) ) ) ).
% relcomp_distrib
thf(fact_7209_relcomp__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,S: set @ ( product_prod @ A @ C ),T4: set @ ( product_prod @ A @ C ),R: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ C ) ) @ S @ T4 ) @ R )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( relcomp @ A @ C @ B @ S @ R ) @ ( relcomp @ A @ C @ B @ T4 @ R ) ) ) ).
% relcomp_distrib2
thf(fact_7210_R__O__Id,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( relcomp @ A @ B @ B @ R @ ( id2 @ B ) )
= R ) ).
% R_O_Id
thf(fact_7211_Id__O__R,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( relcomp @ A @ A @ B @ ( id2 @ A ) @ R )
= R ) ).
% Id_O_R
thf(fact_7212_range__snd,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_snd
thf(fact_7213_snd__image__times,axiom,
! [B: $tType,A: $tType,A4: set @ B,B2: set @ A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu3: B] : B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu3: B] : B2 ) )
= B2 ) ) ) ).
% snd_image_times
thf(fact_7214_relcomp__subset__Sigma,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),A4: set @ A,B2: set @ B,S3: set @ ( product_prod @ B @ C ),C2: set @ C] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S3
@ ( product_Sigma @ B @ C @ B2
@ ^ [Uu3: B] : C2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R2 @ S3 )
@ ( product_Sigma @ A @ C @ A4
@ ^ [Uu3: A] : C2 ) ) ) ) ).
% relcomp_subset_Sigma
thf(fact_7215_relcomp__UNION__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S3: set @ ( product_prod @ A @ C ),R2: D > ( set @ ( product_prod @ C @ B ) ),I5: set @ D] :
( ( relcomp @ A @ C @ B @ S3 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ C @ B ) ) @ ( image2 @ D @ ( set @ ( product_prod @ C @ B ) ) @ R2 @ I5 ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I4: D] : ( relcomp @ A @ C @ B @ S3 @ ( R2 @ I4 ) )
@ I5 ) ) ) ).
% relcomp_UNION_distrib
thf(fact_7216_relcomp__UNION__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R2: D > ( set @ ( product_prod @ A @ C ) ),I5: set @ D,S3: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ C ) ) @ ( image2 @ D @ ( set @ ( product_prod @ A @ C ) ) @ R2 @ I5 ) ) @ S3 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I4: D] : ( relcomp @ A @ C @ B @ ( R2 @ I4 ) @ S3 )
@ I5 ) ) ) ).
% relcomp_UNION_distrib2
thf(fact_7217_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_7218_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A13: A,A24: C,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A13 @ A24 ) @ ( relcomp @ A @ B @ C @ R2 @ S3 ) )
=> ~ ! [B7: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A13 @ B7 ) @ R2 )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B7 @ A24 ) @ S3 ) ) ) ).
% relcomp.cases
thf(fact_7219_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A13: A,A24: C,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A13 @ A24 ) @ ( relcomp @ A @ B @ C @ R2 @ S3 ) )
= ( ? [A5: A,B5: B,C6: C] :
( ( A13 = A5 )
& ( A24 = C6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R2 )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ C6 ) @ S3 ) ) ) ) ).
% relcomp.simps
thf(fact_7220_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A3: A,B3: B,R2: set @ ( product_prod @ A @ B ),C3: C,S3: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B3 @ C3 ) @ S3 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ C3 ) @ ( relcomp @ A @ B @ C @ R2 @ S3 ) ) ) ) ).
% relcomp.relcompI
thf(fact_7221_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R2: set @ ( product_prod @ A @ C ),S3: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R2 @ S3 ) )
=> ~ ! [X3: A,Y2: C,Z3: B] :
( ( Xz
= ( product_Pair @ A @ B @ X3 @ Z3 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y2 ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y2 @ Z3 ) @ S3 ) ) ) ) ).
% relcompE
thf(fact_7222_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A3: A,C3: B,R2: set @ ( product_prod @ A @ C ),S3: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ C3 ) @ ( relcomp @ A @ C @ B @ R2 @ S3 ) )
=> ~ ! [B7: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ B7 ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B7 @ C3 ) @ S3 ) ) ) ).
% relcompEpair
thf(fact_7223_O__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ A @ D ),S: set @ ( product_prod @ D @ C ),T4: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( relcomp @ A @ D @ C @ R @ S ) @ T4 )
= ( relcomp @ A @ D @ B @ R @ ( relcomp @ D @ C @ B @ S @ T4 ) ) ) ).
% O_assoc
thf(fact_7224_relcomp__Image,axiom,
! [A: $tType,C: $tType,B: $tType,X4: set @ ( product_prod @ B @ C ),Y6: set @ ( product_prod @ C @ A ),Z7: set @ B] :
( ( image @ B @ A @ ( relcomp @ B @ C @ A @ X4 @ Y6 ) @ Z7 )
= ( image @ C @ A @ Y6 @ ( image @ B @ C @ X4 @ Z7 ) ) ) ).
% relcomp_Image
thf(fact_7225_min__ext__compat,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S ) @ 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 @ S ) @ ( insert2 @ ( 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_7226_max__ext__compat,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S ) @ 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 @ S ) @ ( insert2 @ ( 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_7227_relcomp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R4: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ B @ C ),S3: set @ ( product_prod @ B @ C )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S5 @ S3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R4 @ S5 ) @ ( relcomp @ A @ B @ C @ R2 @ S3 ) ) ) ) ).
% relcomp_mono
thf(fact_7228_wf__relcomp__compatible,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S ) @ ( relcomp @ A @ A @ A @ S @ R ) )
=> ( wf @ A @ ( relcomp @ A @ A @ A @ S @ R ) ) ) ) ).
% wf_relcomp_compatible
thf(fact_7229_wf__union__compatible,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R )
=> ( ( wf @ A @ S )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S ) @ R )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) ) ) ) ).
% wf_union_compatible
thf(fact_7230_trancl__Int__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 ) @ S3 ) @ R2 ) @ S3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 ) @ S3 ) ) ) ).
% trancl_Int_subset
thf(fact_7231_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
@ ^ [X2: A,Z6: B] :
? [Y3: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X2 @ Y3 ) @ R5 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y3 @ Z6 ) @ S8 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_7232_qc__wf__relto__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S ) @ ( relcomp @ A @ A @ A @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) @ R ) )
=> ( ( wf @ A @ ( relcomp @ A @ A @ A @ ( transitive_rtrancl @ A @ S ) @ ( relcomp @ A @ A @ A @ R @ ( transitive_rtrancl @ A @ S ) ) ) )
= ( wf @ A @ R ) ) ) ).
% qc_wf_relto_iff
thf(fact_7233_rtrancl__Int__subset,axiom,
! [A: $tType,S3: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ S3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R2 ) @ S3 ) @ R2 ) @ S3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R2 ) @ S3 ) ) ) ).
% rtrancl_Int_subset
thf(fact_7234_subset__snd__imageI,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,S: set @ ( product_prod @ A @ B ),X: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 )
@ S )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ B2 @ ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S ) ) ) ) ).
% subset_snd_imageI
thf(fact_7235_reduction__pairI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S ) @ R )
=> ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) ) ) ).
% reduction_pairI
thf(fact_7236_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,X4: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S ) @ X4 )
= ( 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,A14: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z6 ) @ A14 )
@ A14 ) )
@ X4
@ S ) ) ) ).
% insert_relcomp_union_fold
thf(fact_7237_range__fst,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_fst
thf(fact_7238_fst__image__times,axiom,
! [B: $tType,A: $tType,B2: set @ B,A4: set @ A] :
( ( ( B2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_7239_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_7240_Id__fstsnd__eq,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [X2: product_prod @ A @ A] :
( ( product_fst @ A @ A @ X2 )
= ( product_snd @ A @ A @ X2 ) ) ) ) ).
% Id_fstsnd_eq
thf(fact_7241_reduction__pair__def,axiom,
! [A: $tType] :
( ( fun_reduction_pair @ A )
= ( ^ [P3: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )] :
( ( wf @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P3 ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P3 ) @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P3 ) ) @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P3 ) ) ) ) ) ).
% reduction_pair_def
thf(fact_7242_reduction__pair__lemma,axiom,
! [A: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( fun_reduction_pair @ A @ P )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P ) )
=> ( ( wf @ A @ S )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) ) ) ) ) ).
% reduction_pair_lemma
thf(fact_7243_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_7244_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: A > ( set @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ B2 ) )
= ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ( B2 @ X2 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_7245_sorted__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) ) ).
% sorted_enumerate
thf(fact_7246_rel__fun__Collect__case__prodD,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,A4: A > B > $o,B2: C > D > $o,F3: A > C,G2: B > D,X4: set @ ( product_prod @ A @ B ),X: product_prod @ A @ B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 @ F3 @ G2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A4 ) ) )
=> ( ( member @ ( product_prod @ A @ B ) @ X @ X4 )
=> ( B2 @ ( comp @ A @ C @ ( product_prod @ A @ B ) @ F3 @ ( product_fst @ A @ B ) @ X ) @ ( comp @ B @ D @ ( product_prod @ A @ B ) @ G2 @ ( product_snd @ A @ B ) @ X ) ) ) ) ) ).
% rel_fun_Collect_case_prodD
thf(fact_7247_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X4: set @ A,A4: set @ ( product_prod @ A @ B ),Y6: set @ B,P: A > B > $o,Q: A > B > $o] :
( ( X4
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 ) )
=> ( ( Y6
= ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A4 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ! [Xa2: B] :
( ( member @ B @ Xa2 @ Y6 )
=> ( ( P @ X3 @ Xa2 )
=> ( Q @ X3 @ Xa2 ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_7248_subset__fst__imageI,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,S: set @ ( product_prod @ A @ B ),Y: B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 )
@ S )
=> ( ( member @ B @ Y @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) ) ) ) ).
% subset_fst_imageI
thf(fact_7249_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y )
=> ( ( bezw @ X @ Y )
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_7250_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4
@ ( product_Sigma @ A @ B @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 )
@ ^ [Uu3: A] : ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A4 ) ) ) ).
% subset_fst_snd
thf(fact_7251_eventually__prodI,axiom,
! [A: $tType,B: $tType,P: A > $o,F4: filter @ A,Q: B > $o,G3: filter @ B] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ B @ Q @ G3 )
=> ( eventually @ ( product_prod @ A @ B )
@ ^ [X2: product_prod @ A @ B] :
( ( P @ ( product_fst @ A @ B @ X2 ) )
& ( Q @ ( product_snd @ A @ B @ X2 ) ) )
@ ( prod_filter @ A @ B @ F4 @ G3 ) ) ) ) ).
% eventually_prodI
thf(fact_7252_bezw_Osimps,axiom,
( bezw
= ( ^ [X2: nat,Y3: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y3
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X2 @ Y3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X2 @ Y3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X2 @ Y3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X2 @ Y3 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_7253_bezw_Oelims,axiom,
! [X: nat,Xa3: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa3 )
= Y )
=> ( ( ( Xa3
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa3 ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_7254_predicate2__transferD,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > B > $o,R22: C > D > $o,P: A > C > $o,Q: B > D > $o,A3: product_prod @ A @ B,A4: set @ ( product_prod @ A @ B ),B3: product_prod @ C @ D,B2: set @ ( product_prod @ C @ D )] :
( ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ R1
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ R22
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ P
@ Q )
=> ( ( member @ ( product_prod @ A @ B ) @ A3 @ A4 )
=> ( ( member @ ( product_prod @ C @ D ) @ B3 @ B2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R1 ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ C @ D ) ) @ B2 @ ( collect @ ( product_prod @ C @ D ) @ ( product_case_prod @ C @ D @ $o @ R22 ) ) )
=> ( ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_fst @ C @ D @ B3 ) )
= ( Q @ ( product_snd @ A @ B @ A3 ) @ ( product_snd @ C @ D @ B3 ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_7255_nths__shift__lemma,axiom,
! [A: $tType,A4: set @ nat,Xs: list @ A,I: nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P5: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P5 ) @ A4 )
@ ( zip @ A @ nat @ Xs @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P5: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P5 ) @ I ) @ A4 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_7256_nths__def,axiom,
! [A: $tType] :
( ( nths @ A )
= ( ^ [Xs3: list @ A,A6: set @ nat] :
( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P5: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P5 ) @ A6 )
@ ( zip @ A @ nat @ Xs3 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ) ).
% nths_def
thf(fact_7257_in__set__zip,axiom,
! [B: $tType,A: $tType,P6: product_prod @ A @ B,Xs: list @ A,Ys2: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ P6 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
= ( ? [N2: nat] :
( ( ( nth @ A @ Xs @ N2 )
= ( product_fst @ A @ B @ P6 ) )
& ( ( nth @ B @ Ys2 @ N2 )
= ( product_snd @ A @ B @ P6 ) )
& ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ord_less @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ).
% in_set_zip
thf(fact_7258_fun_Oin__rel,axiom,
! [B: $tType,A: $tType,D: $tType,R: A > B > $o,A3: D > A,B3: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y4: D,Z2: D] : Y4 = Z2
@ R
@ A3
@ B3 )
= ( ? [Z6: D > ( product_prod @ A @ B )] :
( ( member @ ( D > ( product_prod @ A @ B ) ) @ Z6
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X2: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X2 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) ) ) )
& ( ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) @ Z6 )
= A3 )
& ( ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) @ Z6 )
= B3 ) ) ) ) ).
% fun.in_rel
thf(fact_7259_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,R: set @ ( product_prod @ C @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S )
=> ( ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ R ) @ S )
= ( 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,A14: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z6 ) @ A14 )
@ A14 ) )
@ ( relcomp @ C @ A @ B @ R @ S )
@ S ) ) ) ).
% insert_relcomp_fold
thf(fact_7260_in__set__enumerate__eq,axiom,
! [A: $tType,P6: product_prod @ nat @ A,N: nat,Xs: list @ A] :
( ( member @ ( product_prod @ nat @ A ) @ P6 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) )
= ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P6 ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P6 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
& ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P6 ) @ N ) )
= ( product_snd @ nat @ A @ P6 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_7261_bezw_Opelims,axiom,
! [X: nat,Xa3: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa3 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) )
=> ~ ( ( ( ( Xa3
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa3 ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) ) ) ) ) ).
% bezw.pelims
thf(fact_7262_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F2: A > nat,G: B > nat,P5: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F2 @ ( product_fst @ A @ B @ P5 ) ) @ ( G @ ( product_snd @ A @ B @ P5 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_7263_Rat_Opositive_Orep__eq,axiom,
( positive
= ( ^ [X2: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ ( rep_Rat @ X2 ) ) @ ( product_snd @ int @ int @ ( rep_Rat @ X2 ) ) ) ) ) ) ).
% Rat.positive.rep_eq
thf(fact_7264_normalize__def,axiom,
( normalize
= ( ^ [P5: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P5 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P5 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P5 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) )
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_snd @ int @ int @ P5 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P5 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P5 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_7265_gcd__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B3: A] :
( ( ( gcd_gcd @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% gcd_eq_0_iff
thf(fact_7266_gcd__pos__int,axiom,
! [M: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ M @ N ) )
= ( ( M
!= ( zero_zero @ int ) )
| ( N
!= ( zero_zero @ int ) ) ) ) ).
% gcd_pos_int
thf(fact_7267_Gcd__insert,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,A4: set @ A] :
( ( gcd_Gcd @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( gcd_gcd @ A @ A3 @ ( gcd_Gcd @ A @ A4 ) ) ) ) ).
% Gcd_insert
thf(fact_7268_Gcd__fin_Oinsert,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A4: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ A4 ) ) ) ) ).
% Gcd_fin.insert
thf(fact_7269_Gcd__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,B3: A] :
( ( gcd_Gcd @ A @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_gcd @ A @ A3 @ B3 ) ) ) ).
% Gcd_2
thf(fact_7270_Gcd__fin_Osubset,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( gcd_gcd @ A @ ( semiring_gcd_Gcd_fin @ A @ B2 ) @ ( semiring_gcd_Gcd_fin @ A @ A4 ) )
= ( semiring_gcd_Gcd_fin @ A @ A4 ) ) ) ) ).
% Gcd_fin.subset
thf(fact_7271_gcd__ge__0__int,axiom,
! [X: int,Y: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ X @ Y ) ) ).
% gcd_ge_0_int
thf(fact_7272_Gcd__fin_Ounion,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( gcd_gcd @ A @ ( semiring_gcd_Gcd_fin @ A @ A4 ) @ ( semiring_gcd_Gcd_fin @ A @ B2 ) ) ) ) ).
% Gcd_fin.union
thf(fact_7273_gcd__le1__int,axiom,
! [A3: int,B3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ A3 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A3 @ B3 ) @ A3 ) ) ).
% gcd_le1_int
thf(fact_7274_gcd__le2__int,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A3 @ B3 ) @ B3 ) ) ).
% gcd_le2_int
thf(fact_7275_gcd__cases__int,axiom,
! [X: int,Y: int,P: int > $o] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P @ ( gcd_gcd @ int @ X @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( P @ ( gcd_gcd @ int @ X @ Y ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_7276_gcd__unique__int,axiom,
! [D2: int,A3: int,B3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
& ( dvd_dvd @ int @ D2 @ A3 )
& ( dvd_dvd @ int @ D2 @ B3 )
& ! [E3: int] :
( ( ( dvd_dvd @ int @ E3 @ A3 )
& ( dvd_dvd @ int @ E3 @ B3 ) )
=> ( dvd_dvd @ int @ E3 @ D2 ) ) )
= ( D2
= ( gcd_gcd @ int @ A3 @ B3 ) ) ) ).
% gcd_unique_int
thf(fact_7277_gcd__non__0__int,axiom,
! [Y: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Y )
=> ( ( gcd_gcd @ int @ X @ Y )
= ( gcd_gcd @ int @ Y @ ( modulo_modulo @ int @ X @ Y ) ) ) ) ).
% gcd_non_0_int
thf(fact_7278_Gcd__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A4: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Gcd_fin.insert_remove
thf(fact_7279_Gcd__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Gcd_fin.remove
thf(fact_7280_Gcd__set__eq__fold,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [Xs: list @ A] :
( ( gcd_Gcd @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ).
% Gcd_set_eq_fold
thf(fact_7281_Gcd__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ).
% Gcd_fin.set_eq_fold
thf(fact_7282_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( ^ [A6: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A6 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A6 ) @ ( one_one @ A ) ) ) ) ) ).
% Gcd_fin.eq_fold
thf(fact_7283_Rat_Opositive_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ $o @ $o @ pcr_rat
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) )
@ positive ) ).
% Rat.positive.transfer
thf(fact_7284_quotient__of__def,axiom,
( quotient_of
= ( ^ [X2: rat] :
( the @ ( product_prod @ int @ int )
@ ^ [Pair: product_prod @ int @ int] :
( ( X2
= ( fract @ ( product_fst @ int @ int @ Pair ) @ ( product_snd @ int @ int @ Pair ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ Pair ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ Pair ) @ ( product_snd @ int @ int @ Pair ) ) ) ) ) ) ).
% quotient_of_def
thf(fact_7285_gcd__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( gcd_gcd @ nat @ A3 @ B3 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B3
= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.eq_neutr_iff
thf(fact_7286_gcd__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ A3 )
= A3 ) ).
% gcd_nat.left_neutral
thf(fact_7287_gcd__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( zero_zero @ nat )
= ( gcd_gcd @ nat @ A3 @ B3 ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B3
= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.neutr_eq_iff
thf(fact_7288_gcd__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( gcd_gcd @ nat @ A3 @ ( zero_zero @ nat ) )
= A3 ) ).
% gcd_nat.right_neutral
thf(fact_7289_gcd__0__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ X @ ( zero_zero @ nat ) )
= X ) ).
% gcd_0_nat
thf(fact_7290_gcd__0__left__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ X )
= X ) ).
% gcd_0_left_nat
thf(fact_7291_gcd__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% gcd_Suc_0
thf(fact_7292_gcd__pos__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_gcd @ nat @ M @ N ) )
= ( ( M
!= ( zero_zero @ nat ) )
| ( N
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_pos_nat
thf(fact_7293_coprime__mod__right__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( algebr8660921524188924756oprime @ A @ A3 @ ( modulo_modulo @ A @ B3 @ A3 ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B3 ) ) ) ) ).
% coprime_mod_right_iff
thf(fact_7294_coprime__mod__left__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( algebr8660921524188924756oprime @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ B3 )
= ( algebr8660921524188924756oprime @ A @ A3 @ B3 ) ) ) ) ).
% coprime_mod_left_iff
thf(fact_7295_coprime__power__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,N: nat,B3: A] :
( ( algebr8660921524188924756oprime @ A @ ( power_power @ A @ A3 @ N ) @ B3 )
= ( ( algebr8660921524188924756oprime @ A @ A3 @ B3 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% coprime_power_left_iff
thf(fact_7296_coprime__power__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( power_power @ A @ B3 @ N ) )
= ( ( algebr8660921524188924756oprime @ A @ A3 @ B3 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% coprime_power_right_iff
thf(fact_7297_coprime__0__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ ( zero_zero @ A ) @ A3 )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_0_left_iff
thf(fact_7298_coprime__0__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( zero_zero @ A ) )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_0_right_iff
thf(fact_7299_normalize__stable,axiom,
! [Q5: int,P6: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Q5 )
=> ( ( algebr8660921524188924756oprime @ int @ P6 @ Q5 )
=> ( ( normalize @ ( product_Pair @ int @ int @ P6 @ Q5 ) )
= ( product_Pair @ int @ int @ P6 @ Q5 ) ) ) ) ).
% normalize_stable
thf(fact_7300_gcd__diff2__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ N @ M ) @ N )
= ( gcd_gcd @ nat @ M @ N ) ) ) ).
% gcd_diff2_nat
thf(fact_7301_gcd__diff1__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ M @ N ) @ N )
= ( gcd_gcd @ nat @ M @ N ) ) ) ).
% gcd_diff1_nat
thf(fact_7302_gcd__le1__nat,axiom,
! [A3: nat,B3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A3 @ B3 ) @ A3 ) ) ).
% gcd_le1_nat
thf(fact_7303_gcd__le2__nat,axiom,
! [B3: nat,A3: nat] :
( ( B3
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A3 @ B3 ) @ B3 ) ) ).
% gcd_le2_nat
thf(fact_7304_gcd__non__0__nat,axiom,
! [Y: nat,X: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ( ( gcd_gcd @ nat @ X @ Y )
= ( gcd_gcd @ nat @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) ) ).
% gcd_non_0_nat
thf(fact_7305_gcd__nat_Osimps,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X2: nat,Y3: nat] :
( if @ nat
@ ( Y3
= ( zero_zero @ nat ) )
@ X2
@ ( gcd_gcd @ nat @ Y3 @ ( modulo_modulo @ nat @ X2 @ Y3 ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_7306_gcd__nat_Oelims,axiom,
! [X: nat,Xa3: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa3 )
= Y )
=> ( ( ( Xa3
= ( zero_zero @ nat ) )
=> ( Y = X ) )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_7307_div__gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B3: A] :
( ( ( A3
!= ( zero_zero @ A ) )
| ( B3
!= ( zero_zero @ A ) ) )
=> ( algebr8660921524188924756oprime @ A @ ( divide_divide @ A @ A3 @ ( gcd_gcd @ A @ A3 @ B3 ) ) @ ( divide_divide @ A @ B3 @ ( gcd_gcd @ A @ A3 @ B3 ) ) ) ) ) ).
% div_gcd_coprime
thf(fact_7308_gcd__coprime__exists,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B3: A] :
( ( ( gcd_gcd @ A @ A3 @ B3 )
!= ( zero_zero @ A ) )
=> ? [A27: A,B15: A] :
( ( A3
= ( times_times @ A @ A27 @ ( gcd_gcd @ A @ A3 @ B3 ) ) )
& ( B3
= ( times_times @ A @ B15 @ ( gcd_gcd @ A @ A3 @ B3 ) ) )
& ( algebr8660921524188924756oprime @ A @ A27 @ B15 ) ) ) ) ).
% gcd_coprime_exists
thf(fact_7309_gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B3: A,A11: A,B10: A] :
( ( ( gcd_gcd @ A @ A3 @ B3 )
!= ( zero_zero @ A ) )
=> ( ( A3
= ( times_times @ A @ A11 @ ( gcd_gcd @ A @ A3 @ B3 ) ) )
=> ( ( B3
= ( times_times @ A @ B10 @ ( gcd_gcd @ A @ A3 @ B3 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A11 @ B10 ) ) ) ) ) ).
% gcd_coprime
thf(fact_7310_bezout__nat,axiom,
! [A3: nat,B3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [X3: nat,Y2: nat] :
( ( times_times @ nat @ A3 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B3 @ Y2 ) @ ( gcd_gcd @ nat @ A3 @ B3 ) ) ) ) ).
% bezout_nat
thf(fact_7311_Gcd__in,axiom,
! [A4: set @ nat] :
( ! [A7: nat,B7: nat] :
( ( member @ nat @ A7 @ A4 )
=> ( ( member @ nat @ B7 @ A4 )
=> ( member @ nat @ ( gcd_gcd @ nat @ A7 @ B7 ) @ A4 ) ) )
=> ( ( A4
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( gcd_Gcd @ nat @ A4 ) @ A4 ) ) ) ).
% Gcd_in
thf(fact_7312_Rat__cases,axiom,
! [Q5: rat] :
~ ! [A7: int,B7: int] :
( ( Q5
= ( fract @ A7 @ B7 ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ~ ( algebr8660921524188924756oprime @ int @ A7 @ B7 ) ) ) ).
% Rat_cases
thf(fact_7313_Rat__induct,axiom,
! [P: rat > $o,Q5: rat] :
( ! [A7: int,B7: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ( ( algebr8660921524188924756oprime @ int @ A7 @ B7 )
=> ( P @ ( fract @ A7 @ B7 ) ) ) )
=> ( P @ Q5 ) ) ).
% Rat_induct
thf(fact_7314_bezout__gcd__nat_H,axiom,
! [B3: nat,A3: nat] :
? [X3: nat,Y2: nat] :
( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B3 @ Y2 ) @ ( times_times @ nat @ A3 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X3 ) @ ( times_times @ nat @ B3 @ Y2 ) )
= ( gcd_gcd @ nat @ A3 @ B3 ) ) )
| ( ( ord_less_eq @ nat @ ( times_times @ nat @ A3 @ Y2 ) @ ( times_times @ nat @ B3 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ B3 @ X3 ) @ ( times_times @ nat @ A3 @ Y2 ) )
= ( gcd_gcd @ nat @ A3 @ B3 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_7315_Gcd__nat__set__eq__fold,axiom,
! [Xs: list @ nat] :
( ( gcd_Gcd @ nat @ ( set2 @ nat @ Xs ) )
= ( fold @ nat @ nat @ ( gcd_gcd @ nat ) @ Xs @ ( zero_zero @ nat ) ) ) ).
% Gcd_nat_set_eq_fold
thf(fact_7316_Rat__cases__nonzero,axiom,
! [Q5: rat] :
( ! [A7: int,B7: int] :
( ( Q5
= ( fract @ A7 @ B7 ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ( ( A7
!= ( zero_zero @ int ) )
=> ~ ( algebr8660921524188924756oprime @ int @ A7 @ B7 ) ) ) )
=> ( Q5
= ( zero_zero @ rat ) ) ) ).
% Rat_cases_nonzero
thf(fact_7317_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat )
@ ^ [M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_7318_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( gcd_gcd @ nat @ M @ N )
= ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D5: nat] :
( ( dvd_dvd @ nat @ D5 @ M )
& ( dvd_dvd @ nat @ D5 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_7319_Rats__cases_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( field_char_0_Rats @ A ) )
=> ~ ! [A7: int,B7: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ( ( algebr8660921524188924756oprime @ int @ A7 @ B7 )
=> ( X
!= ( divide_divide @ A @ ( ring_1_of_int @ A @ A7 ) @ ( ring_1_of_int @ A @ B7 ) ) ) ) ) ) ) ).
% Rats_cases'
thf(fact_7320_quotient__of__unique,axiom,
! [R2: rat] :
? [X3: product_prod @ int @ int] :
( ( R2
= ( fract @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ X3 ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) )
& ! [Y5: product_prod @ int @ int] :
( ( ( R2
= ( fract @ ( product_fst @ int @ int @ Y5 ) @ ( product_snd @ int @ int @ Y5 ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ Y5 ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ Y5 ) @ ( product_snd @ int @ int @ Y5 ) ) )
=> ( Y5 = X3 ) ) ) ).
% quotient_of_unique
thf(fact_7321_gcd__nat_Opelims,axiom,
! [X: nat,Xa3: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa3 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) )
=> ~ ( ( ( ( Xa3
= ( zero_zero @ nat ) )
=> ( Y = X ) )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_7322_list_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R6: A > B > $o,A5: list @ A,B5: list @ B] :
? [Z6: list @ ( product_prod @ A @ B )] :
( ( member @ ( list @ ( product_prod @ A @ B ) ) @ Z6
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X2: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z6 )
= A5 )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z6 )
= B5 ) ) ) ) ).
% list.in_rel
thf(fact_7323_coprime__Suc__0__left,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ).
% coprime_Suc_0_left
thf(fact_7324_coprime__Suc__0__right,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ).
% coprime_Suc_0_right
thf(fact_7325_list_Orel__mono,axiom,
! [B: $tType,A: $tType,R: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra )
=> ( ord_less_eq @ ( ( list @ A ) > ( list @ B ) > $o ) @ ( list_all2 @ A @ B @ R ) @ ( list_all2 @ A @ B @ Ra ) ) ) ).
% list.rel_mono
thf(fact_7326_list__all2__conv__all__nth,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P3: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P3 @ ( nth @ A @ Xs3 @ I4 ) @ ( nth @ B @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_7327_list__all2__all__nthI,axiom,
! [A: $tType,B: $tType,A3: list @ A,B3: list @ B,P: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B3 ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ A3 ) )
=> ( P @ ( nth @ A @ A3 @ N3 ) @ ( nth @ B @ B3 @ N3 ) ) )
=> ( list_all2 @ A @ B @ P @ A3 @ B3 ) ) ) ).
% list_all2_all_nthI
thf(fact_7328_list__all2__nthD2,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys2: list @ B,P6: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys2 )
=> ( ( ord_less @ nat @ P6 @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( P @ ( nth @ A @ Xs @ P6 ) @ ( nth @ B @ Ys2 @ P6 ) ) ) ) ).
% list_all2_nthD2
thf(fact_7329_list__all2__nthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys2: list @ B,P6: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys2 )
=> ( ( ord_less @ nat @ P6 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ P6 ) @ ( nth @ B @ Ys2 @ P6 ) ) ) ) ).
% list_all2_nthD
thf(fact_7330_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% coprime_diff_one_right_nat
thf(fact_7331_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ N ) ) ).
% coprime_diff_one_left_nat
thf(fact_7332_sum__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( monoid_add @ A ) )
=> ! [A4: A > B > $o] :
( ( A4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A4 ) @ A4 @ ( groups8242544230860333062m_list @ A ) @ ( groups8242544230860333062m_list @ B ) ) ) ) ) ).
% sum_list_transfer
thf(fact_7333_Rats__abs__nat__div__natE,axiom,
! [X: real] :
( ( member @ real @ X @ ( field_char_0_Rats @ real ) )
=> ~ ! [M4: nat,N3: nat] :
( ( N3
!= ( zero_zero @ nat ) )
=> ( ( ( abs_abs @ real @ X )
= ( divide_divide @ real @ ( semiring_1_of_nat @ real @ M4 ) @ ( semiring_1_of_nat @ real @ N3 ) ) )
=> ~ ( algebr8660921524188924756oprime @ nat @ M4 @ N3 ) ) ) ) ).
% Rats_abs_nat_div_natE
thf(fact_7334_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A4: A > B > $o,B2: C > D > $o] :
( ( A4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( A > ( list @ C ) > A ) @ ( B > ( list @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ A @ B @ B2 @ A4 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A4 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B2 ) @ A4 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_7335_relImage__Gr,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
=> ( ( bNF_Gr4221423524335903396lImage @ A @ B @ R @ F3 )
= ( relcomp @ B @ A @ B @ ( converse @ A @ B @ ( bNF_Gr @ A @ B @ A4 @ F3 ) ) @ ( relcomp @ A @ A @ B @ R @ ( bNF_Gr @ A @ B @ A4 @ F3 ) ) ) ) ) ).
% relImage_Gr
thf(fact_7336_relInvImage__Gr,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),B2: set @ A,A4: set @ B,F3: B > A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
=> ( ( bNF_Gr7122648621184425601vImage @ B @ A @ A4 @ R @ F3 )
= ( relcomp @ B @ A @ B @ ( bNF_Gr @ B @ A @ A4 @ F3 ) @ ( relcomp @ A @ A @ B @ R @ ( converse @ B @ A @ ( bNF_Gr @ B @ A @ A4 @ F3 ) ) ) ) ) ) ).
% relInvImage_Gr
thf(fact_7337_converse__inject,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S3: set @ ( product_prod @ B @ A )] :
( ( ( converse @ B @ A @ R2 )
= ( converse @ B @ A @ S3 ) )
= ( R2 = S3 ) ) ).
% converse_inject
thf(fact_7338_converse__converse,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( converse @ B @ A @ ( converse @ A @ B @ R2 ) )
= R2 ) ).
% converse_converse
thf(fact_7339_converse__iff,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,R2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( converse @ B @ A @ R2 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A3 ) @ R2 ) ) ).
% converse_iff
thf(fact_7340_Field__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( converse @ A @ A @ R2 ) )
= ( field2 @ A @ R2 ) ) ).
% Field_converse
thf(fact_7341_converse__mono,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S3: set @ ( product_prod @ B @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S3 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S3 ) ) ).
% converse_mono
thf(fact_7342_converse__empty,axiom,
! [B: $tType,A: $tType] :
( ( converse @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% converse_empty
thf(fact_7343_converse__Id,axiom,
! [A: $tType] :
( ( converse @ A @ A @ ( id2 @ A ) )
= ( id2 @ A ) ) ).
% converse_Id
thf(fact_7344_refl__on__converse,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A4 @ ( converse @ A @ A @ R2 ) )
= ( refl_on @ A @ A4 @ R2 ) ) ).
% refl_on_converse
thf(fact_7345_finite__converse,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( converse @ B @ A @ R2 ) )
= ( finite_finite2 @ ( product_prod @ B @ A ) @ R2 ) ) ).
% finite_converse
thf(fact_7346_total__on__converse,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ A4 @ ( converse @ A @ A @ R2 ) )
= ( total_on @ A @ A4 @ R2 ) ) ).
% total_on_converse
thf(fact_7347_antisym__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( antisym @ A @ ( converse @ A @ A @ R2 ) )
= ( antisym @ A @ R2 ) ) ).
% antisym_converse
thf(fact_7348_converse__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( converse @ B @ A @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% converse_UNIV
thf(fact_7349_converse__Id__on,axiom,
! [A: $tType,A4: set @ A] :
( ( converse @ A @ A @ ( id_on @ A @ A4 ) )
= ( id_on @ A @ A4 ) ) ).
% converse_Id_on
thf(fact_7350_card__inverse,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A )] :
( ( finite_card @ ( product_prod @ A @ B ) @ ( converse @ B @ A @ R ) )
= ( finite_card @ ( product_prod @ B @ A ) @ R ) ) ).
% card_inverse
thf(fact_7351_converse__relcomp,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ B @ C ),S3: set @ ( product_prod @ C @ A )] :
( ( converse @ B @ A @ ( relcomp @ B @ C @ A @ R2 @ S3 ) )
= ( relcomp @ A @ C @ B @ ( converse @ C @ A @ S3 ) @ ( converse @ B @ C @ R2 ) ) ) ).
% converse_relcomp
thf(fact_7352_converse__Un,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S3: set @ ( product_prod @ B @ A )] :
( ( converse @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S3 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S3 ) ) ) ).
% converse_Un
thf(fact_7353_converseI,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A3 ) @ ( converse @ A @ B @ R2 ) ) ) ).
% converseI
thf(fact_7354_converseE,axiom,
! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R2 ) )
=> ~ ! [X3: B,Y2: A] :
( ( Yx
= ( product_Pair @ A @ B @ Y2 @ X3 ) )
=> ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y2 ) @ R2 ) ) ) ).
% converseE
thf(fact_7355_converseD,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,R2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( converse @ B @ A @ R2 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A3 ) @ R2 ) ) ).
% converseD
thf(fact_7356_converse_Osimps,axiom,
! [B: $tType,A: $tType,A13: B,A24: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A13 @ A24 ) @ ( converse @ A @ B @ R2 ) )
= ( ? [A5: A,B5: B] :
( ( A13 = B5 )
& ( A24 = A5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_7357_converse_Ocases,axiom,
! [B: $tType,A: $tType,A13: B,A24: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A13 @ A24 ) @ ( converse @ A @ B @ R2 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A24 @ A13 ) @ R2 ) ) ).
% converse.cases
thf(fact_7358_converse__UNION,axiom,
! [B: $tType,A: $tType,C: $tType,R2: C > ( set @ ( product_prod @ B @ A ) ),S: set @ C] :
( ( converse @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X2: C] : ( converse @ B @ A @ ( R2 @ X2 ) )
@ S ) ) ) ).
% converse_UNION
thf(fact_7359_converse__unfold,axiom,
! [A: $tType,B: $tType] :
( ( converse @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ A )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [Y3: A,X2: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y3 ) @ R5 ) ) ) ) ) ).
% converse_unfold
thf(fact_7360_converse__INTER,axiom,
! [B: $tType,A: $tType,C: $tType,R2: C > ( set @ ( product_prod @ B @ A ) ),S: set @ C] :
( ( converse @ B @ A @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S ) ) )
= ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X2: C] : ( converse @ B @ A @ ( R2 @ X2 ) )
@ S ) ) ) ).
% converse_INTER
thf(fact_7361_converse__Int,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S3: set @ ( product_prod @ B @ A )] :
( ( converse @ B @ A @ ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S3 ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S3 ) ) ) ).
% converse_Int
thf(fact_7362_converse__subset__swap,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ ( converse @ B @ A @ S3 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( converse @ A @ B @ R2 ) @ S3 ) ) ).
% converse_subset_swap
thf(fact_7363_Image__subset__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A ),A4: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R2 @ A4 ) @ B2 )
= ( ord_less_eq @ ( set @ B ) @ A4 @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ ( converse @ B @ A @ R2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) ) ) ) ) ).
% Image_subset_eq
thf(fact_7364_refl__on__comp__subset,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A4 @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( relcomp @ A @ A @ A @ ( converse @ A @ A @ R2 ) @ R2 ) ) ) ).
% refl_on_comp_subset
thf(fact_7365_Image__INT__eq,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ B @ A ),A4: set @ C,B2: C > ( set @ B )] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R2 ) )
=> ( ( A4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( image @ B @ A @ R2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B2 @ A4 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X2: C] : ( image @ B @ A @ R2 @ ( B2 @ X2 ) )
@ A4 ) ) ) ) ) ).
% Image_INT_eq
thf(fact_7366_trans__wf__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ( wf @ A @ R2 )
= ( ! [A5: A] :
( wf @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( image @ A @ A @ ( converse @ A @ A @ R2 ) @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) )
@ ^ [Uu3: A] : ( image @ A @ A @ ( converse @ A @ A @ R2 ) @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% trans_wf_iff
thf(fact_7367_trans__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ ( converse @ A @ A @ R2 ) )
= ( trans @ A @ R2 ) ) ).
% trans_converse
thf(fact_7368_trans__O__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ R2 ) @ R2 ) ) ).
% trans_O_subset
thf(fact_7369_single__valued__subset,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 )
=> ( ( single_valued @ A @ B @ S3 )
=> ( single_valued @ A @ B @ R2 ) ) ) ).
% single_valued_subset
thf(fact_7370_trans__Id,axiom,
! [A: $tType] : ( trans @ A @ ( id2 @ A ) ) ).
% trans_Id
thf(fact_7371_single__valued__Id,axiom,
! [A: $tType] : ( single_valued @ A @ A @ ( id2 @ A ) ) ).
% single_valued_Id
thf(fact_7372_single__valued__Id__on,axiom,
! [A: $tType,A4: set @ A] : ( single_valued @ A @ A @ ( id_on @ A @ A4 ) ) ).
% single_valued_Id_on
thf(fact_7373_trans__Id__on,axiom,
! [A: $tType,A4: set @ A] : ( trans @ A @ ( id_on @ A @ A4 ) ) ).
% trans_Id_on
thf(fact_7374_trans__Int,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ( trans @ A @ S3 )
=> ( trans @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ).
% trans_Int
thf(fact_7375_trans__INTER,axiom,
! [B: $tType,A: $tType,S: set @ A,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( trans @ B @ ( R2 @ X3 ) ) )
=> ( trans @ B @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S ) ) ) ) ).
% trans_INTER
thf(fact_7376_transD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_7377_transE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_7378_transI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z3 ) @ R2 ) ) )
=> ( trans @ A @ R2 ) ) ).
% transI
thf(fact_7379_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X2: A,Y3: A,Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z6 ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z6 ) @ R5 ) ) ) ) ) ).
% trans_def
thf(fact_7380_single__valuedD,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),X: A,Y: B,Z: B] :
( ( single_valued @ A @ B @ R2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z ) @ R2 )
=> ( Y = Z ) ) ) ) ).
% single_valuedD
thf(fact_7381_single__valuedI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y2: B,Z3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z3 ) @ R2 )
=> ( Y2 = Z3 ) ) )
=> ( single_valued @ A @ B @ R2 ) ) ).
% single_valuedI
thf(fact_7382_single__valued__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valued @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
! [X2: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R5 )
=> ! [Z6: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z6 ) @ R5 )
=> ( Y3 = Z6 ) ) ) ) ) ).
% single_valued_def
thf(fact_7383_trans__empty,axiom,
! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trans_empty
thf(fact_7384_single__valued__empty,axiom,
! [B: $tType,A: $tType] : ( single_valued @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% single_valued_empty
thf(fact_7385_single__valued__relcomp,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( single_valued @ A @ B @ R2 )
=> ( ( single_valued @ B @ C @ S3 )
=> ( single_valued @ A @ C @ ( relcomp @ A @ B @ C @ R2 @ S3 ) ) ) ) ).
% single_valued_relcomp
thf(fact_7386_under__incr,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R2 @ A3 ) @ ( order_under @ A @ R2 @ B3 ) ) ) ) ).
% under_incr
thf(fact_7387_trans__singleton,axiom,
! [A: $tType,A3: A] : ( trans @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trans_singleton
thf(fact_7388_trans__diff__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ( antisym @ A @ R2 )
=> ( trans @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ).
% trans_diff_Id
thf(fact_7389_trans__join,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Y12: A] :
! [Z6: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ Z6 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y25: A,Aa2: A] :
( ( Y12 = Y25 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Aa2 ) @ R5 ) )
@ Z6 ) )
@ X2 ) ) ) ) ).
% trans_join
thf(fact_7390_underS__incr,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( trans @ A @ R2 )
=> ( ( antisym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( order_underS @ A @ R2 @ B3 ) ) ) ) ) ).
% underS_incr
thf(fact_7391_Image__Int__eq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),A4: set @ B,B2: set @ B] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R ) )
=> ( ( image @ B @ A @ R @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ R @ A4 ) @ ( image @ B @ A @ R @ B2 ) ) ) ) ).
% Image_Int_eq
thf(fact_7392_wf__finite__segments,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R2 )
=> ( ( trans @ A @ R2 )
=> ( ! [X3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R2 ) ) )
=> ( wf @ A @ R2 ) ) ) ) ).
% wf_finite_segments
thf(fact_7393_Rat_Opositive_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o @ ratrel
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) )
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ).
% Rat.positive.rsp
thf(fact_7394_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [A5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R5 ) ) ) ).
% irrefl_def
thf(fact_7395_irreflI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A7: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ A7 ) @ R )
=> ( irrefl @ A @ R ) ) ).
% irreflI
thf(fact_7396_irrefl__diff__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( irrefl @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ).
% irrefl_diff_Id
thf(fact_7397_irrefl__distinct,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [A5: A,B5: A] : A5 != B5
@ X2 ) ) ) ) ).
% irrefl_distinct
thf(fact_7398_Rat_Opositive_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( positive @ ( abs_Rat @ X ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% Rat.positive.abs_eq
thf(fact_7399_irreflp__irrefl__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irreflp @ A
@ ^ [A5: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ R ) )
= ( irrefl @ A @ R ) ) ).
% irreflp_irrefl_eq
thf(fact_7400_irreflp__def,axiom,
! [A: $tType] :
( ( irreflp @ A )
= ( ^ [R6: A > A > $o] :
! [A5: A] :
~ ( R6 @ A5 @ A5 ) ) ) ).
% irreflp_def
thf(fact_7401_irreflpI,axiom,
! [A: $tType,R: A > A > $o] :
( ! [A7: A] :
~ ( R @ A7 @ A7 )
=> ( irreflp @ A @ R ) ) ).
% irreflpI
thf(fact_7402_irreflp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( irreflp @ A @ ( ord_less @ A ) ) ) ).
% irreflp_less
thf(fact_7403_irreflp__greater,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( irreflp @ A
@ ^ [X2: A,Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% irreflp_greater
thf(fact_7404_le__prod__filterI,axiom,
! [A: $tType,B: $tType,F4: filter @ ( product_prod @ A @ B ),A4: filter @ A,B2: filter @ B] :
( ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ F4 ) @ A4 )
=> ( ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ F4 ) @ B2 )
=> ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ F4 @ ( prod_filter @ A @ B @ A4 @ B2 ) ) ) ) ).
% le_prod_filterI
thf(fact_7405_Bseq__monoseq__convergent_H__dec,axiom,
! [F3: nat > real,M5: nat] :
( ( bfun @ nat @ real
@ ^ [N2: nat] : ( F3 @ ( plus_plus @ nat @ N2 @ M5 ) )
@ ( at_top @ nat ) )
=> ( ! [M4: nat,N3: nat] :
( ( ord_less_eq @ nat @ M5 @ M4 )
=> ( ( ord_less_eq @ nat @ M4 @ N3 )
=> ( ord_less_eq @ real @ ( F3 @ N3 ) @ ( F3 @ M4 ) ) ) )
=> ( topolo6863149650580417670ergent @ real @ F3 ) ) ) ).
% Bseq_monoseq_convergent'_dec
thf(fact_7406_filtermap__id_H,axiom,
! [A: $tType] :
( ( filtermap @ A @ A
@ ^ [X2: A] : X2 )
= ( ^ [F8: filter @ A] : F8 ) ) ).
% filtermap_id'
thf(fact_7407_filtermap__bot,axiom,
! [B: $tType,A: $tType,F3: B > A] :
( ( filtermap @ B @ A @ F3 @ ( bot_bot @ ( filter @ B ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ).
% filtermap_bot
thf(fact_7408_filtermap__principal,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B] :
( ( filtermap @ B @ A @ F3 @ ( principal @ B @ A4 ) )
= ( principal @ A @ ( image2 @ B @ A @ F3 @ A4 ) ) ) ).
% filtermap_principal
thf(fact_7409_filtermap__SUP,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,F4: C > ( filter @ B ),B2: set @ C] :
( ( filtermap @ B @ A @ F3 @ ( complete_Sup_Sup @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F4 @ B2 ) ) )
= ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B5: C] : ( filtermap @ B @ A @ F3 @ ( F4 @ B5 ) )
@ B2 ) ) ) ).
% filtermap_SUP
thf(fact_7410_filtermap__bot__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,F4: filter @ B] :
( ( ( filtermap @ B @ A @ F3 @ F4 )
= ( bot_bot @ ( filter @ A ) ) )
= ( F4
= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtermap_bot_iff
thf(fact_7411_filtermap__sup,axiom,
! [A: $tType,B: $tType,F3: B > A,F1: filter @ B,F22: filter @ B] :
( ( filtermap @ B @ A @ F3 @ ( sup_sup @ ( filter @ B ) @ F1 @ F22 ) )
= ( sup_sup @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ F1 ) @ ( filtermap @ B @ A @ F3 @ F22 ) ) ) ).
% filtermap_sup
thf(fact_7412_prod__filter__assoc,axiom,
! [A: $tType,B: $tType,C: $tType,F4: filter @ A,G3: filter @ B,H7: filter @ C] :
( ( prod_filter @ ( product_prod @ A @ B ) @ C @ ( prod_filter @ A @ B @ F4 @ G3 ) @ 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 )
@ ^ [X2: A] :
( product_case_prod @ B @ C @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [Y3: B] : ( product_Pair @ ( product_prod @ A @ B ) @ C @ ( product_Pair @ A @ B @ X2 @ Y3 ) ) ) )
@ ( prod_filter @ A @ ( product_prod @ B @ C ) @ F4 @ ( prod_filter @ B @ C @ G3 @ H7 ) ) ) ) ).
% prod_filter_assoc
thf(fact_7413_convergent__mult__const__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,F3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( topolo6863149650580417670ergent @ A
@ ^ [N2: nat] : ( times_times @ A @ C3 @ ( F3 @ N2 ) ) )
= ( topolo6863149650580417670ergent @ A @ F3 ) ) ) ) ).
% convergent_mult_const_iff
thf(fact_7414_convergent__mult__const__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,F3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( topolo6863149650580417670ergent @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ C3 ) )
= ( topolo6863149650580417670ergent @ A @ F3 ) ) ) ) ).
% convergent_mult_const_right_iff
thf(fact_7415_eventually__filtermap,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: B > A,F4: filter @ B] :
( ( eventually @ A @ P @ ( filtermap @ B @ A @ F3 @ F4 ) )
= ( eventually @ B
@ ^ [X2: B] : ( P @ ( F3 @ X2 ) )
@ F4 ) ) ).
% eventually_filtermap
thf(fact_7416_filterlim__filtermap,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > B,F1: filter @ B,G2: C > A,F22: filter @ C] :
( ( filterlim @ A @ B @ F3 @ F1 @ ( filtermap @ C @ A @ G2 @ F22 ) )
= ( filterlim @ C @ B
@ ^ [X2: C] : ( F3 @ ( G2 @ X2 ) )
@ F1
@ F22 ) ) ).
% filterlim_filtermap
thf(fact_7417_filtermap__ident,axiom,
! [A: $tType,F4: filter @ A] :
( ( filtermap @ A @ A
@ ^ [X2: A] : X2
@ F4 )
= F4 ) ).
% filtermap_ident
thf(fact_7418_filtermap__filtermap,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,G2: C > B,F4: filter @ C] :
( ( filtermap @ B @ A @ F3 @ ( filtermap @ C @ B @ G2 @ F4 ) )
= ( filtermap @ C @ A
@ ^ [X2: C] : ( F3 @ ( G2 @ X2 ) )
@ F4 ) ) ).
% filtermap_filtermap
thf(fact_7419_prod__filtermap2,axiom,
! [B: $tType,A: $tType,C: $tType,F4: filter @ A,G2: C > B,G3: filter @ C] :
( ( prod_filter @ A @ B @ F4 @ ( filtermap @ C @ B @ G2 @ G3 ) )
= ( filtermap @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B ) @ ( product_apsnd @ C @ B @ A @ G2 ) @ ( prod_filter @ A @ C @ F4 @ G3 ) ) ) ).
% prod_filtermap2
thf(fact_7420_map__filter__on__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( map_filter_on @ A @ B @ ( top_top @ ( set @ A ) ) )
= ( filtermap @ A @ B ) ) ).
% map_filter_on_UNIV
thf(fact_7421_filtermap__fun__inverse,axiom,
! [B: $tType,A: $tType,G2: A > B,F4: filter @ B,G3: filter @ A,F3: B > A] :
( ( filterlim @ A @ B @ G2 @ F4 @ G3 )
=> ( ( filterlim @ B @ A @ F3 @ G3 @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F3 @ ( G2 @ X2 ) )
= X2 )
@ G3 )
=> ( ( filtermap @ B @ A @ F3 @ F4 )
= G3 ) ) ) ) ).
% filtermap_fun_inverse
thf(fact_7422_filtermap__eq__strong,axiom,
! [B: $tType,A: $tType,F3: A > B,F4: filter @ A,G3: filter @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( filtermap @ A @ B @ F3 @ F4 )
= ( filtermap @ A @ B @ F3 @ G3 ) )
= ( F4 = G3 ) ) ) ).
% filtermap_eq_strong
thf(fact_7423_filtermap__nhds__times,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filtermap @ A @ A @ ( times_times @ A @ C3 ) @ ( topolo7230453075368039082e_nhds @ A @ A3 ) )
= ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C3 @ A3 ) ) ) ) ) ).
% filtermap_nhds_times
thf(fact_7424_filtercomap__filtermap,axiom,
! [B: $tType,A: $tType,F4: filter @ A,F3: A > B] : ( ord_less_eq @ ( filter @ A ) @ F4 @ ( filtercomap @ A @ B @ F3 @ ( filtermap @ A @ B @ F3 @ F4 ) ) ) ).
% filtercomap_filtermap
thf(fact_7425_filtermap__filtercomap,axiom,
! [B: $tType,A: $tType,F3: B > A,F4: filter @ A] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ ( filtercomap @ B @ A @ F3 @ F4 ) ) @ F4 ) ).
% filtermap_filtercomap
thf(fact_7426_filtermap__le__iff__le__filtercomap,axiom,
! [B: $tType,A: $tType,F3: B > A,F4: filter @ B,G3: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ F4 ) @ G3 )
= ( ord_less_eq @ ( filter @ B ) @ F4 @ ( filtercomap @ B @ A @ F3 @ G3 ) ) ) ).
% filtermap_le_iff_le_filtercomap
thf(fact_7427_filterlim__def,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F2: A > B,F24: filter @ B,F14: filter @ A] : ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ A @ B @ F2 @ F14 ) @ F24 ) ) ) ).
% filterlim_def
thf(fact_7428_lim__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: nat > A,X: A] :
( ( topolo6863149650580417670ergent @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ X )
=> ( ord_less_eq @ A @ ( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat ) @ F3 ) @ X ) ) ) ) ).
% lim_le
thf(fact_7429_filtermap__mono,axiom,
! [B: $tType,A: $tType,F4: filter @ A,F11: filter @ A,F3: A > B] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ F11 )
=> ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ A @ B @ F3 @ F4 ) @ ( filtermap @ A @ B @ F3 @ F11 ) ) ) ).
% filtermap_mono
thf(fact_7430_filtermap__inf,axiom,
! [A: $tType,B: $tType,F3: B > A,F1: filter @ B,F22: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ ( inf_inf @ ( filter @ B ) @ F1 @ F22 ) ) @ ( inf_inf @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ F1 ) @ ( filtermap @ B @ A @ F3 @ F22 ) ) ) ).
% filtermap_inf
thf(fact_7431_filtermap__Pair,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > A,G2: C > B,F4: filter @ C] :
( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) )
@ ( filtermap @ C @ ( product_prod @ A @ B )
@ ^ [X2: C] : ( product_Pair @ A @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ F4 )
@ ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F3 @ F4 ) @ ( filtermap @ C @ B @ G2 @ F4 ) ) ) ).
% filtermap_Pair
thf(fact_7432_filtermap__def,axiom,
! [B: $tType,A: $tType] :
( ( filtermap @ A @ B )
= ( ^ [F2: A > B,F8: filter @ A] :
( abs_filter @ B
@ ^ [P3: B > $o] :
( eventually @ A
@ ^ [X2: A] : ( P3 @ ( F2 @ X2 ) )
@ F8 ) ) ) ) ).
% filtermap_def
thf(fact_7433_filtermap__mono__strong,axiom,
! [B: $tType,A: $tType,F3: A > B,F4: filter @ A,G3: filter @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ A @ B @ F3 @ F4 ) @ ( filtermap @ A @ B @ F3 @ G3 ) )
= ( ord_less_eq @ ( filter @ A ) @ F4 @ G3 ) ) ) ).
% filtermap_mono_strong
thf(fact_7434_filtermap__fst__prod__filter,axiom,
! [B: $tType,A: $tType,A4: filter @ A,B2: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( prod_filter @ A @ B @ A4 @ B2 ) ) @ A4 ) ).
% filtermap_fst_prod_filter
thf(fact_7435_filtermap__snd__prod__filter,axiom,
! [B: $tType,A: $tType,A4: filter @ B,B2: filter @ A] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( prod_filter @ B @ A @ A4 @ B2 ) ) @ B2 ) ).
% filtermap_snd_prod_filter
thf(fact_7436_filtermap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,F4: C > ( filter @ B ),B2: set @ C] :
( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F4 @ B2 ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B5: C] : ( filtermap @ B @ A @ F3 @ ( F4 @ B5 ) )
@ B2 ) ) ) ).
% filtermap_INF
thf(fact_7437_at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A] :
( ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
= ( filtermap @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ X2 @ A3 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_to_0
thf(fact_7438_Bseq__mono__convergent,axiom,
! [X4: nat > real] :
( ( bfun @ nat @ real @ X4 @ ( at_top @ nat ) )
=> ( ! [M4: nat,N3: nat] :
( ( ord_less_eq @ nat @ M4 @ N3 )
=> ( ord_less_eq @ real @ ( X4 @ M4 ) @ ( X4 @ N3 ) ) )
=> ( topolo6863149650580417670ergent @ real @ X4 ) ) ) ).
% Bseq_mono_convergent
thf(fact_7439_convergent__realpow,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( topolo6863149650580417670ergent @ real @ ( power_power @ real @ X ) ) ) ) ).
% convergent_realpow
thf(fact_7440_filterlim__INF__INF,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,J5: set @ A,I5: set @ B,F3: D > C,F4: B > ( filter @ D ),G3: A > ( filter @ C )] :
( ! [M4: A] :
( ( member @ A @ M4 @ J5 )
=> ? [X5: B] :
( ( member @ B @ X5 @ I5 )
& ( ord_less_eq @ ( filter @ C ) @ ( filtermap @ D @ C @ F3 @ ( F4 @ X5 ) ) @ ( G3 @ M4 ) ) ) )
=> ( filterlim @ D @ C @ F3 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ G3 @ J5 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ F4 @ I5 ) ) ) ) ).
% filterlim_INF_INF
thf(fact_7441_filtermap__times__pos__at__right,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [C3: A,P6: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( filtermap @ A @ A @ ( times_times @ A @ C3 ) @ ( topolo174197925503356063within @ A @ P6 @ ( set_ord_greaterThan @ A @ P6 ) ) )
= ( topolo174197925503356063within @ A @ ( times_times @ A @ C3 @ P6 ) @ ( set_ord_greaterThan @ A @ ( times_times @ A @ C3 @ P6 ) ) ) ) ) ) ).
% filtermap_times_pos_at_right
thf(fact_7442_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_7443_cauchy__filter__metric__filtermap,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V768167426530841204y_dist @ B )
& ( topolo7287701948861334536_space @ B ) )
=> ! [F3: A > B,F4: filter @ A] :
( ( topolo6773858410816713723filter @ B @ ( filtermap @ A @ B @ F3 @ F4 ) )
= ( ! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [P3: A > $o] :
( ( eventually @ A @ P3 @ F4 )
& ! [X2: A,Y3: A] :
( ( ( P3 @ X2 )
& ( P3 @ Y3 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) @ E3 ) ) ) ) ) ) ) ).
% cauchy_filter_metric_filtermap
thf(fact_7444_Bseq__monoseq__convergent_H__inc,axiom,
! [F3: nat > real,M5: nat] :
( ( bfun @ nat @ real
@ ^ [N2: nat] : ( F3 @ ( plus_plus @ nat @ N2 @ M5 ) )
@ ( at_top @ nat ) )
=> ( ! [M4: nat,N3: nat] :
( ( ord_less_eq @ nat @ M5 @ M4 )
=> ( ( ord_less_eq @ nat @ M4 @ N3 )
=> ( ord_less_eq @ real @ ( F3 @ M4 ) @ ( F3 @ N3 ) ) ) )
=> ( topolo6863149650580417670ergent @ real @ F3 ) ) ) ).
% Bseq_monoseq_convergent'_inc
thf(fact_7445_sorted__insort__insert__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( linord329482645794927042rt_key @ B @ A @ F3 @ X @ Xs ) ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_7446_pair__lessI2,axiom,
! [A3: nat,B3: nat,S3: nat,T2: nat] :
( ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( ord_less @ nat @ S3 @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S3 ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_7447_pair__less__iff1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ X @ Z ) ) @ fun_pair_less )
= ( ord_less @ nat @ Y @ Z ) ) ).
% pair_less_iff1
thf(fact_7448_filtermap__sequentually__ne__bot,axiom,
! [A: $tType,F3: nat > A] :
( ( filtermap @ nat @ A @ F3 @ ( at_top @ nat ) )
!= ( bot_bot @ ( filter @ A ) ) ) ).
% filtermap_sequentually_ne_bot
thf(fact_7449_filtermap__image__finite__subsets__at__top,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F3 @ A4 )
=> ( ( filtermap @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F3 ) @ ( finite5375528669736107172at_top @ A @ A4 ) )
= ( finite5375528669736107172at_top @ B @ ( image2 @ A @ B @ F3 @ A4 ) ) ) ) ).
% filtermap_image_finite_subsets_at_top
thf(fact_7450_pair__lessI1,axiom,
! [A3: nat,B3: nat,S3: nat,T2: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S3 ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_7451_set__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( set2 @ A
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ Xs ) )
= ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ) ).
% set_insort_insert
thf(fact_7452_sorted__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,X: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ Xs ) ) ) ) ).
% sorted_insort_insert
thf(fact_7453_prod__filter__principal__singleton,axiom,
! [A: $tType,B: $tType,X: A,F4: filter @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ F4 )
= ( filtermap @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ F4 ) ) ).
% prod_filter_principal_singleton
thf(fact_7454_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F4: filter @ A,X: B] :
( ( prod_filter @ A @ B @ F4 @ ( principal @ B @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( filtermap @ A @ ( product_prod @ A @ B )
@ ^ [A5: A] : ( product_Pair @ A @ B @ A5 @ X )
@ F4 ) ) ).
% prod_filter_principal_singleton2
thf(fact_7455_pair__leqI2,axiom,
! [A3: nat,B3: nat,S3: nat,T2: nat] :
( ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( ord_less_eq @ nat @ S3 @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S3 ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_7456_pair__leqI1,axiom,
! [A3: nat,B3: nat,S3: nat,T2: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S3 ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_7457_admissible__chfin,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o] :
( ! [S2: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ S2 )
=> ( finite_finite2 @ A @ S2 ) )
=> ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P ) ) ) ).
% admissible_chfin
thf(fact_7458_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 )
@ ^ [X2: A,Y3: A] : ( ord_less @ A @ Y3 @ X2 )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_7459_ccpo_OadmissibleD,axiom,
! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o,P: A > $o,A4: set @ A] :
( ( comple1908693960933563346ssible @ A @ Lub @ Ord @ P )
=> ( ( comple1602240252501008431_chain @ A @ Ord @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( P @ X3 ) )
=> ( P @ ( Lub @ A4 ) ) ) ) ) ) ).
% ccpo.admissibleD
thf(fact_7460_ccpo_OadmissibleI,axiom,
! [A: $tType,Ord: A > A > $o,P: A > $o,Lub: ( set @ A ) > A] :
( ! [A9: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord @ A9 )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A9 )
=> ( P @ X5 ) )
=> ( P @ ( Lub @ A9 ) ) ) ) )
=> ( comple1908693960933563346ssible @ A @ Lub @ Ord @ P ) ) ).
% ccpo.admissibleI
thf(fact_7461_ccpo_Oadmissible__def,axiom,
! [A: $tType] :
( ( comple1908693960933563346ssible @ A )
= ( ^ [Lub2: ( set @ A ) > A,Ord2: A > A > $o,P3: A > $o] :
! [A6: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord2 @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( P3 @ X2 ) )
=> ( P3 @ ( Lub2 @ A6 ) ) ) ) ) ) ) ).
% ccpo.admissible_def
thf(fact_7462_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A3 )
=> ( A3 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_7463_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( A3 != Top )
= ( Less @ A3 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_7464_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A3 )
= ( A3 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_7465_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A3 ) ) ).
% ordering_top.extremum_strict
thf(fact_7466_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( Less_eq @ A3 @ Top ) ) ).
% ordering_top.extremum
thf(fact_7467_admissible__disj,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o,Q: A > $o] :
( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P )
=> ( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ Q )
=> ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A )
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) ) ) ) ).
% admissible_disj
thf(fact_7468_gcd__nat_Oordering__top__axioms,axiom,
( ordering_top @ nat @ ( dvd_dvd @ nat )
@ ^ [M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ M2 @ N2 )
& ( M2 != N2 ) )
@ ( zero_zero @ nat ) ) ).
% gcd_nat.ordering_top_axioms
thf(fact_7469_top_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ( ordering_top @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ A ) ) ) ).
% top.ordering_top_axioms
thf(fact_7470_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top @ nat
@ ^ [X2: nat,Y3: nat] : ( ord_less_eq @ nat @ Y3 @ X2 )
@ ^ [X2: nat,Y3: nat] : ( ord_less @ nat @ Y3 @ X2 )
@ ( zero_zero @ nat ) ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_7471_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K @ L ) @ N )
= ( ( ( ord_less @ nat @ N @ M )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) )
| ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_7472_span__singleton,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_Vector_span @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image2 @ real @ A
@ ^ [K3: real] : ( real_V8093663219630862766scaleR @ A @ K3 @ X )
@ ( top_top @ ( set @ real ) ) ) ) ) ).
% span_singleton
thf(fact_7473_concat__bit__0,axiom,
! [K: int,L: int] :
( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L )
= L ) ).
% concat_bit_0
thf(fact_7474_span__insert__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A] :
( ( real_Vector_span @ A @ ( insert2 @ A @ ( zero_zero @ A ) @ S ) )
= ( real_Vector_span @ A @ S ) ) ) ).
% span_insert_0
thf(fact_7475_concat__bit__nonnegative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_concat_bit @ N @ K @ L ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ).
% concat_bit_nonnegative_iff
thf(fact_7476_concat__bit__negative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less @ int @ ( bit_concat_bit @ N @ K @ L ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ).
% concat_bit_negative_iff
thf(fact_7477_span__empty,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_span @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% span_empty
thf(fact_7478_span__delete__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A] :
( ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( real_Vector_span @ A @ S ) ) ) ).
% span_delete_0
thf(fact_7479_dependent__insertD,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A,S: set @ A] :
( ~ ( member @ A @ A3 @ ( real_Vector_span @ A @ S ) )
=> ( ( real_V358717886546972837endent @ A @ ( insert2 @ A @ A3 @ S ) )
=> ( real_V358717886546972837endent @ A @ S ) ) ) ) ).
% dependent_insertD
thf(fact_7480_independent__insert,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A,S: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ ( insert2 @ A @ A3 @ S ) ) )
= ( ( ( member @ A @ A3 @ S )
=> ~ ( real_V358717886546972837endent @ A @ S ) )
& ( ~ ( member @ A @ A3 @ S )
=> ( ~ ( real_V358717886546972837endent @ A @ S )
& ~ ( member @ A @ A3 @ ( real_Vector_span @ A @ S ) ) ) ) ) ) ) ).
% independent_insert
thf(fact_7481_independent__insertI,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A,S: set @ A] :
( ~ ( member @ A @ A3 @ ( real_Vector_span @ A @ S ) )
=> ( ~ ( real_V358717886546972837endent @ A @ S )
=> ~ ( real_V358717886546972837endent @ A @ ( insert2 @ A @ A3 @ S ) ) ) ) ) ).
% independent_insertI
thf(fact_7482_span__redundant,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,S: set @ A] :
( ( member @ A @ X @ ( real_Vector_span @ A @ S ) )
=> ( ( real_Vector_span @ A @ ( insert2 @ A @ X @ S ) )
= ( real_Vector_span @ A @ S ) ) ) ) ).
% span_redundant
thf(fact_7483_in__span__insert,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A,B3: A,S: set @ A] :
( ( member @ A @ A3 @ ( real_Vector_span @ A @ ( insert2 @ A @ B3 @ S ) ) )
=> ( ~ ( member @ A @ A3 @ ( real_Vector_span @ A @ S ) )
=> ( member @ A @ B3 @ ( real_Vector_span @ A @ ( insert2 @ A @ A3 @ S ) ) ) ) ) ) ).
% in_span_insert
thf(fact_7484_span__trans,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,S: set @ A,Y: A] :
( ( member @ A @ X @ ( real_Vector_span @ A @ S ) )
=> ( ( member @ A @ Y @ ( real_Vector_span @ A @ ( insert2 @ A @ X @ S ) ) )
=> ( member @ A @ Y @ ( real_Vector_span @ A @ S ) ) ) ) ) ).
% span_trans
thf(fact_7485_eq__span__insert__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y: A,S: set @ A] :
( ( member @ A @ ( minus_minus @ A @ X @ Y ) @ ( real_Vector_span @ A @ S ) )
=> ( ( real_Vector_span @ A @ ( insert2 @ A @ X @ S ) )
= ( real_Vector_span @ A @ ( insert2 @ A @ Y @ S ) ) ) ) ) ).
% eq_span_insert_eq
thf(fact_7486_in__span__delete,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A,S: set @ A,B3: A] :
( ( member @ A @ A3 @ ( real_Vector_span @ A @ S ) )
=> ( ~ ( member @ A @ A3 @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ A @ B3 @ ( real_Vector_span @ A @ ( insert2 @ A @ A3 @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% in_span_delete
thf(fact_7487_span__breakdown__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,A3: A,S: set @ A] :
( ( member @ A @ X @ ( real_Vector_span @ A @ ( insert2 @ A @ A3 @ S ) ) )
= ( ? [K3: real] : ( member @ A @ ( minus_minus @ A @ X @ ( real_V8093663219630862766scaleR @ A @ K3 @ A3 ) ) @ ( real_Vector_span @ A @ S ) ) ) ) ) ).
% span_breakdown_eq
thf(fact_7488_span__induct__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,S: set @ A,H: A > $o] :
( ( member @ A @ X @ ( real_Vector_span @ A @ S ) )
=> ( ( H @ ( zero_zero @ A ) )
=> ( ! [C5: real,X3: A,Y2: A] :
( ( member @ A @ X3 @ S )
=> ( ( H @ Y2 )
=> ( H @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ C5 @ X3 ) @ Y2 ) ) ) )
=> ( H @ X ) ) ) ) ) ).
% span_induct_alt
thf(fact_7489_span__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A] : ( member @ A @ ( zero_zero @ A ) @ ( real_Vector_span @ A @ S ) ) ) ).
% span_0
thf(fact_7490_span__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A,T4: set @ A] :
( ( ( real_Vector_span @ A @ S )
= ( real_Vector_span @ A @ T4 ) )
= ( ( ord_less_eq @ ( set @ A ) @ S @ ( real_Vector_span @ A @ T4 ) )
& ( ord_less_eq @ ( set @ A ) @ T4 @ ( real_Vector_span @ A @ S ) ) ) ) ) ).
% span_eq
thf(fact_7491_span__mono,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( real_Vector_span @ A @ A4 ) @ ( real_Vector_span @ A @ B2 ) ) ) ) ).
% span_mono
thf(fact_7492_span__superset,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A] : ( ord_less_eq @ ( set @ A ) @ S @ ( real_Vector_span @ A @ S ) ) ) ).
% span_superset
thf(fact_7493_maximal__independent__subset__extend,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A,V: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ V )
=> ( ~ ( real_V358717886546972837endent @ A @ S )
=> ~ ! [B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ V )
=> ( ~ ( real_V358717886546972837endent @ A @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B4 ) ) ) ) ) ) ) ) ).
% maximal_independent_subset_extend
thf(fact_7494_spanning__subset__independent,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ~ ( real_V358717886546972837endent @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( real_Vector_span @ A @ B2 ) )
=> ( A4 = B2 ) ) ) ) ) ).
% spanning_subset_independent
thf(fact_7495_maximal__independent__subset,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [V: set @ A] :
~ ! [B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ V )
=> ( ~ ( real_V358717886546972837endent @ A @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B4 ) ) ) ) ) ).
% maximal_independent_subset
thf(fact_7496_span__Un,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A,T4: set @ A] :
( ( real_Vector_span @ A @ ( sup_sup @ ( set @ A ) @ S @ T4 ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [X2: A,Y3: A] :
( ( Uu3
= ( plus_plus @ A @ X2 @ Y3 ) )
& ( member @ A @ X2 @ ( real_Vector_span @ A @ S ) )
& ( member @ A @ Y3 @ ( real_Vector_span @ A @ T4 ) ) ) ) ) ) ).
% span_Un
thf(fact_7497_span__insert,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A,S: set @ A] :
( ( real_Vector_span @ A @ ( insert2 @ A @ A3 @ S ) )
= ( collect @ A
@ ^ [X2: A] :
? [K3: real] : ( member @ A @ ( minus_minus @ A @ X2 @ ( real_V8093663219630862766scaleR @ A @ K3 @ A3 ) ) @ ( real_Vector_span @ A @ S ) ) ) ) ) ).
% span_insert
thf(fact_7498_span__finite,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ( real_Vector_span @ A @ S )
= ( image2 @ ( A > real ) @ A
@ ^ [U2: A > real] :
( groups7311177749621191930dd_sum @ A @ A
@ ^ [V6: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V6 ) @ V6 )
@ S )
@ ( top_top @ ( set @ ( A > real ) ) ) ) ) ) ) ).
% span_finite
thf(fact_7499_dependent__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [P3: set @ A] :
? [X2: A] :
( ( member @ A @ X2 @ P3 )
& ( member @ A @ X2 @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ P3 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% dependent_def
thf(fact_7500_span__image__scale,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A,C3: A > real] :
( ( finite_finite2 @ A @ S )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( C3 @ X3 )
!= ( zero_zero @ real ) ) )
=> ( ( real_Vector_span @ A
@ ( image2 @ A @ A
@ ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( C3 @ X2 ) @ X2 )
@ S ) )
= ( real_Vector_span @ A @ S ) ) ) ) ) ).
% span_image_scale
thf(fact_7501_span__breakdown,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B3: A,S: set @ A,A3: A] :
( ( member @ A @ B3 @ S )
=> ( ( member @ A @ A3 @ ( real_Vector_span @ A @ S ) )
=> ? [K2: real] : ( member @ A @ ( minus_minus @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ K2 @ B3 ) ) @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% span_breakdown
thf(fact_7502_independent__span__bound,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [T4: set @ A,S: set @ A] :
( ( finite_finite2 @ A @ T4 )
=> ( ~ ( real_V358717886546972837endent @ A @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ ( real_Vector_span @ A @ T4 ) )
=> ( ( finite_finite2 @ A @ S )
& ( ord_less_eq @ nat @ ( finite_card @ A @ S ) @ ( finite_card @ A @ T4 ) ) ) ) ) ) ) ).
% independent_span_bound
thf(fact_7503_exchange__lemma,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [T4: set @ A,S: set @ A] :
( ( finite_finite2 @ A @ T4 )
=> ( ~ ( real_V358717886546972837endent @ A @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ ( real_Vector_span @ A @ T4 ) )
=> ? [T13: set @ A] :
( ( ( finite_card @ A @ T13 )
= ( finite_card @ A @ T4 ) )
& ( finite_finite2 @ A @ T13 )
& ( ord_less_eq @ ( set @ A ) @ S @ T13 )
& ( ord_less_eq @ ( set @ A ) @ T13 @ ( sup_sup @ ( set @ A ) @ S @ T4 ) )
& ( ord_less_eq @ ( set @ A ) @ S @ ( real_Vector_span @ A @ T13 ) ) ) ) ) ) ) ).
% exchange_lemma
thf(fact_7504_span__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_span @ A )
= ( ^ [B6: set @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [F2: A > real] :
( ( Uu3
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( F2 @ X2 ) @ X2 )
@ ( collect @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
!= ( zero_zero @ real ) ) ) ) )
& ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
!= ( zero_zero @ real ) ) )
@ B6 )
& ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
!= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% span_alt
thf(fact_7505_span__explicit_H,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_span @ A )
= ( ^ [B5: set @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [F2: A > real] :
( ( Uu3
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V6: A] : ( real_V8093663219630862766scaleR @ A @ ( F2 @ V6 ) @ V6 )
@ ( collect @ A
@ ^ [V6: A] :
( ( F2 @ V6 )
!= ( zero_zero @ real ) ) ) ) )
& ( finite_finite2 @ A
@ ( collect @ A
@ ^ [V6: A] :
( ( F2 @ V6 )
!= ( zero_zero @ real ) ) ) )
& ! [V6: A] :
( ( ( F2 @ V6 )
!= ( zero_zero @ real ) )
=> ( member @ A @ V6 @ B5 ) ) ) ) ) ) ) ).
% span_explicit'
thf(fact_7506_span__explicit,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_span @ A )
= ( ^ [B5: set @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [T3: set @ A,R5: A > real] :
( ( Uu3
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [A5: A] : ( real_V8093663219630862766scaleR @ A @ ( R5 @ A5 ) @ A5 )
@ T3 ) )
& ( finite_finite2 @ A @ T3 )
& ( ord_less_eq @ ( set @ A ) @ T3 @ B5 ) ) ) ) ) ) ).
% span_explicit
thf(fact_7507_representation__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V7696804695334737415tation @ A )
= ( ^ [Basis2: set @ A,V6: A] :
( if @ ( A > real )
@ ( ~ ( real_V358717886546972837endent @ A @ Basis2 )
& ( member @ A @ V6 @ ( real_Vector_span @ A @ Basis2 ) ) )
@ ( fChoice @ ( A > real )
@ ^ [F2: A > real] :
( ! [W3: A] :
( ( ( F2 @ W3 )
!= ( zero_zero @ real ) )
=> ( member @ A @ W3 @ Basis2 ) )
& ( finite_finite2 @ A
@ ( collect @ A
@ ^ [W3: A] :
( ( F2 @ W3 )
!= ( zero_zero @ real ) ) ) )
& ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [W3: A] : ( real_V8093663219630862766scaleR @ A @ ( F2 @ W3 ) @ W3 )
@ ( collect @ A
@ ^ [W3: A] :
( ( F2 @ W3 )
!= ( zero_zero @ real ) ) ) )
= V6 ) ) )
@ ^ [B5: A] : ( zero_zero @ real ) ) ) ) ) ).
% representation_def
thf(fact_7508_extend__basis__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V4986007116245087402_basis @ A )
= ( ^ [B6: set @ A] :
( fChoice @ ( set @ A )
@ ^ [B16: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ B16 )
& ~ ( real_V358717886546972837endent @ A @ B16 )
& ( ( real_Vector_span @ A @ B16 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% extend_basis_def
thf(fact_7509_representation__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A] :
( ( real_V7696804695334737415tation @ A @ Basis @ ( zero_zero @ A ) )
= ( ^ [B5: A] : ( zero_zero @ real ) ) ) ) ).
% representation_zero
thf(fact_7510_finite__representation,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [B5: A] :
( ( real_V7696804695334737415tation @ A @ Basis @ V2 @ B5 )
!= ( zero_zero @ real ) ) ) ) ) ).
% finite_representation
thf(fact_7511_representation__extend,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A,Basis3: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ Basis3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ Basis3 @ Basis )
=> ( ( real_V7696804695334737415tation @ A @ Basis @ V2 )
= ( real_V7696804695334737415tation @ A @ Basis3 @ V2 ) ) ) ) ) ) ).
% representation_extend
thf(fact_7512_extend__basis__superset,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ ( real_V4986007116245087402_basis @ A @ B2 ) ) ) ) ).
% extend_basis_superset
thf(fact_7513_sum__representation__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A,B2: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ Basis ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ Basis @ B2 )
=> ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [B5: A] : ( real_V8093663219630862766scaleR @ A @ ( real_V7696804695334737415tation @ A @ Basis @ V2 @ B5 ) @ B5 )
@ B2 )
= V2 ) ) ) ) ) ) ).
% sum_representation_eq
thf(fact_7514_representation__eqI,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A,F3: A > real] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ Basis ) )
=> ( ! [B7: A] :
( ( ( F3 @ B7 )
!= ( zero_zero @ real ) )
=> ( member @ A @ B7 @ Basis ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [B5: A] :
( ( F3 @ B5 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [B5: A] : ( real_V8093663219630862766scaleR @ A @ ( F3 @ B5 ) @ B5 )
@ ( collect @ A
@ ^ [B5: A] :
( ( F3 @ B5 )
!= ( zero_zero @ real ) ) ) )
= V2 )
=> ( ( real_V7696804695334737415tation @ A @ Basis @ V2 )
= F3 ) ) ) ) ) ) ) ).
% representation_eqI
thf(fact_7515_construct__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ( ( real_V4425403222259421789struct @ A @ B )
= ( ^ [B6: set @ A,G: A > B,V6: A] :
( groups7311177749621191930dd_sum @ A @ B
@ ^ [B5: A] : ( real_V8093663219630862766scaleR @ B @ ( real_V7696804695334737415tation @ A @ ( real_V4986007116245087402_basis @ A @ B6 ) @ V6 @ B5 ) @ ( if @ B @ ( member @ A @ B5 @ B6 ) @ ( G @ B5 ) @ ( zero_zero @ B ) ) )
@ ( collect @ A
@ ^ [B5: A] :
( ( real_V7696804695334737415tation @ A @ ( real_V4986007116245087402_basis @ A @ B6 ) @ V6 @ B5 )
!= ( zero_zero @ real ) ) ) ) ) ) ) ).
% construct_def
thf(fact_7516_dim__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_dim @ A )
= ( ^ [V4: set @ A] :
( if @ nat
@ ? [B5: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ B5 )
& ( ( real_Vector_span @ A @ B5 )
= ( real_Vector_span @ A @ V4 ) ) )
@ ( finite_card @ A
@ ( fChoice @ ( set @ A )
@ ^ [B5: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ B5 )
& ( ( real_Vector_span @ A @ B5 )
= ( real_Vector_span @ A @ V4 ) ) ) ) )
@ ( zero_zero @ nat ) ) ) ) ) ).
% dim_def
thf(fact_7517_dim__le__card_H,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ord_less_eq @ nat @ ( real_Vector_dim @ A @ S3 ) @ ( finite_card @ A @ S3 ) ) ) ) ).
% dim_le_card'
thf(fact_7518_dim__unique,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A,V: set @ A,N: nat] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ V )
=> ( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B2 ) )
=> ( ~ ( real_V358717886546972837endent @ A @ B2 )
=> ( ( ( finite_card @ A @ B2 )
= N )
=> ( ( real_Vector_dim @ A @ V )
= N ) ) ) ) ) ) ).
% dim_unique
thf(fact_7519_basis__exists,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [V: set @ A] :
~ ! [B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ V )
=> ( ~ ( real_V358717886546972837endent @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B4 ) )
=> ( ( finite_card @ A @ B4 )
!= ( real_Vector_dim @ A @ V ) ) ) ) ) ) ).
% basis_exists
thf(fact_7520_basis__card__eq__dim,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A,V: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ V )
=> ( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B2 ) )
=> ( ~ ( real_V358717886546972837endent @ A @ B2 )
=> ( ( finite_card @ A @ B2 )
= ( real_Vector_dim @ A @ V ) ) ) ) ) ) ).
% basis_card_eq_dim
thf(fact_7521_dim__le__card,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [V: set @ A,W4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ W4 ) )
=> ( ( finite_finite2 @ A @ W4 )
=> ( ord_less_eq @ nat @ ( real_Vector_dim @ A @ V ) @ ( finite_card @ A @ W4 ) ) ) ) ) ).
% dim_le_card
thf(fact_7522_span__card__ge__dim,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A,V: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ V )
=> ( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B2 ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ nat @ ( real_Vector_dim @ A @ V ) @ ( finite_card @ A @ B2 ) ) ) ) ) ) ).
% span_card_ge_dim
thf(fact_7523_construct__outside,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [B2: set @ A,V2: A,F3: A > B] :
( ~ ( real_V358717886546972837endent @ A @ B2 )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ ( real_V4986007116245087402_basis @ A @ B2 ) @ B2 ) ) )
=> ( ( real_V4425403222259421789struct @ A @ B @ B2 @ F3 @ V2 )
= ( zero_zero @ B ) ) ) ) ) ).
% construct_outside
thf(fact_7524_linear__indep__image__lemma,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F3: A > B,B2: set @ A,X: A] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ~ ( real_V358717886546972837endent @ B @ ( image2 @ A @ B @ F3 @ B2 ) )
=> ( ( inj_on @ A @ B @ F3 @ B2 )
=> ( ( member @ A @ X @ ( real_Vector_span @ A @ B2 ) )
=> ( ( ( F3 @ X )
= ( zero_zero @ B ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ) ) ) ) ).
% linear_indep_image_lemma
thf(fact_7525_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_7526_linear__eq__0__on__span,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [F3: A > B,B3: set @ A,X: A] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B3 )
=> ( ( F3 @ X3 )
= ( zero_zero @ B ) ) )
=> ( ( member @ A @ X @ ( real_Vector_span @ A @ B3 ) )
=> ( ( F3 @ X )
= ( zero_zero @ B ) ) ) ) ) ) ).
% linear_eq_0_on_span
thf(fact_7527_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_7528_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_7529_module__hom__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ( real_Vector_linear @ A @ B
@ ^ [X2: A] : ( zero_zero @ B ) ) ) ).
% module_hom_zero
thf(fact_7530_linear__0,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [F3: A > B] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ( F3 @ ( zero_zero @ A ) )
= ( zero_zero @ B ) ) ) ) ).
% linear_0
thf(fact_7531_linear__injective__0,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F3: A > B] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
= ( ! [X2: A] :
( ( ( F3 @ X2 )
= ( zero_zero @ B ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% linear_injective_0
thf(fact_7532_linear__spans__image,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F3: A > B,V: set @ A,B2: set @ A] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ V ) @ ( real_Vector_span @ B @ ( image2 @ A @ B @ F3 @ B2 ) ) ) ) ) ) ).
% linear_spans_image
thf(fact_7533_linear__surj__right__inverse,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [F3: A > B,T4: set @ B,S: set @ A] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( real_Vector_span @ B @ T4 ) @ ( image2 @ A @ B @ F3 @ ( real_Vector_span @ A @ S ) ) )
=> ? [G9: B > A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G9 @ ( top_top @ ( set @ B ) ) ) @ ( real_Vector_span @ A @ S ) )
& ( real_Vector_linear @ B @ A @ G9 )
& ! [X5: B] :
( ( member @ B @ X5 @ ( real_Vector_span @ B @ T4 ) )
=> ( ( F3 @ ( G9 @ X5 ) )
= X5 ) ) ) ) ) ) ).
% linear_surj_right_inverse
thf(fact_7534_linear__spanning__surjective__image,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F3: A > B,S: set @ A] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( real_Vector_span @ A @ S ) )
=> ( ( ( image2 @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( top_top @ ( set @ B ) ) @ ( real_Vector_span @ B @ ( image2 @ A @ B @ F3 @ S ) ) ) ) ) ) ) ).
% linear_spanning_surjective_image
thf(fact_7535_linear__inj__on__left__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F3: A > B,S: set @ A] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ( inj_on @ A @ B @ F3 @ ( real_Vector_span @ A @ S ) )
=> ? [G9: B > A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G9 @ ( top_top @ ( set @ B ) ) ) @ ( real_Vector_span @ A @ S ) )
& ( real_Vector_linear @ B @ A @ G9 )
& ! [X5: A] :
( ( member @ A @ X5 @ ( real_Vector_span @ A @ S ) )
=> ( ( G9 @ ( F3 @ X5 ) )
= X5 ) ) ) ) ) ) ).
% linear_inj_on_left_inverse
thf(fact_7536_finite__basis__to__basis__subspace__isomorphism,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [S: set @ A,T4: set @ B,B2: set @ A,C2: set @ B] :
( ( real_Vector_subspace @ A @ S )
=> ( ( real_Vector_subspace @ B @ T4 )
=> ( ( ( real_Vector_dim @ A @ S )
= ( real_Vector_dim @ B @ T4 ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ S )
=> ( ~ ( real_V358717886546972837endent @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ ( real_Vector_span @ A @ B2 ) )
=> ( ( ( finite_card @ A @ B2 )
= ( real_Vector_dim @ A @ S ) )
=> ( ( finite_finite2 @ B @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ C2 @ T4 )
=> ( ~ ( real_V358717886546972837endent @ B @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ T4 @ ( real_Vector_span @ B @ C2 ) )
=> ( ( ( finite_card @ B @ C2 )
= ( real_Vector_dim @ B @ T4 ) )
=> ? [F6: A > B] :
( ( real_Vector_linear @ A @ B @ F6 )
& ( ( image2 @ A @ B @ F6 @ B2 )
= C2 )
& ( ( image2 @ A @ B @ F6 @ S )
= T4 )
& ( inj_on @ A @ B @ F6 @ S ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% finite_basis_to_basis_subspace_isomorphism
thf(fact_7537_linear__exists__left__inverse__on,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F3: A > B,V: set @ A] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ( real_Vector_subspace @ A @ V )
=> ( ( inj_on @ A @ B @ F3 @ V )
=> ? [G9: B > A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G9 @ ( top_top @ ( set @ B ) ) ) @ V )
& ( real_Vector_linear @ B @ A @ G9 )
& ! [X5: A] :
( ( member @ A @ X5 @ V )
=> ( ( G9 @ ( F3 @ X5 ) )
= X5 ) ) ) ) ) ) ) ).
% linear_exists_left_inverse_on
thf(fact_7538_linear__subspace__kernel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F3: A > B] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( real_Vector_subspace @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( F3 @ X2 )
= ( zero_zero @ B ) ) ) ) ) ) ).
% linear_subspace_kernel
thf(fact_7539_subspace__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A] :
( ( real_Vector_subspace @ A @ S )
=> ( member @ A @ ( zero_zero @ A ) @ S ) ) ) ).
% subspace_0
thf(fact_7540_span__subspace,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( real_Vector_span @ A @ A4 ) )
=> ( ( real_Vector_subspace @ A @ B2 )
=> ( ( real_Vector_span @ A @ A4 )
= B2 ) ) ) ) ) ).
% span_subspace
thf(fact_7541_span__minimal,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A,T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T4 )
=> ( ( real_Vector_subspace @ A @ T4 )
=> ( ord_less_eq @ ( set @ A ) @ ( real_Vector_span @ A @ S ) @ T4 ) ) ) ) ).
% span_minimal
thf(fact_7542_span__unique,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A,T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T4 )
=> ( ( real_Vector_subspace @ A @ T4 )
=> ( ! [T14: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T14 )
=> ( ( real_Vector_subspace @ A @ T14 )
=> ( ord_less_eq @ ( set @ A ) @ T4 @ T14 ) ) )
=> ( ( real_Vector_span @ A @ S )
= T4 ) ) ) ) ) ).
% span_unique
thf(fact_7543_subspace__single__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( real_Vector_subspace @ A @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% subspace_single_0
thf(fact_7544_subspace__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_subspace @ A )
= ( ^ [S7: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ S7 )
& ! [X2: A] :
( ( member @ A @ X2 @ S7 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ S7 )
=> ( member @ A @ ( plus_plus @ A @ X2 @ Y3 ) @ S7 ) ) )
& ! [C6: real,X2: A] :
( ( member @ A @ X2 @ S7 )
=> ( member @ A @ ( real_V8093663219630862766scaleR @ A @ C6 @ X2 ) @ S7 ) ) ) ) ) ) ).
% subspace_def
thf(fact_7545_subspaceI,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ S )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ S )
=> ( ( member @ A @ Y2 @ S )
=> ( member @ A @ ( plus_plus @ A @ X3 @ Y2 ) @ S ) ) )
=> ( ! [C5: real,X3: A] :
( ( member @ A @ X3 @ S )
=> ( member @ A @ ( real_V8093663219630862766scaleR @ A @ C5 @ X3 ) @ S ) )
=> ( real_Vector_subspace @ A @ S ) ) ) ) ) ).
% subspaceI
thf(fact_7546_linear__injective__on__subspace__0,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F3: A > B,S3: set @ A] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ( real_Vector_subspace @ A @ S3 )
=> ( ( inj_on @ A @ B @ F3 @ S3 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ S3 )
=> ( ( ( F3 @ X2 )
= ( zero_zero @ B ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ) ) ) ) ).
% linear_injective_on_subspace_0
thf(fact_7547_linear__exists__right__inverse__on,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [F3: A > B,V: set @ A] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ( real_Vector_subspace @ A @ V )
=> ? [G9: B > A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G9 @ ( top_top @ ( set @ B ) ) ) @ V )
& ( real_Vector_linear @ B @ A @ G9 )
& ! [X5: B] :
( ( member @ B @ X5 @ ( image2 @ A @ B @ F3 @ V ) )
=> ( ( F3 @ ( G9 @ X5 ) )
= X5 ) ) ) ) ) ) ).
% linear_exists_right_inverse_on
thf(fact_7548_less__eq__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V6: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V6: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_7549_less__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V6: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V6: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% less_int.rsp
thf(fact_7550_zero__int_Orsp,axiom,
intrel @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ).
% zero_int.rsp
thf(fact_7551_one__int_Orsp,axiom,
intrel @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ).
% one_int.rsp
thf(fact_7552_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_7553_less__eq__enat__def,axiom,
( ( ord_less_eq @ extended_enat )
= ( ^ [M2: extended_enat] :
( extended_case_enat @ $o
@ ^ [N1: nat] :
( extended_case_enat @ $o
@ ^ [M12: nat] : ( ord_less_eq @ nat @ M12 @ N1 )
@ $false
@ M2 )
@ $true ) ) ) ).
% less_eq_enat_def
thf(fact_7554_size__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( zero_zero @ A ) )
= ( zero_zero @ nat ) ) ) ).
% size_0
thf(fact_7555_euclidean__size__eq__0__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A] :
( ( ( euclid6346220572633701492n_size @ A @ B3 )
= ( zero_zero @ nat ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% euclidean_size_eq_0_iff
thf(fact_7556_euclidean__size__greater__0__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
= ( B3
!= ( zero_zero @ A ) ) ) ) ).
% euclidean_size_greater_0_iff
thf(fact_7557_dvd__euclidean__size__eq__imp__dvd,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( dvd_dvd @ A @ B3 @ A3 )
=> ( dvd_dvd @ A @ A3 @ B3 ) ) ) ) ) ).
% dvd_euclidean_size_eq_imp_dvd
thf(fact_7558_unit__iff__euclidean__size,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
= ( ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) )
& ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_iff_euclidean_size
thf(fact_7559_size__mult__mono,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% size_mult_mono
thf(fact_7560_size__mult__mono_H,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ B3 @ A3 ) ) ) ) ) ).
% size_mult_mono'
thf(fact_7561_dvd__proper__imp__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ~ ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) ) ) ) ) ) ).
% dvd_proper_imp_size_less
thf(fact_7562_dvd__imp__size__le,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) ) ) ) ) ).
% dvd_imp_size_le
thf(fact_7563_mod__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ ( modulo_modulo @ A @ A3 @ B3 ) ) @ ( euclid6346220572633701492n_size @ A @ B3 ) ) ) ) ).
% mod_size_less
thf(fact_7564_less__enat__def,axiom,
( ( ord_less @ extended_enat )
= ( ^ [M2: extended_enat,N2: extended_enat] :
( extended_case_enat @ $o
@ ^ [M12: nat] : ( extended_case_enat @ $o @ ( ord_less @ nat @ M12 ) @ $true @ N2 )
@ $false
@ M2 ) ) ) ).
% less_enat_def
thf(fact_7565_divmod__cases,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,A3: A] :
( ( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( modulo_modulo @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
=> ( A3
!= ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ B3 ) ) ) )
=> ( ( ( B3
!= ( zero_zero @ A ) )
=> ! [Q3: A,R3: A] :
( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( R3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B3 )
= Q3 )
=> ( ( ( modulo_modulo @ A @ A3 @ B3 )
= R3 )
=> ( A3
!= ( plus_plus @ A @ ( times_times @ A @ Q3 @ B3 ) @ R3 ) ) ) ) ) ) ) )
=> ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divmod_cases
thf(fact_7566_mod__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,R2: A,Q5: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q5 @ B3 ) @ R2 )
= A3 )
=> ( ( modulo_modulo @ A @ A3 @ B3 )
= R2 ) ) ) ) ) ) ).
% mod_eqI
thf(fact_7567_division__segment__not__0,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A] :
( ( euclid7384307370059645450egment @ A @ A3 )
!= ( zero_zero @ A ) ) ) ).
% division_segment_not_0
thf(fact_7568_division__segment__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( euclid7384307370059645450egment @ A @ ( times_times @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( euclid7384307370059645450egment @ A @ B3 ) ) ) ) ) ) ).
% division_segment_mult
thf(fact_7569_division__segment__mod,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( euclid7384307370059645450egment @ A @ ( modulo_modulo @ A @ A3 @ B3 ) )
= ( euclid7384307370059645450egment @ A @ B3 ) ) ) ) ) ).
% division_segment_mod
thf(fact_7570_division__segment__int__def,axiom,
( ( euclid7384307370059645450egment @ int )
= ( ^ [K3: int] : ( if @ int @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ K3 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% division_segment_int_def
thf(fact_7571_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B3: A] :
( ( ( euclid7384307370059645450egment @ A @ A3 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ) ).
% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_7572_div__bounded,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,R2: A,Q5: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q5 @ B3 ) @ R2 ) @ B3 )
= Q5 ) ) ) ) ) ).
% div_bounded
thf(fact_7573_div__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,R2: A,Q5: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q5 @ B3 ) @ R2 )
= A3 )
=> ( ( divide_divide @ A @ A3 @ B3 )
= Q5 ) ) ) ) ) ) ).
% div_eqI
thf(fact_7574_sorted__wrt__iff__nth__Suc__transp,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( transp @ A @ P )
=> ( ( sorted_wrt @ A @ P @ Xs )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ ( suc @ I4 ) ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_Suc_transp
thf(fact_7575_total__inv__image,axiom,
! [B: $tType,A: $tType,F3: A > B,R2: set @ ( product_prod @ B @ B )] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ R2 )
=> ( total_on @ A @ ( top_top @ ( set @ A ) ) @ ( inv_image @ B @ A @ R2 @ F3 ) ) ) ) ).
% total_inv_image
thf(fact_7576_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R2: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R2 @ F3 ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_7577_converse__inv__image,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( converse @ A @ A @ ( inv_image @ B @ A @ R @ F3 ) )
= ( inv_image @ B @ A @ ( converse @ B @ B @ R ) @ F3 ) ) ).
% converse_inv_image
thf(fact_7578_transp__INF,axiom,
! [B: $tType,A: $tType,S: set @ A,R2: A > B > B > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( transp @ B @ ( R2 @ X3 ) ) )
=> ( transp @ B @ ( complete_Inf_Inf @ ( B > B > $o ) @ ( image2 @ A @ ( B > B > $o ) @ R2 @ S ) ) ) ) ).
% transp_INF
thf(fact_7579_transp__trans__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( transp @ A
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) )
= ( trans @ A @ R2 ) ) ).
% transp_trans_eq
thf(fact_7580_transp__trans,axiom,
! [A: $tType] :
( ( transp @ A )
= ( ^ [R5: A > A > $o] : ( trans @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ).
% transp_trans
thf(fact_7581_trans__inv__image,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),F3: B > A] :
( ( trans @ A @ R2 )
=> ( trans @ B @ ( inv_image @ A @ B @ R2 @ F3 ) ) ) ).
% trans_inv_image
thf(fact_7582_transp__inf,axiom,
! [A: $tType,R2: A > A > $o,S3: A > A > $o] :
( ( transp @ A @ R2 )
=> ( ( transp @ A @ S3 )
=> ( transp @ A @ ( inf_inf @ ( A > A > $o ) @ R2 @ S3 ) ) ) ) ).
% transp_inf
thf(fact_7583_transp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less @ A ) ) ) ).
% transp_less
thf(fact_7584_transpD,axiom,
! [A: $tType,R2: A > A > $o,X: A,Y: A,Z: A] :
( ( transp @ A @ R2 )
=> ( ( R2 @ X @ Y )
=> ( ( R2 @ Y @ Z )
=> ( R2 @ X @ Z ) ) ) ) ).
% transpD
thf(fact_7585_transpE,axiom,
! [A: $tType,R2: A > A > $o,X: A,Y: A,Z: A] :
( ( transp @ A @ R2 )
=> ( ( R2 @ X @ Y )
=> ( ( R2 @ Y @ Z )
=> ( R2 @ X @ Z ) ) ) ) ).
% transpE
thf(fact_7586_transpI,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [X3: A,Y2: A,Z3: A] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ Z3 )
=> ( R2 @ X3 @ Z3 ) ) )
=> ( transp @ A @ R2 ) ) ).
% transpI
thf(fact_7587_transp__def,axiom,
! [A: $tType] :
( ( transp @ A )
= ( ^ [R5: A > A > $o] :
! [X2: A,Y3: A,Z6: A] :
( ( R5 @ X2 @ Y3 )
=> ( ( R5 @ Y3 @ Z6 )
=> ( R5 @ X2 @ Z6 ) ) ) ) ) ).
% transp_def
thf(fact_7588_transp__equality,axiom,
! [A: $tType] :
( transp @ A
@ ^ [Y4: A,Z2: A] : Y4 = Z2 ) ).
% transp_equality
thf(fact_7589_transp__empty,axiom,
! [A: $tType] :
( transp @ A
@ ^ [X2: A,Y3: A] : $false ) ).
% transp_empty
thf(fact_7590_transp__singleton,axiom,
! [A: $tType,A3: A] :
( transp @ A
@ ^ [X2: A,Y3: A] :
( ( X2 = A3 )
& ( Y3 = A3 ) ) ) ).
% transp_singleton
thf(fact_7591_transp__gr,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X2: A,Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% transp_gr
thf(fact_7592_transp__ge,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% transp_ge
thf(fact_7593_transp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less_eq @ A ) ) ) ).
% transp_le
thf(fact_7594_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ B ),F2: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ R5 ) ) ) ) ) ).
% inv_image_def
thf(fact_7595_of__rat__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ rat @ R2 @ ( one_one @ rat ) ) ) ) ).
% of_rat_le_1_iff
thf(fact_7596_one__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R2 ) )
= ( ord_less_eq @ rat @ ( one_one @ rat ) @ R2 ) ) ) ).
% one_le_of_rat_iff
thf(fact_7597_zero__eq__of__rat__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( zero_zero @ A )
= ( field_char_0_of_rat @ A @ A3 ) )
= ( ( zero_zero @ rat )
= A3 ) ) ) ).
% zero_eq_of_rat_iff
thf(fact_7598_of__rat__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( field_char_0_of_rat @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ rat ) ) ) ) ).
% of_rat_eq_0_iff
thf(fact_7599_of__rat__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A @ ( zero_zero @ rat ) )
= ( zero_zero @ A ) ) ) ).
% of_rat_0
thf(fact_7600_of__rat__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( zero_zero @ A ) )
= ( ord_less @ rat @ R2 @ ( zero_zero @ rat ) ) ) ) ).
% of_rat_less_0_iff
thf(fact_7601_zero__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R2 ) )
= ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 ) ) ) ).
% zero_less_of_rat_iff
thf(fact_7602_one__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R2 ) )
= ( ord_less @ rat @ ( one_one @ rat ) @ R2 ) ) ) ).
% one_less_of_rat_iff
thf(fact_7603_of__rat__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( one_one @ A ) )
= ( ord_less @ rat @ R2 @ ( one_one @ rat ) ) ) ) ).
% of_rat_less_1_iff
thf(fact_7604_zero__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R2 ) )
= ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ R2 ) ) ) ).
% zero_le_of_rat_iff
thf(fact_7605_of__rat__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ rat @ R2 @ ( zero_zero @ rat ) ) ) ) ).
% of_rat_le_0_iff
thf(fact_7606_of__rat__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat,S3: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( field_char_0_of_rat @ A @ S3 ) )
= ( ord_less_eq @ rat @ R2 @ S3 ) ) ) ).
% of_rat_less_eq
thf(fact_7607_of__rat__dense,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ? [Q3: rat] :
( ( ord_less @ real @ X @ ( field_char_0_of_rat @ real @ Q3 ) )
& ( ord_less @ real @ ( field_char_0_of_rat @ real @ Q3 ) @ Y ) ) ) ).
% of_rat_dense
thf(fact_7608_of__rat__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat,S3: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( field_char_0_of_rat @ A @ S3 ) )
= ( ord_less @ rat @ R2 @ S3 ) ) ) ).
% of_rat_less
thf(fact_7609_less__RealD,axiom,
! [Y6: nat > rat,X: real] :
( ( cauchy @ Y6 )
=> ( ( ord_less @ real @ X @ ( real2 @ Y6 ) )
=> ? [N3: nat] : ( ord_less @ real @ X @ ( field_char_0_of_rat @ real @ ( Y6 @ N3 ) ) ) ) ) ).
% less_RealD
thf(fact_7610_le__RealI,axiom,
! [Y6: nat > rat,X: real] :
( ( cauchy @ Y6 )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ X @ ( field_char_0_of_rat @ real @ ( Y6 @ N3 ) ) )
=> ( ord_less_eq @ real @ X @ ( real2 @ Y6 ) ) ) ) ).
% le_RealI
thf(fact_7611_Real__leI,axiom,
! [X4: nat > rat,Y: real] :
( ( cauchy @ X4 )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( field_char_0_of_rat @ real @ ( X4 @ N3 ) ) @ Y )
=> ( ord_less_eq @ real @ ( real2 @ X4 ) @ Y ) ) ) ).
% Real_leI
thf(fact_7612_sym__trans__comp__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( sym @ A @ R2 )
=> ( ( trans @ A @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( converse @ A @ A @ R2 ) @ R2 ) @ R2 ) ) ) ).
% sym_trans_comp_subset
thf(fact_7613_bounded__bilinear__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ( ( real_V2442710119149674383linear @ A @ B @ C )
= ( ^ [Prod: A > B > C] :
( ! [A5: A,A28: A,B5: B] :
( ( Prod @ ( plus_plus @ A @ A5 @ A28 ) @ B5 )
= ( plus_plus @ C @ ( Prod @ A5 @ B5 ) @ ( Prod @ A28 @ B5 ) ) )
& ! [A5: A,B5: B,B17: B] :
( ( Prod @ A5 @ ( plus_plus @ B @ B5 @ B17 ) )
= ( plus_plus @ C @ ( Prod @ A5 @ B5 ) @ ( Prod @ A5 @ B17 ) ) )
& ! [R5: real,A5: A,B5: B] :
( ( Prod @ ( real_V8093663219630862766scaleR @ A @ R5 @ A5 ) @ B5 )
= ( real_V8093663219630862766scaleR @ C @ R5 @ ( Prod @ A5 @ B5 ) ) )
& ! [A5: A,R5: real,B5: B] :
( ( Prod @ A5 @ ( real_V8093663219630862766scaleR @ B @ R5 @ B5 ) )
= ( real_V8093663219630862766scaleR @ C @ R5 @ ( Prod @ A5 @ B5 ) ) )
& ? [K5: real] :
! [A5: A,B5: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod @ A5 @ B5 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A5 ) @ ( real_V7770717601297561774m_norm @ B @ B5 ) ) @ K5 ) ) ) ) ) ) ).
% bounded_bilinear_def
thf(fact_7614_sym__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( sym @ A @ ( converse @ A @ A @ R2 ) )
= ( sym @ A @ R2 ) ) ).
% sym_converse
thf(fact_7615_sym__UNION,axiom,
! [B: $tType,A: $tType,S: set @ A,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( sym @ B @ ( R2 @ X3 ) ) )
=> ( sym @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S ) ) ) ) ).
% sym_UNION
thf(fact_7616_symD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B3: A,A3: A] :
( ( sym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 ) ) ) ).
% symD
thf(fact_7617_symE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B3: A,A3: A] :
( ( sym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 ) ) ) ).
% symE
thf(fact_7618_symI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [A7: A,B7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B7 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B7 @ A7 ) @ R2 ) )
=> ( sym @ A @ R2 ) ) ).
% symI
thf(fact_7619_sym__def,axiom,
! [A: $tType] :
( ( sym @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X2: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R5 ) ) ) ) ).
% sym_def
thf(fact_7620_sym__Un,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( sym @ A @ R2 )
=> ( ( sym @ A @ S3 )
=> ( sym @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ).
% sym_Un
thf(fact_7621_sym__Int,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( sym @ A @ R2 )
=> ( ( sym @ A @ S3 )
=> ( sym @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ).
% sym_Int
thf(fact_7622_sym__Id__on,axiom,
! [A: $tType,A4: set @ A] : ( sym @ A @ ( id_on @ A @ A4 ) ) ).
% sym_Id_on
thf(fact_7623_sym__Id,axiom,
! [A: $tType] : ( sym @ A @ ( id2 @ A ) ) ).
% sym_Id
thf(fact_7624_bounded__bilinear_Ozero__right,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [Prod2: A > B > C,A3: A] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ( ( Prod2 @ A3 @ ( zero_zero @ B ) )
= ( zero_zero @ C ) ) ) ) ).
% bounded_bilinear.zero_right
thf(fact_7625_bounded__bilinear_Ozero__left,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [Prod2: A > B > C,B3: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ( ( Prod2 @ ( zero_zero @ A ) @ B3 )
= ( zero_zero @ C ) ) ) ) ).
% bounded_bilinear.zero_left
thf(fact_7626_sym__conv__converse__eq,axiom,
! [A: $tType] :
( ( sym @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( ( converse @ A @ A @ R5 )
= R5 ) ) ) ).
% sym_conv_converse_eq
thf(fact_7627_sym__Int__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( sym @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( converse @ A @ A @ R2 ) ) ) ).
% sym_Int_converse
thf(fact_7628_sym__Un__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( sym @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( converse @ A @ A @ R2 ) ) ) ).
% sym_Un_converse
thf(fact_7629_sym__inv__image,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),F3: B > A] :
( ( sym @ A @ R2 )
=> ( sym @ B @ ( inv_image @ A @ B @ R2 @ F3 ) ) ) ).
% sym_inv_image
thf(fact_7630_bounded__bilinear_Obounded,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod2: A > B > C] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ? [K9: real] :
! [A10: A,B11: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod2 @ A10 @ B11 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A10 ) @ ( real_V7770717601297561774m_norm @ B @ B11 ) ) @ K9 ) ) ) ) ).
% bounded_bilinear.bounded
thf(fact_7631_sym__INTER,axiom,
! [B: $tType,A: $tType,S: set @ A,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( sym @ B @ ( R2 @ X3 ) ) )
=> ( sym @ B @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S ) ) ) ) ).
% sym_INTER
thf(fact_7632_bounded__bilinear_Otendsto__right__zero,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod2: A > B > C,F3: D > B,F4: filter @ D,C3: A] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ( ( filterlim @ D @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ D @ C
@ ^ [X2: D] : ( Prod2 @ C3 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) ) ) ) ).
% bounded_bilinear.tendsto_right_zero
thf(fact_7633_bounded__bilinear_Otendsto__left__zero,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod2: A > B > C,F3: D > A,F4: filter @ D,C3: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D @ C
@ ^ [X2: D] : ( Prod2 @ ( F3 @ X2 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) ) ) ) ).
% bounded_bilinear.tendsto_left_zero
thf(fact_7634_bounded__bilinear_Otendsto__zero,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod2: A > B > C,F3: D > A,F4: filter @ D,G2: D > B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( ( filterlim @ D @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ D @ C
@ ^ [X2: D] : ( Prod2 @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) ) ) ) ) ).
% bounded_bilinear.tendsto_zero
thf(fact_7635_bounded__bilinear_Ononneg__bounded,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod2: A > B > C] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ? [K9: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ K9 )
& ! [A10: A,B11: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod2 @ A10 @ B11 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A10 ) @ ( real_V7770717601297561774m_norm @ B @ B11 ) ) @ K9 ) ) ) ) ) ).
% bounded_bilinear.nonneg_bounded
thf(fact_7636_bounded__bilinear_Opos__bounded,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod2: A > B > C] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ? [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
& ! [A10: A,B11: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod2 @ A10 @ B11 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A10 ) @ ( real_V7770717601297561774m_norm @ B @ B11 ) ) @ K9 ) ) ) ) ) ).
% bounded_bilinear.pos_bounded
thf(fact_7637_bounded__bilinear_Ointro,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod2: A > B > C] :
( ! [A7: A,A27: A,B7: B] :
( ( Prod2 @ ( plus_plus @ A @ A7 @ A27 ) @ B7 )
= ( plus_plus @ C @ ( Prod2 @ A7 @ B7 ) @ ( Prod2 @ A27 @ B7 ) ) )
=> ( ! [A7: A,B7: B,B15: B] :
( ( Prod2 @ A7 @ ( plus_plus @ B @ B7 @ B15 ) )
= ( plus_plus @ C @ ( Prod2 @ A7 @ B7 ) @ ( Prod2 @ A7 @ B15 ) ) )
=> ( ! [R3: real,A7: A,B7: B] :
( ( Prod2 @ ( real_V8093663219630862766scaleR @ A @ R3 @ A7 ) @ B7 )
= ( real_V8093663219630862766scaleR @ C @ R3 @ ( Prod2 @ A7 @ B7 ) ) )
=> ( ! [A7: A,R3: real,B7: B] :
( ( Prod2 @ A7 @ ( real_V8093663219630862766scaleR @ B @ R3 @ B7 ) )
= ( real_V8093663219630862766scaleR @ C @ R3 @ ( Prod2 @ A7 @ B7 ) ) )
=> ( ? [K6: real] :
! [A7: A,B7: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod2 @ A7 @ B7 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A7 ) @ ( real_V7770717601297561774m_norm @ B @ B7 ) ) @ K6 ) )
=> ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 ) ) ) ) ) ) ) ).
% bounded_bilinear.intro
thf(fact_7638_coinduct3__lemma,axiom,
! [A: $tType,X4: set @ A,F3: ( set @ A ) > ( set @ A )] :
( ( ord_less_eq @ ( set @ A ) @ X4
@ ( F3
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X2: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F3 @ X2 ) @ X4 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ord_less_eq @ ( set @ A )
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X2: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F3 @ X2 ) @ X4 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) )
@ ( F3
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X2: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F3 @ X2 ) @ X4 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) ) ) ) ).
% coinduct3_lemma
thf(fact_7639_def__coinduct3,axiom,
! [A: $tType,A4: set @ A,F3: ( set @ A ) > ( set @ A ),A3: A,X4: set @ A] :
( ( A4
= ( complete_lattice_gfp @ ( set @ A ) @ F3 ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ( member @ A @ A3 @ X4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4
@ ( F3
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X2: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F3 @ X2 ) @ X4 ) @ A4 ) ) ) )
=> ( member @ A @ A3 @ A4 ) ) ) ) ) ).
% def_coinduct3
thf(fact_7640_gfp__gfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A > A] :
( ! [X3: A,Y2: A,W: A,Z3: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ W @ Z3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 @ W ) @ ( F3 @ Y2 @ Z3 ) ) ) )
=> ( ( complete_lattice_gfp @ A
@ ^ [X2: A] : ( complete_lattice_gfp @ A @ ( F3 @ X2 ) ) )
= ( complete_lattice_gfp @ A
@ ^ [X2: A] : ( F3 @ X2 @ X2 ) ) ) ) ) ).
% gfp_gfp
thf(fact_7641_weak__coinduct,axiom,
! [A: $tType,A3: A,X4: set @ A,F3: ( set @ A ) > ( set @ A )] :
( ( member @ A @ A3 @ X4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ ( F3 @ X4 ) )
=> ( member @ A @ A3 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) ).
% weak_coinduct
thf(fact_7642_gfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,G2: A > A] :
( ! [Z10: A] : ( ord_less_eq @ A @ ( F3 @ Z10 ) @ ( G2 @ Z10 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F3 ) @ ( complete_lattice_gfp @ A @ G2 ) ) ) ) ).
% gfp_mono
thf(fact_7643_gfp__upperbound,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X4: A,F3: A > A] :
( ( ord_less_eq @ A @ X4 @ ( F3 @ X4 ) )
=> ( ord_less_eq @ A @ X4 @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ).
% gfp_upperbound
thf(fact_7644_gfp__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,X4: A] :
( ! [U4: A] :
( ( ord_less_eq @ A @ U4 @ ( F3 @ U4 ) )
=> ( ord_less_eq @ A @ U4 @ X4 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F3 ) @ X4 ) ) ) ).
% gfp_least
thf(fact_7645_gfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F4: A > A,X: A] :
( ( order_mono @ A @ A @ F4 )
=> ( ( ( F4 @ X )
= X )
=> ( ! [Z3: A] :
( ( ( F4 @ Z3 )
= Z3 )
=> ( ord_less_eq @ A @ Z3 @ X ) )
=> ( ( complete_lattice_gfp @ A @ F4 )
= X ) ) ) ) ) ).
% gfp_eqI
thf(fact_7646_gfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_gfp @ A )
= ( ^ [F2: A > A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [U2: A] : ( ord_less_eq @ A @ U2 @ ( F2 @ U2 ) ) ) ) ) ) ) ).
% gfp_def
thf(fact_7647_def__Collect__coinduct,axiom,
! [A: $tType,A4: set @ A,P: ( set @ A ) > A > $o,A3: A,X4: set @ A] :
( ( A4
= ( complete_lattice_gfp @ ( set @ A )
@ ^ [W3: set @ A] : ( collect @ A @ ( P @ W3 ) ) ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A )
@ ^ [W3: set @ A] : ( collect @ A @ ( P @ W3 ) ) )
=> ( ( member @ A @ A3 @ X4 )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ X4 )
=> ( P @ ( sup_sup @ ( set @ A ) @ X4 @ A4 ) @ Z3 ) )
=> ( member @ A @ A3 @ A4 ) ) ) ) ) ).
% def_Collect_coinduct
thf(fact_7648_gfp__fun__UnI2,axiom,
! [A: $tType,F3: ( set @ A ) > ( set @ A ),A3: A,X4: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ( member @ A @ A3 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) )
=> ( member @ A @ A3 @ ( F3 @ ( sup_sup @ ( set @ A ) @ X4 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) ) ) ).
% gfp_fun_UnI2
thf(fact_7649_weak__coinduct__image,axiom,
! [A: $tType,B: $tType,A3: A,X4: set @ A,G2: A > B,F3: ( set @ B ) > ( set @ B )] :
( ( member @ A @ A3 @ X4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G2 @ X4 ) @ ( F3 @ ( image2 @ A @ B @ G2 @ X4 ) ) )
=> ( member @ B @ ( G2 @ A3 ) @ ( complete_lattice_gfp @ ( set @ B ) @ F3 ) ) ) ) ).
% weak_coinduct_image
thf(fact_7650_coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,X4: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ X4 @ ( F3 @ ( sup_sup @ A @ X4 @ ( complete_lattice_gfp @ A @ F3 ) ) ) )
=> ( ord_less_eq @ A @ X4 @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ) ).
% coinduct
thf(fact_7651_def__coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: A,F3: A > A,X4: A] :
( ( A4
= ( complete_lattice_gfp @ A @ F3 ) )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ X4 @ ( F3 @ ( sup_sup @ A @ X4 @ A4 ) ) )
=> ( ord_less_eq @ A @ X4 @ A4 ) ) ) ) ) ).
% def_coinduct
thf(fact_7652_coinduct__lemma,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X4: A,F3: A > A] :
( ( ord_less_eq @ A @ X4 @ ( F3 @ ( sup_sup @ A @ X4 @ ( complete_lattice_gfp @ A @ F3 ) ) ) )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ X4 @ ( complete_lattice_gfp @ A @ F3 ) ) @ ( F3 @ ( sup_sup @ A @ X4 @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ) ) ) ).
% coinduct_lemma
thf(fact_7653_coinduct__set,axiom,
! [A: $tType,F3: ( set @ A ) > ( set @ A ),A3: A,X4: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ( member @ A @ A3 @ X4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ ( F3 @ ( sup_sup @ ( set @ A ) @ X4 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) )
=> ( member @ A @ A3 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) ) ).
% coinduct_set
thf(fact_7654_def__coinduct__set,axiom,
! [A: $tType,A4: set @ A,F3: ( set @ A ) > ( set @ A ),A3: A,X4: set @ A] :
( ( A4
= ( complete_lattice_gfp @ ( set @ A ) @ F3 ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ( member @ A @ A3 @ X4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ ( F3 @ ( sup_sup @ ( set @ A ) @ X4 @ A4 ) ) )
=> ( member @ A @ A3 @ A4 ) ) ) ) ) ).
% def_coinduct_set
thf(fact_7655_gfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,P: A > $o] :
( ( order_mono @ A @ A @ F3 )
=> ( ! [S2: A] :
( ( P @ S2 )
=> ( ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F3 ) @ S2 )
=> ( P @ ( F3 @ S2 ) ) ) )
=> ( ! [M9: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ M9 )
=> ( P @ X5 ) )
=> ( P @ ( complete_Inf_Inf @ A @ M9 ) ) )
=> ( P @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ) ) ).
% gfp_ordinal_induct
thf(fact_7656_gfp__funpow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,N: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( complete_lattice_gfp @ A @ ( compow @ ( A > A ) @ ( suc @ N ) @ F3 ) )
= ( complete_lattice_gfp @ A @ F3 ) ) ) ) ).
% gfp_funpow
thf(fact_7657_lfp__le__gfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A] :
( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F3 ) @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ).
% lfp_le_gfp
thf(fact_7658_gfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,K: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F3 @ ( top_top @ A ) )
= ( compow @ ( A > A ) @ K @ F3 @ ( top_top @ A ) ) )
=> ( ( complete_lattice_gfp @ A @ F3 )
= ( compow @ ( A > A ) @ K @ F3 @ ( top_top @ A ) ) ) ) ) ) ).
% gfp_Kleene_iter
thf(fact_7659_coinduct3,axiom,
! [A: $tType,F3: ( set @ A ) > ( set @ A ),A3: A,X4: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ( member @ A @ A3 @ X4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4
@ ( F3
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X2: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F3 @ X2 ) @ X4 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) )
=> ( member @ A @ A3 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) ) ).
% coinduct3
thf(fact_7660_gfp__transfer__bounded,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [P: A > $o,F3: A > A,Alpha: A > B,G2: B > B] :
( ( P @ ( F3 @ ( top_top @ A ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( P @ ( F3 @ X3 ) ) )
=> ( ! [M9: nat > A] :
( ( order_antimono @ nat @ A @ M9 )
=> ( ! [I3: nat] : ( P @ ( M9 @ I3 ) )
=> ( P @ ( complete_Inf_Inf @ A @ ( image2 @ nat @ A @ M9 @ ( top_top @ ( set @ nat ) ) ) ) ) ) )
=> ( ! [M9: nat > A] :
( ( order_antimono @ nat @ A @ M9 )
=> ( ! [I3: nat] : ( P @ ( M9 @ I3 ) )
=> ( ( Alpha @ ( complete_Inf_Inf @ A @ ( image2 @ nat @ A @ M9 @ ( top_top @ ( set @ nat ) ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image2 @ nat @ B
@ ^ [I4: nat] : ( Alpha @ ( M9 @ I4 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) )
=> ( ( order_inf_continuous @ A @ A @ F3 )
=> ( ( order_inf_continuous @ B @ B @ G2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ( Alpha @ ( F3 @ X3 ) )
= ( G2 @ ( Alpha @ X3 ) ) ) )
=> ( ! [X3: B] : ( ord_less_eq @ B @ ( G2 @ X3 ) @ ( Alpha @ ( F3 @ ( top_top @ A ) ) ) )
=> ( ( Alpha @ ( complete_lattice_gfp @ A @ F3 ) )
= ( complete_lattice_gfp @ B @ G2 ) ) ) ) ) ) ) ) ) ) ) ).
% gfp_transfer_bounded
thf(fact_7661_these__insert__Some,axiom,
! [A: $tType,X: A,A4: set @ ( option @ A )] :
( ( these @ A @ ( insert2 @ ( option @ A ) @ ( some @ A @ X ) @ A4 ) )
= ( insert2 @ A @ X @ ( these @ A @ A4 ) ) ) ).
% these_insert_Some
thf(fact_7662_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_7663_inf__continuous__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( counta4013691401010221786attice @ B ) )
=> ! [P: A > B,Q: A > B] :
( ( order_inf_continuous @ A @ B @ P )
=> ( ( order_inf_continuous @ A @ B @ Q )
=> ( order_inf_continuous @ A @ B
@ ^ [X2: A] : ( sup_sup @ B @ ( P @ X2 ) @ ( Q @ X2 ) ) ) ) ) ) ).
% inf_continuous_sup
thf(fact_7664_these__empty__eq,axiom,
! [A: $tType,B2: set @ ( option @ A )] :
( ( ( these @ A @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B2
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B2
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_7665_these__not__empty__eq,axiom,
! [A: $tType,B2: set @ ( option @ A )] :
( ( ( these @ A @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B2
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B2
!= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_7666_Max_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_max @ A )
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 )
@ ^ [X2: A,Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% Max.semilattice_order_set_axioms
thf(fact_7667_Gcd__fin__def,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( bounde2362111253966948842tice_F @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ) ).
% Gcd_fin_def
thf(fact_7668_Inf__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( lattic4895041142388067077er_set @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_7669_Min_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_min @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% Min.semilattice_order_set_axioms
thf(fact_7670_Sup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic4895041142388067077er_set @ A @ ( sup_sup @ A )
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 )
@ ^ [X2: A,Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_7671_bounded__quasi__semilattice__set_Oremove,axiom,
! [A: $tType,F3: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,A4: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F3 @ Top @ Bot @ Normalize )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ A4 )
= ( F3 @ A3 @ ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% bounded_quasi_semilattice_set.remove
thf(fact_7672_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F3: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,A4: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F3 @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ ( insert2 @ A @ A3 @ A4 ) )
= ( F3 @ A3 @ ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% bounded_quasi_semilattice_set.insert_remove
thf(fact_7673_bounded__quasi__semilattice__set_Oempty,axiom,
! [A: $tType,F3: A > A > A,Top: A,Bot: A,Normalize: A > A] :
( ( bounde6485984586167503788ce_set @ A @ F3 @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ ( bot_bot @ ( set @ A ) ) )
= Top ) ) ).
% bounded_quasi_semilattice_set.empty
thf(fact_7674_bounded__quasi__semilattice__set_Oinfinite,axiom,
! [A: $tType,F3: A > A > A,Top: A,Bot: A,Normalize: A > A,A4: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F3 @ Top @ Bot @ Normalize )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ( ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ A4 )
= Bot ) ) ) ).
% bounded_quasi_semilattice_set.infinite
thf(fact_7675_bounded__quasi__semilattice__set_Osubset,axiom,
! [A: $tType,F3: A > A > A,Top: A,Bot: A,Normalize: A > A,B2: set @ A,A4: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F3 @ Top @ Bot @ Normalize )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( F3 @ ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ B2 ) @ ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ A4 ) )
= ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ A4 ) ) ) ) ).
% bounded_quasi_semilattice_set.subset
thf(fact_7676_bounded__quasi__semilattice__set_Oinsert,axiom,
! [A: $tType,F3: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,A4: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F3 @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ ( insert2 @ A @ A3 @ A4 ) )
= ( F3 @ A3 @ ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ A4 ) ) ) ) ).
% bounded_quasi_semilattice_set.insert
thf(fact_7677_bounded__quasi__semilattice__set_Ounion,axiom,
! [A: $tType,F3: A > A > A,Top: A,Bot: A,Normalize: A > A,A4: set @ A,B2: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F3 @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( F3 @ ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ A4 ) @ ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ B2 ) ) ) ) ).
% bounded_quasi_semilattice_set.union
thf(fact_7678_bounded__quasi__semilattice__set_Oeq__fold,axiom,
! [A: $tType,F3: A > A > A,Top: A,Bot: A,Normalize: A > A,A4: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F3 @ Top @ Bot @ Normalize )
=> ( ( ( finite_finite2 @ A @ A4 )
=> ( ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ A4 )
= ( finite_fold @ A @ A @ F3 @ Top @ A4 ) ) )
& ( ~ ( finite_finite2 @ A @ A4 )
=> ( ( bounde2362111253966948842tice_F @ A @ F3 @ Top @ Bot @ A4 )
= Bot ) ) ) ) ).
% bounded_quasi_semilattice_set.eq_fold
thf(fact_7679_char__of__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,M: A] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N )
=> ( ( unique5772411509450598832har_of @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ M ) )
= ( unique5772411509450598832har_of @ A @ M ) ) ) ) ).
% char_of_take_bit_eq
thf(fact_7680_compute__powr__real,axiom,
( powr_real
= ( ^ [B5: real,I4: real] :
( if @ real @ ( ord_less_eq @ real @ B5 @ ( zero_zero @ real ) )
@ ( abort @ real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ^ [Uu3: product_unit] : ( powr_real @ B5 @ I4 ) )
@ ( if @ real
@ ( ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ I4 ) )
= I4 )
@ ( if @ real @ ( ord_less_eq @ real @ ( zero_zero @ real ) @ I4 ) @ ( power_power @ real @ B5 @ ( nat2 @ ( archim6421214686448440834_floor @ real @ I4 ) ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ B5 @ ( nat2 @ ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ I4 ) ) ) ) ) )
@ ( abort @ real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ^ [Uu3: product_unit] : ( powr_real @ B5 @ I4 ) ) ) ) ) ) ).
% compute_powr_real
thf(fact_7681_UNIV__char__of__nat,axiom,
( ( top_top @ ( set @ char ) )
= ( image2 @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% UNIV_char_of_nat
thf(fact_7682_inj__on__char__of__nat,axiom,
inj_on @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% inj_on_char_of_nat
thf(fact_7683_range__nat__of__char,axiom,
( ( image2 @ char @ nat @ ( comm_s6883823935334413003f_char @ nat ) @ ( top_top @ ( set @ char ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% range_nat_of_char
thf(fact_7684_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_7685_nat__of__char__less__256,axiom,
! [C3: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C3 ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% nat_of_char_less_256
thf(fact_7686_length__code,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% length_code
thf(fact_7687_card__def,axiom,
! [B: $tType] :
( ( finite_card @ B )
= ( finite_folding_F @ B @ nat
@ ^ [Uu3: B] : suc
@ ( zero_zero @ nat ) ) ) ).
% card_def
thf(fact_7688_folding__on_OF_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding_F @ A @ B )
= ( finite_folding_F @ A @ B ) ) ).
% folding_on.F.cong
thf(fact_7689_folding__on_Oinsert__remove,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,A4: set @ A,Z: B] :
( ( finite_folding_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z @ ( insert2 @ A @ X @ A4 ) )
= ( F3 @ X @ ( finite_folding_F @ A @ B @ F3 @ Z @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% folding_on.insert_remove
thf(fact_7690_folding__on_Oremove,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,A4: set @ A,X: A,Z: B] :
( ( finite_folding_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z @ A4 )
= ( F3 @ X @ ( finite_folding_F @ A @ B @ F3 @ Z @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% folding_on.remove
thf(fact_7691_folding__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding_on @ A @ B )
= ( ^ [S7: set @ A,F2: A > B > B] :
! [X2: A,Y3: A] :
( ( member @ A @ X2 @ S7 )
=> ( ( member @ A @ Y3 @ S7 )
=> ( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X2 ) )
= ( comp @ B @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ) ) ) ).
% folding_on_def
thf(fact_7692_folding__on_Ocomp__fun__commute__on,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,Y: A] :
( ( finite_folding_on @ A @ B @ S @ F3 )
=> ( ( member @ A @ X @ S )
=> ( ( member @ A @ Y @ S )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y ) @ ( F3 @ X ) )
= ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ) ).
% folding_on.comp_fun_commute_on
thf(fact_7693_folding__on_Ointro,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B] :
( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ S )
=> ( ( member @ A @ Y2 @ S )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y2 ) @ ( F3 @ X3 ) )
= ( comp @ B @ B @ B @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) ) ) )
=> ( finite_folding_on @ A @ B @ S @ F3 ) ) ).
% folding_on.intro
thf(fact_7694_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_7695_folding__on_Oempty,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,Z: B] :
( ( finite_folding_on @ A @ B @ S @ F3 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z @ ( bot_bot @ ( set @ A ) ) )
= Z ) ) ).
% folding_on.empty
thf(fact_7696_folding__on_Oinfinite,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,A4: set @ A,Z: B] :
( ( finite_folding_on @ A @ B @ S @ F3 )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z @ A4 )
= Z ) ) ) ).
% folding_on.infinite
thf(fact_7697_folding__on_Oeq__fold,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,Z: B,A4: set @ A] :
( ( finite_folding_on @ A @ B @ S @ F3 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z @ A4 )
= ( finite_fold @ A @ B @ F3 @ Z @ A4 ) ) ) ).
% folding_on.eq_fold
thf(fact_7698_folding__on_Oinsert,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,A4: set @ A,Z: B] :
( ( finite_folding_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z @ ( insert2 @ A @ X @ A4 ) )
= ( F3 @ X @ ( finite_folding_F @ A @ B @ F3 @ Z @ A4 ) ) ) ) ) ) ) ).
% folding_on.insert
thf(fact_7699_folding__idem__on_Oinsert__idem,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,A4: set @ A,Z: B] :
( ( finite1890593828518410140dem_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z @ ( insert2 @ A @ X @ A4 ) )
= ( F3 @ X @ ( finite_folding_F @ A @ B @ F3 @ Z @ A4 ) ) ) ) ) ) ).
% folding_idem_on.insert_idem
thf(fact_7700_rel__fun__iff__geq__image2p,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType] :
( ( bNF_rel_fun @ A @ B @ C @ D )
= ( ^ [R6: A > B > $o,S7: C > D > $o,F2: A > C,G: B > D] : ( ord_less_eq @ ( C > D > $o ) @ ( bNF_Greatest_image2p @ A @ C @ B @ D @ F2 @ G @ R6 ) @ S7 ) ) ) ).
% rel_fun_iff_geq_image2p
thf(fact_7701_folding__idem__on_Oaxioms_I1_J,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B] :
( ( finite1890593828518410140dem_on @ A @ B @ S @ F3 )
=> ( finite_folding_on @ A @ B @ S @ F3 ) ) ).
% folding_idem_on.axioms(1)
thf(fact_7702_folding__idem__on_Ocomp__fun__idem__on,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,Y: A] :
( ( finite1890593828518410140dem_on @ A @ B @ S @ F3 )
=> ( ( member @ A @ X @ S )
=> ( ( member @ A @ Y @ S )
=> ( ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ X ) )
= ( F3 @ X ) ) ) ) ) ).
% folding_idem_on.comp_fun_idem_on
thf(fact_7703_folding__idem__on_Ointro,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B] :
( ( finite_folding_on @ A @ B @ S @ F3 )
=> ( ( finite6916993218817215295axioms @ A @ B @ S @ F3 )
=> ( finite1890593828518410140dem_on @ A @ B @ S @ F3 ) ) ) ).
% folding_idem_on.intro
thf(fact_7704_folding__idem__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite1890593828518410140dem_on @ A @ B )
= ( ^ [S7: set @ A,F2: A > B > B] :
( ( finite_folding_on @ A @ B @ S7 @ F2 )
& ( finite6916993218817215295axioms @ A @ B @ S7 @ F2 ) ) ) ) ).
% folding_idem_on_def
thf(fact_7705_folding__idem__on__axioms_Ointro,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B] :
( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ S )
=> ( ( member @ A @ Y2 @ S )
=> ( ( comp @ B @ B @ B @ ( F3 @ X3 ) @ ( F3 @ X3 ) )
= ( F3 @ X3 ) ) ) )
=> ( finite6916993218817215295axioms @ A @ B @ S @ F3 ) ) ).
% folding_idem_on_axioms.intro
thf(fact_7706_folding__idem__on__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite6916993218817215295axioms @ A @ B )
= ( ^ [S7: set @ A,F2: A > B > B] :
! [X2: A,Y3: A] :
( ( member @ A @ X2 @ S7 )
=> ( ( member @ A @ Y3 @ S7 )
=> ( ( comp @ B @ B @ B @ ( F2 @ X2 ) @ ( F2 @ X2 ) )
= ( F2 @ X2 ) ) ) ) ) ) ).
% folding_idem_on_axioms_def
thf(fact_7707_folding__idem__on_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B] :
( ( finite1890593828518410140dem_on @ A @ B @ S @ F3 )
=> ( finite6916993218817215295axioms @ A @ B @ S @ F3 ) ) ).
% folding_idem_on.axioms(2)
thf(fact_7708_folding__idem__def_H,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding_idem @ A @ B )
= ( finite1890593828518410140dem_on @ A @ B @ ( top_top @ ( set @ A ) ) ) ) ).
% folding_idem_def'
thf(fact_7709_Func__empty,axiom,
! [B: $tType,A: $tType,B2: set @ B] :
( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( insert2 @ ( A > B )
@ ^ [X2: A] : ( undefined @ B )
@ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ).
% Func_empty
thf(fact_7710_folding__idem_Ocomp__fun__idem,axiom,
! [B: $tType,A: $tType,F3: A > B > B,X: A] :
( ( finite_folding_idem @ A @ B @ F3 )
=> ( ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ X ) )
= ( F3 @ X ) ) ) ).
% folding_idem.comp_fun_idem
thf(fact_7711_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa3: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa3 )
= Y )
=> ( ! [X3: A] :
( ( Xa3
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( Y != X3 ) )
=> ( ! [X3: A,Y2: A,Zs2: list @ A] :
( ( Xa3
= ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Zs2 ) ) )
=> ( Y
!= ( if @ A @ ( ord_less_eq @ B @ ( X @ X3 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) )
=> ~ ( ( Xa3
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_7712_prod__filtermap1,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,F4: filter @ C,G3: filter @ B] :
( ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F3 @ F4 ) @ G3 )
= ( filtermap @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B ) @ ( product_apfst @ C @ A @ B @ F3 ) @ ( prod_filter @ C @ B @ F4 @ G3 ) ) ) ).
% prod_filtermap1
thf(fact_7713_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,X: A,Y: A,Zs: list @ A] :
( ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( if @ A @ ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ Y @ Zs ) ) ) ) @ X @ ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_7714_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa3: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa3 )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ Xa3 ) )
=> ( ! [X3: A] :
( ( Xa3
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ( Y = X3 )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) )
=> ( ! [X3: A,Y2: A,Zs2: list @ A] :
( ( Xa3
= ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Zs2 ) ) )
=> ( ( Y
= ( if @ A @ ( ord_less_eq @ B @ ( X @ X3 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y2 @ Zs2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) ) )
=> ~ ( ( Xa3
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
thf(fact_7715_single__valuedp__single__valued__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( single_valuedp @ A @ B
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R2 ) )
= ( single_valued @ A @ B @ R2 ) ) ).
% single_valuedp_single_valued_eq
thf(fact_7716_single__valuedp__less__eq,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,S3: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R2 @ S3 )
=> ( ( single_valuedp @ A @ B @ S3 )
=> ( single_valuedp @ A @ B @ R2 ) ) ) ).
% single_valuedp_less_eq
thf(fact_7717_single__valuedp__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valuedp @ A @ B )
= ( ^ [R5: A > B > $o] :
! [X2: A,Y3: B] :
( ( R5 @ X2 @ Y3 )
=> ! [Z6: B] :
( ( R5 @ X2 @ Z6 )
=> ( Y3 = Z6 ) ) ) ) ) ).
% single_valuedp_def
thf(fact_7718_single__valuedpI,axiom,
! [B: $tType,A: $tType,R2: A > B > $o] :
( ! [X3: A,Y2: B,Z3: B] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ X3 @ Z3 )
=> ( Y2 = Z3 ) ) )
=> ( single_valuedp @ A @ B @ R2 ) ) ).
% single_valuedpI
thf(fact_7719_single__valuedpD,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X: A,Y: B,Z: B] :
( ( single_valuedp @ A @ B @ R2 )
=> ( ( R2 @ X @ Y )
=> ( ( R2 @ X @ Z )
=> ( Y = Z ) ) ) ) ).
% single_valuedpD
thf(fact_7720_single__valuedp__bot,axiom,
! [B: $tType,A: $tType] : ( single_valuedp @ A @ B @ ( bot_bot @ ( A > B > $o ) ) ) ).
% single_valuedp_bot
thf(fact_7721_eq__subset,axiom,
! [A: $tType,P: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y4: A,Z2: A] : Y4 = Z2
@ ^ [A5: A,B5: A] :
( ( P @ A5 @ B5 )
| ( A5 = B5 ) ) ) ).
% eq_subset
thf(fact_7722_less__than__iff,axiom,
! [X: nat,Y: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ less_than )
= ( ord_less @ nat @ X @ Y ) ) ).
% less_than_iff
thf(fact_7723_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X: A,Y: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
& R )
=> ( R
& ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ) ).
% predicate2D_conj
thf(fact_7724_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F3: ( A > B ) > C,G2: C] :
( ( F3
= ( ^ [X2: A > B] : G2 ) )
=> ( ( F3
@ ^ [X2: A] : ( zero_zero @ B ) )
= G2 ) ) ) ).
% fun_cong_unused_0
thf(fact_7725_finite__subset__Union__chain,axiom,
! [A: $tType,A4: set @ A,B12: set @ ( set @ A ),A20: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ B12 ) )
=> ( ( B12
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ B12 )
=> ~ ! [B4: set @ A] :
( ( member @ ( set @ A ) @ B4 @ B12 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_7726_elimnum,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).
% elimnum
thf(fact_7727_enat__ord__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% enat_ord_simps(2)
thf(fact_7728_enat__ord__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% enat_ord_simps(1)
thf(fact_7729_idiff__enat__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( extended_enat2 @ ( zero_zero @ nat ) ) )
= N ) ).
% idiff_enat_0_right
thf(fact_7730_idiff__enat__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ ( zero_zero @ nat ) ) @ N )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% idiff_enat_0
thf(fact_7731_numeral__less__enat__iff,axiom,
! [M: num,N: nat] :
( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ N ) ) ).
% numeral_less_enat_iff
thf(fact_7732_numeral__le__enat__iff,axiom,
! [M: num,N: nat] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ N ) ) ).
% numeral_le_enat_iff
thf(fact_7733_subset__Zorn_H,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ! [C7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C7 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C7 ) @ A4 ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% subset_Zorn'
thf(fact_7734_pred__on_OchainI,axiom,
! [A: $tType,C2: set @ A,A4: set @ A,P: A > A > $o] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ A4 )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ C2 )
=> ( ( member @ A @ Y2 @ C2 )
=> ( ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y4: A,Z2: A] : Y4 = Z2
@ X3
@ Y2 )
| ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y4: A,Z2: A] : Y4 = Z2
@ Y2
@ X3 ) ) ) )
=> ( pred_chain @ A @ A4 @ P @ C2 ) ) ) ).
% pred_on.chainI
thf(fact_7735_pred__on_Ochain__def,axiom,
! [A: $tType] :
( ( pred_chain @ A )
= ( ^ [A6: set @ A,P3: A > A > $o,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ A6 )
& ! [X2: A] :
( ( member @ A @ X2 @ C4 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ C4 )
=> ( ( sup_sup @ ( A > A > $o ) @ P3
@ ^ [Y4: A,Z2: A] : Y4 = Z2
@ X2
@ Y3 )
| ( sup_sup @ ( A > A > $o ) @ P3
@ ^ [Y4: A,Z2: A] : Y4 = Z2
@ Y3
@ X2 ) ) ) ) ) ) ) ).
% pred_on.chain_def
thf(fact_7736_subset_Ochain__def,axiom,
! [A: $tType,A4: set @ ( set @ A ),C2: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
= ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C2 @ A4 )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C2 )
=> ! [Y3: set @ A] :
( ( member @ ( set @ A ) @ Y3 @ C2 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y4: set @ A,Z2: set @ A] : Y4 = Z2
@ X2
@ Y3 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y4: set @ A,Z2: set @ A] : Y4 = Z2
@ Y3
@ X2 ) ) ) ) ) ) ).
% subset.chain_def
thf(fact_7737_subset_OchainI,axiom,
! [A: $tType,C2: set @ ( set @ A ),A4: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C2 @ A4 )
=> ( ! [X3: set @ A,Y2: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C2 )
=> ( ( member @ ( set @ A ) @ Y2 @ C2 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y4: set @ A,Z2: set @ A] : Y4 = Z2
@ X3
@ Y2 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y4: set @ A,Z2: set @ A] : Y4 = Z2
@ Y2
@ X3 ) ) ) )
=> ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 ) ) ) ).
% subset.chainI
thf(fact_7738_finite__enat__bounded,axiom,
! [A4: set @ extended_enat,N: nat] :
( ! [Y2: extended_enat] :
( ( member @ extended_enat @ Y2 @ A4 )
=> ( ord_less_eq @ extended_enat @ Y2 @ ( extended_enat2 @ N ) ) )
=> ( finite_finite2 @ extended_enat @ A4 ) ) ).
% finite_enat_bounded
thf(fact_7739_enat__ile,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less_eq @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K2: nat] :
( N
= ( extended_enat2 @ K2 ) ) ) ).
% enat_ile
thf(fact_7740_subset__Zorn,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ! [C7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C7 )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A4 )
& ! [Xa2: set @ A] :
( ( member @ ( set @ A ) @ Xa2 @ C7 )
=> ( ord_less_eq @ ( set @ A ) @ Xa2 @ X5 ) ) ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% subset_Zorn
thf(fact_7741_chains__alt__def,axiom,
! [A: $tType] :
( ( chains2 @ A )
= ( ^ [A6: set @ ( set @ A )] : ( collect @ ( set @ ( set @ A ) ) @ ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ).
% chains_alt_def
thf(fact_7742_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
! [A3: $o,B3: $o,Uu: extended_enat] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Leaf @ A3 @ B3 ) @ Uu )
= ( vEBT_Leaf @ A3 @ B3 ) ) ).
% VEBT_internal.elim_dead.simps(1)
thf(fact_7743_chain__incr,axiom,
! [A: $tType,Y6: A > extended_enat,K: nat] :
( ! [I2: A] :
? [J6: A] : ( ord_less @ extended_enat @ ( Y6 @ I2 ) @ ( Y6 @ J6 ) )
=> ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K ) @ ( Y6 @ J2 ) ) ) ).
% chain_incr
thf(fact_7744_enat__iless,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K2: nat] :
( N
= ( extended_enat2 @ K2 ) ) ) ).
% enat_iless
thf(fact_7745_less__enatE,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ~ ! [K2: nat] :
( ( N
= ( extended_enat2 @ K2 ) )
=> ~ ( ord_less @ nat @ K2 @ M ) ) ) ).
% less_enatE
thf(fact_7746_subset_Ochain__total,axiom,
! [A: $tType,A4: set @ ( set @ A ),C2: set @ ( set @ A ),X: set @ A,Y: set @ A] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
=> ( ( member @ ( set @ A ) @ X @ C2 )
=> ( ( member @ ( set @ A ) @ Y @ C2 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y4: set @ A,Z2: set @ A] : Y4 = Z2
@ X
@ Y )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y4: set @ A,Z2: set @ A] : Y4 = Z2
@ Y
@ X ) ) ) ) ) ).
% subset.chain_total
thf(fact_7747_subset__chain__def,axiom,
! [A: $tType,A20: set @ ( set @ A ),C10: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ C10 )
= ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C10 @ A20 )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C10 )
=> ! [Y3: set @ A] :
( ( member @ ( set @ A ) @ Y3 @ C10 )
=> ( ( ord_less_eq @ ( set @ A ) @ X2 @ Y3 )
| ( ord_less_eq @ ( set @ A ) @ Y3 @ X2 ) ) ) ) ) ) ).
% subset_chain_def
thf(fact_7748_Suc__ile__eq,axiom,
! [M: nat,N: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
= ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% Suc_ile_eq
thf(fact_7749_pred__on_Ochain__empty,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o] : ( pred_chain @ A @ A4 @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pred_on.chain_empty
thf(fact_7750_subset_Ochain__empty,axiom,
! [A: $tType,A4: set @ ( set @ A )] : ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% subset.chain_empty
thf(fact_7751_zero__enat__def,axiom,
( ( zero_zero @ extended_enat )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% zero_enat_def
thf(fact_7752_enat__0__iff_I1_J,axiom,
! [X: nat] :
( ( ( extended_enat2 @ X )
= ( zero_zero @ extended_enat ) )
= ( X
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(1)
thf(fact_7753_enat__0__iff_I2_J,axiom,
! [X: nat] :
( ( ( zero_zero @ extended_enat )
= ( extended_enat2 @ X ) )
= ( X
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(2)
thf(fact_7754_subset__chain__insert,axiom,
! [A: $tType,A20: set @ ( set @ A ),B2: set @ A,B12: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ ( insert2 @ ( set @ A ) @ B2 @ B12 ) )
= ( ( member @ ( set @ A ) @ B2 @ A20 )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ B12 )
=> ( ( ord_less_eq @ ( set @ A ) @ X2 @ B2 )
| ( ord_less_eq @ ( set @ A ) @ B2 @ X2 ) ) )
& ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ B12 ) ) ) ).
% subset_chain_insert
thf(fact_7755_chain__subset__alt__def,axiom,
! [A: $tType] :
( ( chain_subset @ A )
= ( pred_chain @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) @ ( ord_less @ ( set @ A ) ) ) ) ).
% chain_subset_alt_def
thf(fact_7756_iadd__le__enat__iff,axiom,
! [X: extended_enat,Y: extended_enat,N: nat] :
( ( ord_less_eq @ extended_enat @ ( plus_plus @ extended_enat @ X @ Y ) @ ( extended_enat2 @ N ) )
= ( ? [Y9: nat,X10: nat] :
( ( X
= ( extended_enat2 @ X10 ) )
& ( Y
= ( extended_enat2 @ Y9 ) )
& ( ord_less_eq @ nat @ ( plus_plus @ nat @ X10 @ Y9 ) @ N ) ) ) ) ).
% iadd_le_enat_iff
thf(fact_7757_subset__Zorn__nonempty,axiom,
! [A: $tType,A20: set @ ( set @ A )] :
( ( A20
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [C11: set @ ( set @ A )] :
( ( C11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ C11 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C11 ) @ A20 ) ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A20 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A20 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_7758_Union__in__chain,axiom,
! [A: $tType,B12: set @ ( set @ A ),A20: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B12 )
=> ( ( B12
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ B12 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B12 ) @ B12 ) ) ) ) ).
% Union_in_chain
thf(fact_7759_Inter__in__chain,axiom,
! [A: $tType,B12: set @ ( set @ A ),A20: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B12 )
=> ( ( B12
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ B12 )
=> ( member @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) @ B12 ) ) ) ) ).
% Inter_in_chain
thf(fact_7760_subset_Ochain__extend,axiom,
! [A: $tType,A4: set @ ( set @ A ),C2: set @ ( set @ A ),Z: set @ A] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
=> ( ( member @ ( set @ A ) @ Z @ A4 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C2 )
=> ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y4: set @ A,Z2: set @ A] : Y4 = Z2
@ X3
@ Z ) )
=> ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert2 @ ( set @ A ) @ Z @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C2 ) ) ) ) ) ).
% subset.chain_extend
thf(fact_7761_pred__on_Ochain__extend,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o,C2: set @ A,Z: A] :
( ( pred_chain @ A @ A4 @ P @ C2 )
=> ( ( member @ A @ Z @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ C2 )
=> ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y4: A,Z2: A] : Y4 = Z2
@ X3
@ Z ) )
=> ( pred_chain @ A @ A4 @ P @ ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ Z @ ( bot_bot @ ( set @ A ) ) ) @ C2 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_7762_Chains__subset_H,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) ) )
@ ( chains @ A @ R2 ) ) ) ).
% Chains_subset'
thf(fact_7763_VEBT__internal_Oelim__dead_Oelims,axiom,
! [X: vEBT_VEBT,Xa3: extended_enat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_elim_dead @ X @ Xa3 )
= Y )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ( Y
!= ( vEBT_Leaf @ A7 @ B7 ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( Xa3
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ Deg2
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList2 )
@ ( vEBT_VEBT_elim_dead @ Summary3 @ ( extend4730790105801354508finity @ extended_enat ) ) ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
=> ! [L7: nat] :
( ( Xa3
= ( extended_enat2 @ L7 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ Deg2
@ ( take @ vEBT_VEBT @ ( divide_divide @ nat @ L7 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList2 ) )
@ ( vEBT_VEBT_elim_dead @ Summary3 @ ( extended_enat2 @ ( divide_divide @ nat @ L7 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.elims
thf(fact_7764_elimcomplete,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).
% elimcomplete
thf(fact_7765_enat__ord__simps_I6_J,axiom,
! [Q5: extended_enat] :
~ ( ord_less @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ Q5 ) ).
% enat_ord_simps(6)
thf(fact_7766_enat__ord__simps_I4_J,axiom,
! [Q5: extended_enat] :
( ( ord_less @ extended_enat @ Q5 @ ( extend4730790105801354508finity @ extended_enat ) )
= ( Q5
!= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% enat_ord_simps(4)
thf(fact_7767_enat__ord__code_I3_J,axiom,
! [Q5: extended_enat] : ( ord_less_eq @ extended_enat @ Q5 @ ( extend4730790105801354508finity @ extended_enat ) ) ).
% enat_ord_code(3)
thf(fact_7768_enat__ord__simps_I5_J,axiom,
! [Q5: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ Q5 )
= ( Q5
= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% enat_ord_simps(5)
thf(fact_7769_times__enat__simps_I4_J,axiom,
! [M: nat] :
( ( ( M
= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% times_enat_simps(4)
thf(fact_7770_times__enat__simps_I3_J,axiom,
! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ N ) )
= ( zero_zero @ extended_enat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ N ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% times_enat_simps(3)
thf(fact_7771_enat__ord__code_I5_J,axiom,
! [N: nat] :
~ ( ord_less_eq @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ N ) ) ).
% enat_ord_code(5)
thf(fact_7772_infinity__ileE,axiom,
! [M: nat] :
~ ( ord_less_eq @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ M ) ) ).
% infinity_ileE
thf(fact_7773_enat__ord__code_I4_J,axiom,
! [M: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extend4730790105801354508finity @ extended_enat ) ) ).
% enat_ord_code(4)
thf(fact_7774_less__infinityE,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ N @ ( extend4730790105801354508finity @ extended_enat ) )
=> ~ ! [K2: nat] :
( N
!= ( extended_enat2 @ K2 ) ) ) ).
% less_infinityE
thf(fact_7775_infinity__ilessE,axiom,
! [M: nat] :
~ ( ord_less @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ M ) ) ).
% infinity_ilessE
thf(fact_7776_VEBT__internal_Oelim__dead_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ extended_enat] :
( ! [A7: $o,B7: $o,Uu2: extended_enat] :
( X
!= ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Leaf @ A7 @ B7 ) @ Uu2 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( X
!= ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ ( extend4730790105801354508finity @ extended_enat ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT,L7: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ ( extended_enat2 @ L7 ) ) ) ) ) ).
% VEBT_internal.elim_dead.cases
thf(fact_7777_enat__add__left__cancel__less,axiom,
! [A3: extended_enat,B3: extended_enat,C3: extended_enat] :
( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ A3 @ B3 ) @ ( plus_plus @ extended_enat @ A3 @ C3 ) )
= ( ( A3
!= ( extend4730790105801354508finity @ extended_enat ) )
& ( ord_less @ extended_enat @ B3 @ C3 ) ) ) ).
% enat_add_left_cancel_less
thf(fact_7778_enat__ord__simps_I3_J,axiom,
! [Q5: extended_enat] : ( ord_less_eq @ extended_enat @ Q5 @ ( extend4730790105801354508finity @ extended_enat ) ) ).
% enat_ord_simps(3)
thf(fact_7779_enat__add__left__cancel__le,axiom,
! [A3: extended_enat,B3: extended_enat,C3: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( plus_plus @ extended_enat @ A3 @ B3 ) @ ( plus_plus @ extended_enat @ A3 @ C3 ) )
= ( ( A3
= ( extend4730790105801354508finity @ extended_enat ) )
| ( ord_less_eq @ extended_enat @ B3 @ C3 ) ) ) ).
% enat_add_left_cancel_le
thf(fact_7780_imult__infinity__right,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
=> ( ( times_times @ extended_enat @ N @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% imult_infinity_right
thf(fact_7781_imult__infinity,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
=> ( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ N )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% imult_infinity
thf(fact_7782_mono__Chains,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( chains @ A @ R2 ) @ ( chains @ A @ S3 ) ) ) ).
% mono_Chains
thf(fact_7783_Chains__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( set @ A ) ) @ ( chains @ A @ R2 )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) ) ) ) ).
% Chains_subset
thf(fact_7784_VEBT__internal_Oelim__dead_Opelims,axiom,
! [X: vEBT_VEBT,Xa3: extended_enat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_elim_dead @ X @ Xa3 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ X @ Xa3 ) )
=> ( ! [A7: $o,B7: $o] :
( ( X
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ( ( Y
= ( vEBT_Leaf @ A7 @ B7 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Leaf @ A7 @ B7 ) @ Xa3 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( Xa3
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ Deg2
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList2 )
@ ( vEBT_VEBT_elim_dead @ Summary3 @ ( extend4730790105801354508finity @ extended_enat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ ( extend4730790105801354508finity @ extended_enat ) ) ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
=> ! [L7: nat] :
( ( Xa3
= ( extended_enat2 @ L7 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ Deg2
@ ( take @ vEBT_VEBT @ ( divide_divide @ nat @ L7 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList2 ) )
@ ( vEBT_VEBT_elim_dead @ Summary3 @ ( extended_enat2 @ ( divide_divide @ nat @ L7 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ ( extended_enat2 @ L7 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.pelims
thf(fact_7785_times__enat__def,axiom,
( ( times_times @ extended_enat )
= ( ^ [M2: extended_enat,N2: extended_enat] :
( extended_case_enat @ extended_enat
@ ^ [O: nat] :
( extended_case_enat @ extended_enat
@ ^ [P5: nat] : ( extended_enat2 @ ( times_times @ nat @ O @ P5 ) )
@ ( if @ extended_enat
@ ( O
= ( zero_zero @ nat ) )
@ ( zero_zero @ extended_enat )
@ ( extend4730790105801354508finity @ extended_enat ) )
@ N2 )
@ ( if @ extended_enat
@ ( N2
= ( zero_zero @ extended_enat ) )
@ ( zero_zero @ extended_enat )
@ ( extend4730790105801354508finity @ extended_enat ) )
@ M2 ) ) ) ).
% times_enat_def
thf(fact_7786_pred__on_Onot__maxchain__Some,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o,C2: set @ A] :
( ( pred_chain @ A @ A4 @ P @ C2 )
=> ( ~ ( pred_maxchain @ A @ A4 @ P @ C2 )
=> ( ( pred_chain @ A @ A4 @ P
@ ( fChoice @ ( set @ A )
@ ^ [D7: set @ A] :
( ( pred_chain @ A @ A4 @ P @ D7 )
& ( ord_less @ ( set @ A ) @ C2 @ D7 ) ) ) )
& ( ord_less @ ( set @ A ) @ C2
@ ( fChoice @ ( set @ A )
@ ^ [D7: set @ A] :
( ( pred_chain @ A @ A4 @ P @ D7 )
& ( ord_less @ ( set @ A ) @ C2 @ D7 ) ) ) ) ) ) ) ).
% pred_on.not_maxchain_Some
thf(fact_7787_Range__Union,axiom,
! [A: $tType,B: $tType,S: set @ ( set @ ( product_prod @ B @ A ) )] :
( ( range @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ B @ A ) ) @ ( set @ A ) @ ( range @ B @ A ) @ S ) ) ) ).
% Range_Union
thf(fact_7788_Range__Id__on,axiom,
! [A: $tType,A4: set @ A] :
( ( range @ A @ A @ ( id_on @ A @ A4 ) )
= A4 ) ).
% Range_Id_on
thf(fact_7789_Range__empty,axiom,
! [B: $tType,A: $tType] :
( ( range @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Range_empty
thf(fact_7790_Range__Id,axiom,
! [A: $tType] :
( ( range @ A @ A @ ( id2 @ A ) )
= ( top_top @ ( set @ A ) ) ) ).
% Range_Id
thf(fact_7791_Range__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: B > A > $o] :
( ( range @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) )
= ( collect @ A
@ ^ [Y3: A] :
? [X2: B] : ( P @ X2 @ Y3 ) ) ) ).
% Range_Collect_case_prod
thf(fact_7792_Range__insert,axiom,
! [A: $tType,B: $tType,A3: B,B3: A,R2: set @ ( product_prod @ B @ A )] :
( ( range @ B @ A @ ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B3 ) @ R2 ) )
= ( insert2 @ A @ B3 @ ( range @ B @ A @ R2 ) ) ) ).
% Range_insert
thf(fact_7793_Range_Ocases,axiom,
! [B: $tType,A: $tType,A3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range @ A @ B @ R2 ) )
=> ~ ! [A7: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ A3 ) @ R2 ) ) ).
% Range.cases
thf(fact_7794_Range_Osimps,axiom,
! [B: $tType,A: $tType,A3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range @ A @ B @ R2 ) )
= ( ? [A5: A,B5: B] :
( ( A3 = B5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R2 ) ) ) ) ).
% Range.simps
thf(fact_7795_Range_Ointros,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 )
=> ( member @ B @ B3 @ ( range @ A @ B @ R2 ) ) ) ).
% Range.intros
thf(fact_7796_RangeE,axiom,
! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A )] :
( ( member @ A @ B3 @ ( range @ B @ A @ R2 ) )
=> ~ ! [A7: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A7 @ B3 ) @ R2 ) ) ).
% RangeE
thf(fact_7797_Range__iff,axiom,
! [A: $tType,B: $tType,A3: A,R2: set @ ( product_prod @ B @ A )] :
( ( member @ A @ A3 @ ( range @ B @ A @ R2 ) )
= ( ? [Y3: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y3 @ A3 ) @ R2 ) ) ) ).
% Range_iff
thf(fact_7798_Range__empty__iff,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( ( range @ B @ A @ R2 )
= ( bot_bot @ ( set @ A ) ) )
= ( R2
= ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) ) ).
% Range_empty_iff
thf(fact_7799_subset_OHausdorff,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
? [X_1: set @ ( set @ A )] : ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X_1 ) ).
% subset.Hausdorff
thf(fact_7800_Range__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 )
=> ( ord_less_eq @ ( set @ B ) @ ( range @ A @ B @ R2 ) @ ( range @ A @ B @ S3 ) ) ) ).
% Range_mono
thf(fact_7801_Range__snd,axiom,
! [A: $tType,B: $tType] :
( ( range @ B @ A )
= ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) ) ) ).
% Range_snd
thf(fact_7802_snd__eq__Range,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ R )
= ( range @ B @ A @ R ) ) ).
% snd_eq_Range
thf(fact_7803_finite__Range,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R2 )
=> ( finite_finite2 @ B @ ( range @ A @ B @ R2 ) ) ) ).
% finite_Range
thf(fact_7804_Range__Un__eq,axiom,
! [A: $tType,B: $tType,A4: set @ ( product_prod @ B @ A ),B2: set @ ( product_prod @ B @ A )] :
( ( range @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( range @ B @ A @ A4 ) @ ( range @ B @ A @ B2 ) ) ) ).
% Range_Un_eq
thf(fact_7805_subset_Omaxchain__def,axiom,
! [A: $tType,A4: set @ ( set @ A ),C2: set @ ( set @ A )] :
( ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
= ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
& ~ ? [S7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ S7 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C2 @ S7 ) ) ) ) ).
% subset.maxchain_def
thf(fact_7806_subset_Omaxchain__imp__chain,axiom,
! [A: $tType,A4: set @ ( set @ A ),C2: set @ ( set @ A )] :
( ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
=> ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 ) ) ).
% subset.maxchain_imp_chain
thf(fact_7807_Range__Int__subset,axiom,
! [A: $tType,B: $tType,A4: set @ ( product_prod @ B @ A ),B2: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( range @ B @ A @ ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) @ A4 @ B2 ) ) @ ( inf_inf @ ( set @ A ) @ ( range @ B @ A @ A4 ) @ ( range @ B @ A @ B2 ) ) ) ).
% Range_Int_subset
thf(fact_7808_Range__Diff__subset,axiom,
! [A: $tType,B: $tType,A4: set @ ( product_prod @ B @ A ),B2: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( range @ B @ A @ A4 ) @ ( range @ B @ A @ B2 ) ) @ ( range @ B @ A @ ( minus_minus @ ( set @ ( product_prod @ B @ A ) ) @ A4 @ B2 ) ) ) ).
% Range_Diff_subset
thf(fact_7809_subset_Onot__maxchain__Some,axiom,
! [A: $tType,A4: set @ ( set @ A ),C2: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
=> ( ~ ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
=> ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) )
@ ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ D7 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C2 @ D7 ) ) ) )
& ( ord_less @ ( set @ ( set @ A ) ) @ C2
@ ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ D7 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C2 @ D7 ) ) ) ) ) ) ) ).
% subset.not_maxchain_Some
thf(fact_7810_pred__on_Omaxchain__def,axiom,
! [A: $tType] :
( ( pred_maxchain @ A )
= ( ^ [A6: set @ A,P3: A > A > $o,C4: set @ A] :
( ( pred_chain @ A @ A6 @ P3 @ C4 )
& ~ ? [S7: set @ A] :
( ( pred_chain @ A @ A6 @ P3 @ S7 )
& ( ord_less @ ( set @ A ) @ C4 @ S7 ) ) ) ) ) ).
% pred_on.maxchain_def
thf(fact_7811_subset__maxchain__max,axiom,
! [A: $tType,A4: set @ ( set @ A ),C2: set @ ( set @ A ),X4: set @ A] :
( ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
=> ( ( member @ ( set @ A ) @ X4 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) @ X4 )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ C2 )
= X4 ) ) ) ) ).
% subset_maxchain_max
thf(fact_7812_wf__UN,axiom,
! [B: $tType,A: $tType,I5: set @ A,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( wf @ B @ ( R2 @ I2 ) ) )
=> ( ! [I2: A,J2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ( member @ A @ J2 @ I5 )
=> ( ( ( R2 @ I2 )
!= ( R2 @ J2 ) )
=> ( ( inf_inf @ ( set @ B ) @ ( domain @ B @ B @ ( R2 @ I2 ) ) @ ( range @ B @ B @ ( R2 @ J2 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( wf @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ I5 ) ) ) ) ) ).
% wf_UN
thf(fact_7813_pred__on_Osuc__def,axiom,
! [A: $tType] :
( ( pred_suc @ A )
= ( ^ [A6: set @ A,P3: A > A > $o,C4: set @ A] :
( if @ ( set @ A )
@ ( ~ ( pred_chain @ A @ A6 @ P3 @ C4 )
| ( pred_maxchain @ A @ A6 @ P3 @ C4 ) )
@ C4
@ ( fChoice @ ( set @ A )
@ ^ [D7: set @ A] :
( ( pred_chain @ A @ A6 @ P3 @ D7 )
& ( ord_less @ ( set @ A ) @ C4 @ D7 ) ) ) ) ) ) ).
% pred_on.suc_def
thf(fact_7814_subset_Onot__chain__suc,axiom,
! [A: $tType,A4: set @ ( set @ A ),X4: set @ ( set @ A )] :
( ~ ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X4 )
=> ( ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X4 )
= X4 ) ) ).
% subset.not_chain_suc
thf(fact_7815_Domain__Id__on,axiom,
! [A: $tType,A4: set @ A] :
( ( domain @ A @ A @ ( id_on @ A @ A4 ) )
= A4 ) ).
% Domain_Id_on
thf(fact_7816_subset_Omaxchain__suc,axiom,
! [A: $tType,A4: set @ ( set @ A ),X4: set @ ( set @ A )] :
( ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X4 )
=> ( ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X4 )
= X4 ) ) ).
% subset.maxchain_suc
thf(fact_7817_Domain__empty,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Domain_empty
thf(fact_7818_Domain__Id,axiom,
! [A: $tType] :
( ( domain @ A @ A @ ( id2 @ A ) )
= ( top_top @ ( set @ A ) ) ) ).
% Domain_Id
thf(fact_7819_Range__converse,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( range @ B @ A @ ( converse @ A @ B @ R2 ) )
= ( domain @ A @ B @ R2 ) ) ).
% Range_converse
thf(fact_7820_Domain__converse,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( domain @ A @ B @ ( converse @ B @ A @ R2 ) )
= ( range @ B @ A @ R2 ) ) ).
% Domain_converse
thf(fact_7821_Domain__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( domain @ A @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
= ( collect @ A
@ ^ [X2: A] :
? [X8: B] : ( P @ X2 @ X8 ) ) ) ).
% Domain_Collect_case_prod
thf(fact_7822_Domain__insert,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 ) )
= ( insert2 @ A @ A3 @ ( domain @ A @ B @ R2 ) ) ) ).
% Domain_insert
thf(fact_7823_subset_Osuc__def,axiom,
! [A: $tType,A4: set @ ( set @ A ),C2: set @ ( set @ A )] :
( ( ( ~ ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
| ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 ) )
=> ( ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
= C2 ) )
& ( ~ ( ~ ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
| ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 ) )
=> ( ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
= ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ D7 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C2 @ D7 ) ) ) ) ) ) ).
% subset.suc_def
thf(fact_7824_subset_Osuc__not__equals,axiom,
! [A: $tType,A4: set @ ( set @ A ),C2: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
=> ( ~ ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
=> ( ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C2 )
!= C2 ) ) ) ).
% subset.suc_not_equals
thf(fact_7825_pred__on_Ochain__sucD,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o,X4: set @ A] :
( ( pred_chain @ A @ A4 @ P @ X4 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( pred_suc @ A @ A4 @ P @ X4 ) @ A4 )
& ( pred_chain @ A @ A4 @ P @ ( pred_suc @ A @ A4 @ P @ X4 ) ) ) ) ).
% pred_on.chain_sucD
thf(fact_7826_subset_Ochain__suc,axiom,
! [A: $tType,A4: set @ ( set @ A ),X4: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X4 )
=> ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X4 ) ) ) ).
% subset.chain_suc
thf(fact_7827_Domain__Un__eq,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B ),B2: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( domain @ A @ B @ A4 ) @ ( domain @ A @ B @ B2 ) ) ) ).
% Domain_Un_eq
thf(fact_7828_finite__Domain,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R2 )
=> ( finite_finite2 @ A @ ( domain @ A @ B @ R2 ) ) ) ).
% finite_Domain
thf(fact_7829_Domain__fst,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) ).
% Domain_fst
thf(fact_7830_fst__eq__Domain,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ R )
= ( domain @ A @ B @ R ) ) ).
% fst_eq_Domain
thf(fact_7831_Domain__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 )
=> ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R2 ) @ ( domain @ A @ B @ S3 ) ) ) ).
% Domain_mono
thf(fact_7832_pred__on_Osuc__in__carrier,axiom,
! [A: $tType,X4: set @ A,A4: set @ A,P: A > A > $o] :
( ( ord_less_eq @ ( set @ A ) @ X4 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( pred_suc @ A @ A4 @ P @ X4 ) @ A4 ) ) ).
% pred_on.suc_in_carrier
thf(fact_7833_pred__on_Osuc__subset,axiom,
! [A: $tType,X4: set @ A,A4: set @ A,P: A > A > $o] : ( ord_less_eq @ ( set @ A ) @ X4 @ ( pred_suc @ A @ A4 @ P @ X4 ) ) ).
% pred_on.suc_subset
thf(fact_7834_pred__on_Osubset__suc,axiom,
! [A: $tType,X4: set @ A,Y6: set @ A,A4: set @ A,P: A > A > $o] :
( ( ord_less_eq @ ( set @ A ) @ X4 @ Y6 )
=> ( ord_less_eq @ ( set @ A ) @ X4 @ ( pred_suc @ A @ A4 @ P @ Y6 ) ) ) ).
% pred_on.subset_suc
thf(fact_7835_subset_Osuc__in__carrier,axiom,
! [A: $tType,X4: set @ ( set @ A ),A4: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X4 @ A4 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X4 ) @ A4 ) ) ).
% subset.suc_in_carrier
thf(fact_7836_subset_Osuc__subset,axiom,
! [A: $tType,X4: set @ ( set @ A ),A4: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ X4 @ ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X4 ) ) ).
% subset.suc_subset
thf(fact_7837_subset_Osubset__suc,axiom,
! [A: $tType,X4: set @ ( set @ A ),Y6: set @ ( set @ A ),A4: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X4 @ Y6 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ X4 @ ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ Y6 ) ) ) ).
% subset.subset_suc
thf(fact_7838_Domain__empty__iff,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( ( domain @ A @ B @ R2 )
= ( bot_bot @ ( set @ A ) ) )
= ( R2
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% Domain_empty_iff
thf(fact_7839_Domain__iff,axiom,
! [A: $tType,B: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
= ( ? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y3 ) @ R2 ) ) ) ).
% Domain_iff
thf(fact_7840_DomainE,axiom,
! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
=> ~ ! [B7: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B7 ) @ R2 ) ) ).
% DomainE
thf(fact_7841_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 )
=> ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) ) ) ).
% Domain.DomainI
thf(fact_7842_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
= ( ? [A5: A,B5: B] :
( ( A3 = A5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R2 ) ) ) ) ).
% Domain.simps
thf(fact_7843_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
=> ~ ! [B7: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B7 ) @ R2 ) ) ).
% Domain.cases
thf(fact_7844_Domain__unfold,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ^ [X2: A] :
? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R5 ) ) ) ) ).
% Domain_unfold
thf(fact_7845_subset_Ochain__sucD,axiom,
! [A: $tType,A4: set @ ( set @ A ),X4: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X4 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X4 ) @ A4 )
& ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X4 ) ) ) ) ).
% subset.chain_sucD
thf(fact_7846_Domain__Int__subset,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B ),B2: set @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ B2 ) ) @ ( inf_inf @ ( set @ A ) @ ( domain @ A @ B @ A4 ) @ ( domain @ A @ B @ B2 ) ) ) ).
% Domain_Int_subset
thf(fact_7847_Field__def,axiom,
! [A: $tType] :
( ( field2 @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] : ( sup_sup @ ( set @ A ) @ ( domain @ A @ A @ R5 ) @ ( range @ A @ A @ R5 ) ) ) ) ).
% Field_def
thf(fact_7848_Domain__Diff__subset,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B ),B2: set @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ B @ A4 ) @ ( domain @ A @ B @ B2 ) ) @ ( domain @ A @ B @ ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ B2 ) ) ) ).
% Domain_Diff_subset
thf(fact_7849_Domain__Union,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( domain @ A @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ A ) @ ( domain @ A @ B ) @ S ) ) ) ).
% Domain_Union
thf(fact_7850_wf__Un,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( ( wf @ A @ S3 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R2 ) @ ( range @ A @ A @ S3 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ) ).
% wf_Un
thf(fact_7851_wf__Union,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ! [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R )
=> ( wf @ A @ X3 ) )
=> ( ! [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R )
=> ! [Xa2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa2 @ R )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ X3 ) @ ( range @ A @ A @ Xa2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( wf @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R ) ) ) ) ).
% wf_Union
thf(fact_7852_prod__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= ( insert2 @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) ) ).
% prod_set_simps(2)
thf(fact_7853_prod__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X: A,Y: B] :
( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% prod_set_simps(1)
thf(fact_7854_prod__set__defs_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( basic_fsts @ A @ B )
= ( ^ [P5: product_prod @ A @ B] : ( insert2 @ A @ ( product_fst @ A @ B @ P5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% prod_set_defs(1)
thf(fact_7855_prod__set__defs_I2_J,axiom,
! [D: $tType,C: $tType] :
( ( basic_snds @ C @ D )
= ( ^ [P5: product_prod @ C @ D] : ( insert2 @ D @ ( product_snd @ C @ D @ P5 ) @ ( bot_bot @ ( set @ D ) ) ) ) ) ).
% prod_set_defs(2)
thf(fact_7856_iless__Suc__eq,axiom,
! [M: nat,N: extended_enat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_eSuc @ N ) )
= ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% iless_Suc_eq
thf(fact_7857_eSuc__mono,axiom,
! [N: extended_enat,M: extended_enat] :
( ( ord_less @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
= ( ord_less @ extended_enat @ N @ M ) ) ).
% eSuc_mono
thf(fact_7858_eSuc__ile__mono,axiom,
! [N: extended_enat,M: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
= ( ord_less_eq @ extended_enat @ N @ M ) ) ).
% eSuc_ile_mono
thf(fact_7859_iless__eSuc0,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ N @ ( extended_eSuc @ ( zero_zero @ extended_enat ) ) )
= ( N
= ( zero_zero @ extended_enat ) ) ) ).
% iless_eSuc0
thf(fact_7860_sup__filter__parametric,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > ( filter @ A ) ) @ ( ( filter @ B ) > ( filter @ B ) ) @ ( rel_filter @ A @ B @ A4 ) @ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A4 ) @ ( rel_filter @ A @ B @ A4 ) ) @ ( sup_sup @ ( filter @ A ) ) @ ( sup_sup @ ( filter @ B ) ) ) ).
% sup_filter_parametric
thf(fact_7861_top__filter__parametric,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_total @ A @ B @ A4 )
=> ( rel_filter @ A @ B @ A4 @ ( top_top @ ( filter @ A ) ) @ ( top_top @ ( filter @ B ) ) ) ) ).
% top_filter_parametric
thf(fact_7862_rel__filter__mono,axiom,
! [B: $tType,A: $tType,A4: A > B > $o,B2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A4 @ B2 )
=> ( ord_less_eq @ ( ( filter @ A ) > ( filter @ B ) > $o ) @ ( rel_filter @ A @ B @ A4 ) @ ( rel_filter @ A @ B @ B2 ) ) ) ).
% rel_filter_mono
thf(fact_7863_ile__eSuc,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ N @ ( extended_eSuc @ N ) ) ).
% ile_eSuc
thf(fact_7864_rel__filter__eq,axiom,
! [A: $tType] :
( ( rel_filter @ A @ A
@ ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [Y4: filter @ A,Z2: filter @ A] : Y4 = Z2 ) ) ).
% rel_filter_eq
thf(fact_7865_bi__total__rel__filter,axiom,
! [B: $tType,A: $tType,A4: A > B > $o] :
( ( bi_total @ A @ B @ A4 )
=> ( bi_total @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A4 ) ) ) ).
% bi_total_rel_filter
thf(fact_7866_eventually__parametric,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A4 )
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( eventually @ A )
@ ( eventually @ B ) ) ).
% eventually_parametric
thf(fact_7867_filtermap__parametric,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A4: A > C > $o,B2: B > D > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( filter @ A ) > ( filter @ B ) ) @ ( ( filter @ C ) > ( filter @ D ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A4 @ B2 ) @ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ C ) @ ( filter @ B ) @ ( filter @ D ) @ ( rel_filter @ A @ C @ A4 ) @ ( rel_filter @ B @ D @ B2 ) ) @ ( filtermap @ A @ B ) @ ( filtermap @ C @ D ) ) ).
% filtermap_parametric
thf(fact_7868_bot__filter__parametric,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( rel_filter @ A @ B @ A4 @ ( bot_bot @ ( filter @ A ) ) @ ( bot_bot @ ( filter @ B ) ) ) ).
% bot_filter_parametric
thf(fact_7869_not__eSuc__ilei0,axiom,
! [N: extended_enat] :
~ ( ord_less_eq @ extended_enat @ ( extended_eSuc @ N ) @ ( zero_zero @ extended_enat ) ) ).
% not_eSuc_ilei0
thf(fact_7870_i0__iless__eSuc,axiom,
! [N: extended_enat] : ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ ( extended_eSuc @ N ) ) ).
% i0_iless_eSuc
thf(fact_7871_ileI1,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_less @ extended_enat @ M @ N )
=> ( ord_less_eq @ extended_enat @ ( extended_eSuc @ M ) @ N ) ) ).
% ileI1
thf(fact_7872_rel__filter_Ocases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,F4: filter @ A,G3: filter @ B] :
( ( rel_filter @ A @ B @ R @ F4 @ G3 )
=> ~ ! [Z10: filter @ ( product_prod @ A @ B )] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) @ Z10 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) @ ( product_fst @ A @ B ) @ Z10 )
= F4 )
=> ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) @ ( product_snd @ A @ B ) @ Z10 )
!= G3 ) ) ) ) ).
% rel_filter.cases
thf(fact_7873_rel__filter_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( rel_filter @ A @ B )
= ( ^ [R6: A > B > $o,F8: filter @ A,G8: filter @ B] :
? [Z9: filter @ ( product_prod @ A @ B )] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) @ Z9 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) @ ( product_fst @ A @ B ) @ Z9 )
= F8 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) @ ( product_snd @ A @ B ) @ Z9 )
= G8 ) ) ) ) ).
% rel_filter.simps
thf(fact_7874_rel__filter_Ointros,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Z7: filter @ ( product_prod @ A @ B ),F4: filter @ A,G3: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) @ Z7 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) @ ( product_fst @ A @ B ) @ Z7 )
= F4 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) @ ( product_snd @ A @ B ) @ Z7 )
= G3 )
=> ( rel_filter @ A @ B @ R @ F4 @ G3 ) ) ) ) ).
% rel_filter.intros
thf(fact_7875_subset__code_I3_J,axiom,
! [C: $tType] :
~ ( ord_less_eq @ ( set @ C ) @ ( coset @ C @ ( nil @ C ) ) @ ( set2 @ C @ ( nil @ C ) ) ) ).
% subset_code(3)
thf(fact_7876_wo__rel_Oofilter__def,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R2 @ X2 ) @ A4 ) ) ) ) ) ).
% wo_rel.ofilter_def
thf(fact_7877_wo__rel_Oofilter__linord,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
| ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ) ) ) ).
% wo_rel.ofilter_linord
thf(fact_7878_subset__code_I2_J,axiom,
! [B: $tType,A4: set @ B,Ys2: list @ B] :
( ( ord_less_eq @ ( set @ B ) @ A4 @ ( coset @ B @ Ys2 ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ ( set2 @ B @ Ys2 ) )
=> ~ ( member @ B @ X2 @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_7879_ofilter__def,axiom,
! [A: $tType] :
( ( order_ofilter @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R5 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R5 @ X2 ) @ A6 ) ) ) ) ) ).
% ofilter_def
thf(fact_7880_insert__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( insert2 @ A @ X @ ( coset @ A @ Xs ) )
= ( coset @ A @ ( removeAll @ A @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_7881_union__coset__filter,axiom,
! [A: $tType,Xs: list @ A,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( coset @ A @ Xs ) @ A4 )
= ( coset @ A
@ ( filter2 @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ A4 )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_7882_ofilterIncl__def,axiom,
! [A: $tType] :
( ( bNF_We413866401316099525erIncl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A6: set @ A,B6: set @ A] :
( ( order_ofilter @ A @ R5 @ A6 )
& ( A6
!= ( field2 @ A @ R5 ) )
& ( order_ofilter @ A @ R5 @ B6 )
& ( B6
!= ( field2 @ A @ R5 ) )
& ( ord_less @ ( set @ A ) @ A6 @ B6 ) ) ) ) ) ) ).
% ofilterIncl_def
thf(fact_7883_bsqr__ofilter,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),D3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ ( product_prod @ A @ A ) @ ( bNF_Wellorder_bsqr @ A @ R2 ) @ D3 )
=> ( ( ord_less @ ( set @ ( product_prod @ A @ A ) ) @ D3
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R2 )
@ ^ [Uu3: A] : ( field2 @ A @ R2 ) ) )
=> ( ~ ? [A7: A] :
( ( field2 @ A @ R2 )
= ( order_under @ A @ R2 @ A7 ) )
=> ? [A9: set @ A] :
( ( order_ofilter @ A @ R2 @ A9 )
& ( ord_less @ ( set @ A ) @ A9 @ ( field2 @ A @ R2 ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ D3
@ ( product_Sigma @ A @ A @ A9
@ ^ [Uu3: A] : A9 ) ) ) ) ) ) ) ).
% bsqr_ofilter
thf(fact_7884_natLeq__on__well__order__on,axiom,
! [N: nat] :
( order_well_order_on @ nat
@ ( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X2 @ Y3 ) ) ) ) ) ).
% natLeq_on_well_order_on
thf(fact_7885_well__order__on__empty,axiom,
! [A: $tType] : ( order_well_order_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% well_order_on_empty
thf(fact_7886_well__order__on__Restr,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R2 ) )
=> ( order_well_order_on @ A @ A4
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) ) ) ) ) ).
% well_order_on_Restr
thf(fact_7887_natLeq__on__Well__order,axiom,
! [N: nat] :
( order_well_order_on @ nat
@ ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X2 @ Y3 ) ) ) ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X2 @ Y3 ) ) ) ) ) ).
% natLeq_on_Well_order
thf(fact_7888_Linear__order__Well__order__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
= ( ! [A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R2 ) )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A6 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_7889_ofilter__Restr__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( order_ofilter @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
@ A4 ) ) ) ) ).
% ofilter_Restr_subset
thf(fact_7890_ofilter__subset__ordLess,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ B2 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLess
thf(fact_7891_ofilter__ordLess,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ ( field2 @ A @ R2 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ R2 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ).
% ofilter_ordLess
thf(fact_7892_finite__ordLess__infinite,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ~ ( finite_finite2 @ B @ ( field2 @ B @ R4 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ).
% finite_ordLess_infinite
thf(fact_7893_underS__Restr__ordLess,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( field2 @ A @ R2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R2 @ A3 )
@ ^ [Uu3: A] : ( order_underS @ A @ R2 @ A3 ) ) )
@ R2 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% underS_Restr_ordLess
thf(fact_7894_ofilter__subset__ordLeq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLeq
thf(fact_7895_wo__rel_Oofilter__AboveS__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ ( order_AboveS @ A @ R2 @ A4 ) )
= ( field2 @ A @ R2 ) ) ) ) ).
% wo_rel.ofilter_AboveS_Field
thf(fact_7896_AboveS__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( order_AboveS @ A @ R2 @ A4 ) @ ( field2 @ A @ R2 ) ) ).
% AboveS_Field
thf(fact_7897_AboveS__disjoint,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( order_AboveS @ A @ R2 @ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% AboveS_disjoint
thf(fact_7898_wo__rel_Osuc__greater,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ( order_AboveS @ A @ R2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ B3 @ B2 )
=> ( ( ( bNF_Wellorder_wo_suc @ A @ R2 @ B2 )
!= B3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_Wellorder_wo_suc @ A @ R2 @ B2 ) ) @ R2 ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_7899_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( ( order_AboveS @ A @ R2 @ A4 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_Wellorder_wo_suc @ A @ R2 @ A4 ) ) @ R2 )
=> ( ( B3
!= ( bNF_Wellorder_wo_suc @ A @ R2 @ A4 ) )
=> ( member @ A @ B3 @ A4 ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_7900_wo__rel_Oequals__suc__AboveS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ A3 @ ( order_AboveS @ A @ R2 @ B2 ) )
=> ( ! [A27: A] :
( ( member @ A @ A27 @ ( order_AboveS @ A @ R2 @ B2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A27 ) @ R2 ) )
=> ( A3
= ( bNF_Wellorder_wo_suc @ A @ R2 @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_7901_wo__rel_Osuc__inField,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ( order_AboveS @ A @ R2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_Wellorder_wo_suc @ A @ R2 @ B2 ) @ ( field2 @ A @ R2 ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_7902_wo__rel_Osuc__AboveS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ( order_AboveS @ A @ R2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_Wellorder_wo_suc @ A @ R2 @ B2 ) @ ( order_AboveS @ A @ R2 @ B2 ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_7903_inf__filter__parametric,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( ( bi_total @ A @ B @ A4 )
=> ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > ( filter @ A ) ) @ ( ( filter @ B ) > ( filter @ B ) ) @ ( rel_filter @ A @ B @ A4 ) @ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A4 ) @ ( rel_filter @ A @ B @ A4 ) ) @ ( inf_inf @ ( filter @ A ) ) @ ( inf_inf @ ( filter @ B ) ) ) ) ) ).
% inf_filter_parametric
thf(fact_7904_frequently__parametric,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A4 )
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( frequently @ A )
@ ( frequently @ B ) ) ).
% frequently_parametric
thf(fact_7905_frequently__const,axiom,
! [A: $tType,F4: filter @ A,P: $o] :
( ( F4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( frequently @ A
@ ^ [X2: A] : P
@ F4 )
= P ) ) ).
% frequently_const
thf(fact_7906_frequently__bex__finite__distrib,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: B > A > $o,F4: filter @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( frequently @ B
@ ^ [X2: B] :
? [Y3: A] :
( ( member @ A @ Y3 @ A4 )
& ( P @ X2 @ Y3 ) )
@ F4 )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( frequently @ B
@ ^ [Y3: B] : ( P @ Y3 @ X2 )
@ F4 ) ) ) ) ) ).
% frequently_bex_finite_distrib
thf(fact_7907_frequently__bex__finite,axiom,
! [A: $tType,B: $tType,A4: set @ A,P: B > A > $o,F4: filter @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( frequently @ B
@ ^ [X2: B] :
? [Y3: A] :
( ( member @ A @ Y3 @ A4 )
& ( P @ X2 @ Y3 ) )
@ F4 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( frequently @ B
@ ^ [Y3: B] : ( P @ Y3 @ X3 )
@ F4 ) ) ) ) ).
% frequently_bex_finite
thf(fact_7908_frequently__all,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F4: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] :
! [X8: B] : ( P @ X2 @ X8 )
@ F4 )
= ( ! [Y7: A > B] :
( frequently @ A
@ ^ [X2: A] : ( P @ X2 @ ( Y7 @ X2 ) )
@ F4 ) ) ) ).
% frequently_all
thf(fact_7909_not__frequently__False,axiom,
! [A: $tType,F4: filter @ A] :
~ ( frequently @ A
@ ^ [X2: A] : $false
@ F4 ) ).
% not_frequently_False
thf(fact_7910_frequently__disj__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) )
@ F4 )
= ( ( frequently @ A @ P @ F4 )
| ( frequently @ A @ Q @ F4 ) ) ) ).
% frequently_disj_iff
thf(fact_7911_frequently__elim1,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( frequently @ A @ P @ F4 )
=> ( ! [I2: A] :
( ( P @ I2 )
=> ( Q @ I2 ) )
=> ( frequently @ A @ Q @ F4 ) ) ) ).
% frequently_elim1
thf(fact_7912_frequently__disj,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( frequently @ A @ P @ F4 )
=> ( ( frequently @ A @ Q @ F4 )
=> ( frequently @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) )
@ F4 ) ) ) ).
% frequently_disj
thf(fact_7913_frequently__ex,axiom,
! [A: $tType,P: A > $o,F4: filter @ A] :
( ( frequently @ A @ P @ F4 )
=> ? [X_1: A] : ( P @ X_1 ) ) ).
% frequently_ex
thf(fact_7914_eventually__frequently__const__simps_I1_J,axiom,
! [A: $tType,P: A > $o,C2: $o,F4: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] :
( ( P @ X2 )
& C2 )
@ F4 )
= ( ( frequently @ A @ P @ F4 )
& C2 ) ) ).
% eventually_frequently_const_simps(1)
thf(fact_7915_eventually__frequently__const__simps_I2_J,axiom,
! [A: $tType,C2: $o,P: A > $o,F4: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] :
( C2
& ( P @ X2 ) )
@ F4 )
= ( C2
& ( frequently @ A @ P @ F4 ) ) ) ).
% eventually_frequently_const_simps(2)
thf(fact_7916_frequently__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ( frequently @ A @ P @ F4 )
=> ( frequently @ A @ Q @ F4 ) ) ) ).
% frequently_mono
thf(fact_7917_frequentlyE,axiom,
! [A: $tType,P: A > $o,F4: filter @ A] :
( ( frequently @ A @ P @ F4 )
=> ~ ! [X3: A] :
~ ( P @ X3 ) ) ).
% frequentlyE
thf(fact_7918_eventually__frequently__const__simps_I5_J,axiom,
! [A: $tType,P: A > $o,C2: $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> C2 )
@ F4 )
= ( ( frequently @ A @ P @ F4 )
=> C2 ) ) ).
% eventually_frequently_const_simps(5)
thf(fact_7919_frequently__mp,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F4 )
=> ( ( frequently @ A @ P @ F4 )
=> ( frequently @ A @ Q @ F4 ) ) ) ).
% frequently_mp
thf(fact_7920_frequently__def,axiom,
! [A: $tType] :
( ( frequently @ A )
= ( ^ [P3: A > $o,F8: filter @ A] :
~ ( eventually @ A
@ ^ [X2: A] :
~ ( P3 @ X2 )
@ F8 ) ) ) ).
% frequently_def
thf(fact_7921_not__eventually,axiom,
! [A: $tType,P: A > $o,F4: filter @ A] :
( ( ~ ( eventually @ A @ P @ F4 ) )
= ( frequently @ A
@ ^ [X2: A] :
~ ( P @ X2 )
@ F4 ) ) ).
% not_eventually
thf(fact_7922_not__frequently,axiom,
! [A: $tType,P: A > $o,F4: filter @ A] :
( ( ~ ( frequently @ A @ P @ F4 ) )
= ( eventually @ A
@ ^ [X2: A] :
~ ( P @ X2 )
@ F4 ) ) ).
% not_frequently
thf(fact_7923_frequently__rev__mp,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( frequently @ A @ P @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F4 )
=> ( frequently @ A @ Q @ F4 ) ) ) ).
% frequently_rev_mp
thf(fact_7924_frequently__imp__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F4 )
= ( ( eventually @ A @ P @ F4 )
=> ( frequently @ A @ Q @ F4 ) ) ) ).
% frequently_imp_iff
thf(fact_7925_eventually__frequentlyE,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ~ ( P @ X2 )
| ( Q @ X2 ) )
@ F4 )
=> ( ( F4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( frequently @ A @ Q @ F4 ) ) ) ) ).
% eventually_frequentlyE
thf(fact_7926_eventually__frequently,axiom,
! [A: $tType,F4: filter @ A,P: A > $o] :
( ( F4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A @ P @ F4 )
=> ( frequently @ A @ P @ F4 ) ) ) ).
% eventually_frequently
thf(fact_7927_frequently__const__iff,axiom,
! [A: $tType,P: $o,F4: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] : P
@ F4 )
= ( P
& ( F4
!= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% frequently_const_iff
thf(fact_7928_bi__unique__rel__filter,axiom,
! [B: $tType,A: $tType,A4: A > B > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( bi_unique @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A4 ) ) ) ).
% bi_unique_rel_filter
thf(fact_7929_le__filter__parametric,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o ) @ ( rel_filter @ A @ B @ A4 )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A4 )
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( ord_less_eq @ ( filter @ A ) )
@ ( ord_less_eq @ ( filter @ B ) ) ) ) ).
% le_filter_parametric
thf(fact_7930_frequently__at,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A3: A,S: set @ A] :
( ( frequently @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ S ) )
= ( ! [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
=> ? [X2: A] :
( ( member @ A @ X2 @ S )
& ( X2 != A3 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A3 ) @ D5 )
& ( P @ X2 ) ) ) ) ) ) ).
% frequently_at
thf(fact_7931_less__filter__parametric,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o ) @ ( rel_filter @ A @ B @ A4 )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A4 )
@ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
@ ( ord_less @ ( filter @ A ) )
@ ( ord_less @ ( filter @ B ) ) ) ) ).
% less_filter_parametric
thf(fact_7932_card__of__UNION__ordLeq__infinite,axiom,
! [B: $tType,A: $tType,C: $tType,B2: set @ A,I5: set @ B,A4: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I5 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite
thf(fact_7933_List_Oset__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( set2 @ A @ ( insert @ A @ X @ Xs ) )
= ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% List.set_insert
thf(fact_7934_card__of__mono1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_mono1
thf(fact_7935_ex__bij__betw,axiom,
! [B: $tType,A: $tType,A4: set @ A,R2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ? [F6: B > A,B4: set @ B] : ( bij_betw @ B @ A @ F6 @ B4 @ A4 ) ) ).
% ex_bij_betw
thf(fact_7936_card__of__Times2,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ).
% card_of_Times2
thf(fact_7937_card__of__Times1,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu3: B] : A4 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ B @ A ) ) ) ) ).
% card_of_Times1
thf(fact_7938_card__of__empty,axiom,
! [B: $tType,A: $tType,A4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_empty
thf(fact_7939_card__of__empty3,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty3
thf(fact_7940_card__of__Sigma__ordLeq__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,B2: set @ A,I5: set @ B,A4: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C ) @ ( product_Sigma @ B @ C @ I5 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite
thf(fact_7941_infinite__iff__natLeq__ordLeq,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
!= ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Ca8665028551170535155natLeq @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_natLeq_ordLeq
thf(fact_7942_card__of__ordLeq__finite,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% card_of_ordLeq_finite
thf(fact_7943_card__of__ordLeq__infinite,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ~ ( finite_finite2 @ B @ B2 ) ) ) ).
% card_of_ordLeq_infinite
thf(fact_7944_infinite__iff__card__of__nat,axiom,
! [A: $tType,A4: set @ A] :
( ( ~ ( finite_finite2 @ A @ A4 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( top_top @ ( set @ nat ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_card_of_nat
thf(fact_7945_finite__iff__ordLess__natLeq,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ A @ nat ) ) ) ) ).
% finite_iff_ordLess_natLeq
thf(fact_7946_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [G: B > A] :
( ( image2 @ B @ A @ G @ B2 )
= A4 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeq2
thf(fact_7947_surj__imp__ordLeq,axiom,
! [B: $tType,A: $tType,B2: set @ A,F3: B > A,A4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% surj_imp_ordLeq
thf(fact_7948_card__of__singl__ordLeq,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert2 @ B @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_singl_ordLeq
thf(fact_7949_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B2: set @ A,A4: set @ B] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ? [F2: B > A] :
( ( image2 @ B @ A @ F2 @ A4 )
= B2 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ) ).
% card_of_ordLess2
thf(fact_7950_card__of__ordLeq,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( ? [F2: A > B] :
( ( inj_on @ A @ B @ F2 @ A4 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B2 ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_ordLeq
thf(fact_7951_card__of__ordLess,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( ~ ? [F2: A > B] :
( ( inj_on @ A @ B @ F2 @ A4 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B2 ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ).
% card_of_ordLess
thf(fact_7952_ordLeq3__finite__infinite,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( finite_finite2 @ B @ B2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% ordLeq3_finite_infinite
thf(fact_7953_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A13: A,A24: A,A4: set @ A,B2: set @ B] :
( ( ( A13 != A24 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A13 @ ( insert2 @ A @ A24 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times_aux
thf(fact_7954_finite__Plus__iff,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
= ( ( finite_finite2 @ A @ A4 )
& ( finite_finite2 @ B @ B2 ) ) ) ).
% finite_Plus_iff
thf(fact_7955_card__of__Un__Plus__ordLeq,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A4 @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% card_of_Un_Plus_ordLeq
thf(fact_7956_card__Plus,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B2 ) ) ) ) ) ).
% card_Plus
thf(fact_7957_finite__Plus,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) ) ) ) ).
% finite_Plus
thf(fact_7958_finite__PlusD_I1_J,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
=> ( finite_finite2 @ A @ A4 ) ) ).
% finite_PlusD(1)
thf(fact_7959_finite__PlusD_I2_J,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
=> ( finite_finite2 @ B @ B2 ) ) ).
% finite_PlusD(2)
thf(fact_7960_card__Plus__conv__if,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( ( ( finite_finite2 @ A @ A4 )
& ( finite_finite2 @ B @ B2 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B2 ) ) ) )
& ( ~ ( ( finite_finite2 @ A @ A4 )
& ( finite_finite2 @ B @ B2 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
= ( zero_zero @ nat ) ) ) ) ).
% card_Plus_conv_if
thf(fact_7961_card__of__Plus__ordLess__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,C2: set @ A,A4: set @ B,B2: set @ C] :
( ~ ( finite_finite2 @ A @ C2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C2 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C2 ) ) @ ( bNF_We4044943003108391690rdLess @ C @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A4 @ B2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C2 ) ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite
thf(fact_7962_card__of__Plus__Times,axiom,
! [B: $tType,A: $tType,A13: A,A24: A,A4: set @ A,B18: B,B24: B,B2: set @ B] :
( ( ( A13 != A24 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A13 @ ( insert2 @ A @ A24 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) )
=> ( ( ( B18 != B24 )
& ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ B18 @ ( insert2 @ B @ B24 @ ( bot_bot @ ( set @ B ) ) ) ) @ B2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times
thf(fact_7963_Plus__eq__empty__conv,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( ( sum_Plus @ A @ B @ A4 @ B2 )
= ( bot_bot @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Plus_eq_empty_conv
thf(fact_7964_card__of__Times__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ B,B2: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B2 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ A4
@ ^ [Uu3: B] : B2 ) )
@ R2 )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Times_ordLeq_infinite_Field
thf(fact_7965_Card__order__trans,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( X != Y )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( Y != Z )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R2 )
=> ( ( X != Z )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R2 ) ) ) ) ) ) ) ).
% Card_order_trans
thf(fact_7966_infinite__Card__order__limit,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
& ( A3 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X3 ) @ R2 ) ) ) ) ) ).
% infinite_Card_order_limit
thf(fact_7967_Card__order__wo__rel,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( bNF_Wellorder_wo_rel @ A @ R2 ) ) ).
% Card_order_wo_rel
thf(fact_7968_Card__order__infinite__not__under,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ~ ? [A10: A] :
( ( field2 @ A @ R2 )
= ( order_under @ A @ R2 @ A10 ) ) ) ) ).
% Card_order_infinite_not_under
thf(fact_7969_Card__order__empty,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% Card_order_empty
thf(fact_7970_Card__order__singl__ordLeq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),B3: B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( field2 @ A @ R2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert2 @ B @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% Card_order_singl_ordLeq
thf(fact_7971_card__of__Un__ordLeq__infinite__Field,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ B,B2: set @ B] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ) ).
% card_of_Un_ordLeq_infinite_Field
thf(fact_7972_card__of__empty1,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
| ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_empty1
thf(fact_7973_Card__order__Times2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ R2
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu3: B] : ( field2 @ A @ R2 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ) ) ).
% Card_order_Times2
thf(fact_7974_Card__order__Times1,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ R2
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R2 )
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ B ) ) ) ) ) ).
% Card_order_Times1
thf(fact_7975_Card__order__Times__same__infinite,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R2 )
@ ^ [Uu3: A] : ( field2 @ A @ R2 ) ) )
@ R2 )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ A ) @ A ) ) ) ) ).
% Card_order_Times_same_infinite
thf(fact_7976_card__of__UNION__ordLeq__infinite__Field,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set @ ( product_prod @ A @ A ),I5: set @ B,A4: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X3 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I5 ) ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite_Field
thf(fact_7977_card__of__Plus__ordLess__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ B,B2: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R2 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B2 ) @ R2 ) @ ( bNF_We4044943003108391690rdLess @ C @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A4 @ B2 ) ) @ R2 ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite_Field
thf(fact_7978_card__of__Plus__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ B,B2: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B2 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A4 @ B2 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLeq_infinite_Field
thf(fact_7979_card__of__Sigma__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),I5: set @ B,A4: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X3 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C ) @ ( product_Sigma @ B @ C @ I5 @ A4 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite_Field
thf(fact_7980_regularCard__UNION,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),As2: A > ( set @ B ),B2: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca7133664381575040944arCard @ A @ R2 )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ ( set @ B ) @ R2 @ As2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ As2 @ ( field2 @ A @ R2 ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ R2 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
& ( ord_less_eq @ ( set @ B ) @ B2 @ ( As2 @ X3 ) ) ) ) ) ) ) ) ).
% regularCard_UNION
thf(fact_7981_toCard__pred__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr1419584066657907630d_pred @ A @ B )
= ( ^ [A6: set @ A,R5: set @ ( product_prod @ B @ B ),F2: A > B] :
( ( inj_on @ A @ B @ F2 @ A6 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A6 ) @ ( field2 @ B @ R5 ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 ) ) ) ) ).
% toCard_pred_def
thf(fact_7982_cardSuc__UNION,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),As2: ( set @ A ) > ( set @ B ),B2: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) @ As2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As2 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) )
& ( ord_less_eq @ ( set @ B ) @ B2 @ ( As2 @ X3 ) ) ) ) ) ) ) ) ).
% cardSuc_UNION
thf(fact_7983_Card__order__Times__infinite,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),P6: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( field2 @ B @ P6 )
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P6 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R2 )
@ ^ [Uu3: A] : ( field2 @ B @ P6 ) ) )
@ R2 )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ ( field2 @ B @ P6 )
@ ^ [Uu3: B] : ( field2 @ A @ R2 ) ) )
@ R2 )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ) ).
% Card_order_Times_infinite
thf(fact_7984_finite__well__order__on__ordIso,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( order_well_order_on @ A @ A4 @ R2 )
=> ( ( order_well_order_on @ A @ A4 @ R4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% finite_well_order_on_ordIso
thf(fact_7985_card__of__empty2,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty2
thf(fact_7986_card__of__empty__ordIso,axiom,
! [B: $tType,A: $tType] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ).
% card_of_empty_ordIso
thf(fact_7987_card__of__ordIso__finite,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ A4 )
= ( finite_finite2 @ B @ B2 ) ) ) ).
% card_of_ordIso_finite
thf(fact_7988_internalize__ordLeq,axiom,
! [A: $tType,B: $tType,R4: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R4 @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [P5: set @ ( product_prod @ B @ B )] :
( ( ord_less_eq @ ( set @ B ) @ ( field2 @ B @ P5 ) @ ( field2 @ B @ R2 ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R4 @ P5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P5 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_ordLeq
thf(fact_7989_infinite__cardSuc__regularCard,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( bNF_Ca7133664381575040944arCard @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) ) ) ).
% infinite_cardSuc_regularCard
thf(fact_7990_internalize__card__of__ordLeq2,axiom,
! [A: $tType,B: $tType,A4: set @ A,C2: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ C2 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C2 ) ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq2
thf(fact_7991_internalize__ordLess,axiom,
! [A: $tType,B: $tType,R4: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R4 @ R2 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( ? [P5: set @ ( product_prod @ B @ B )] :
( ( ord_less @ ( set @ B ) @ ( field2 @ B @ P5 ) @ ( field2 @ B @ R2 ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R4 @ P5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P5 @ R2 ) @ ( bNF_We4044943003108391690rdLess @ B @ B ) ) ) ) ) ).
% internalize_ordLess
thf(fact_7992_card__of__cardSuc__finite,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ ( set @ A ) @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) ) )
= ( finite_finite2 @ A @ A4 ) ) ).
% card_of_cardSuc_finite
thf(fact_7993_internalize__card__of__ordLeq,axiom,
! [A: $tType,B: $tType,A4: set @ A,R2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ ( field2 @ B @ R2 ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq
thf(fact_7994_card__of__ordIso__finite__Field,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
= ( finite_finite2 @ B @ A4 ) ) ) ) ).
% card_of_ordIso_finite_Field
thf(fact_7995_cardSuc__finite,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( finite_finite2 @ ( set @ A ) @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) )
= ( finite_finite2 @ A @ ( field2 @ A @ R2 ) ) ) ) ).
% cardSuc_finite
thf(fact_7996_card__of__Times__same__infinite,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% card_of_Times_same_infinite
thf(fact_7997_card__of__Times__infinite,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu3: B] : A4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ).
% card_of_Times_infinite
thf(fact_7998_card__of__Times__infinite__simps_I1_J,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(1)
thf(fact_7999_card__of__Times__infinite__simps_I3_J,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu3: B] : A4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(3)
thf(fact_8000_regularCard__def,axiom,
! [A: $tType] :
( ( bNF_Ca7133664381575040944arCard @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [K5: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ K5 @ ( field2 @ A @ R5 ) )
& ( bNF_Ca7293521722713021262ofinal @ A @ K5 @ R5 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ K5 ) @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% regularCard_def
thf(fact_8001_cardSuc__UNION__Cinfinite,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),As2: ( set @ A ) > ( set @ B ),B2: set @ B] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) @ As2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As2 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) )
& ( ord_less_eq @ ( set @ B ) @ B2 @ ( As2 @ X3 ) ) ) ) ) ) ) ).
% cardSuc_UNION_Cinfinite
thf(fact_8002_cinfinite__def,axiom,
! [A: $tType] :
( ( bNF_Ca4139267488887388095finite @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
~ ( finite_finite2 @ A @ ( field2 @ A @ R5 ) ) ) ) ).
% cinfinite_def
thf(fact_8003_Cinfinite__limit,axiom,
! [A: $tType,X: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
& ( X != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X3 ) @ R2 ) ) ) ) ).
% Cinfinite_limit
thf(fact_8004_Cinfinite__limit2,axiom,
! [A: $tType,X15: A,R2: set @ ( product_prod @ A @ A ),X23: A] :
( ( member @ A @ X15 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ X23 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
& ( X15 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X15 @ X3 ) @ R2 )
& ( X23 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X23 @ X3 ) @ R2 ) ) ) ) ) ).
% Cinfinite_limit2
thf(fact_8005_card__of__Func__UNIV,axiom,
! [B: $tType,A: $tType,B2: set @ B] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > B ) @ ( bNF_Wellorder_Func @ A @ B @ ( top_top @ ( set @ A ) ) @ B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F2: A > B] : ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) @ B2 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > B ) @ ( A > B ) ) ) ).
% card_of_Func_UNIV
thf(fact_8006_card__of__bool,axiom,
! [A: $tType,A13: A,A24: A] :
( ( A13 != A24 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ $o @ ( top_top @ ( set @ $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( insert2 @ A @ A13 @ ( insert2 @ A @ A24 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ) ).
% card_of_bool
thf(fact_8007_Un__Cinfinite__bound,axiom,
! [B: $tType,A: $tType,A4: set @ A,R2: set @ ( product_prod @ B @ B ),B2: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% Un_Cinfinite_bound
thf(fact_8008_card__of__Plus__empty2,axiom,
! [B: $tType,A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( bot_bot @ ( set @ B ) ) @ A4 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_empty2
thf(fact_8009_card__of__Plus__empty1,axiom,
! [B: $tType,A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus_empty1
thf(fact_8010_Cinfinite__limit__finite,axiom,
! [A: $tType,X4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
& ! [Xa: A] :
( ( member @ A @ Xa @ X4 )
=> ( ( Xa != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% Cinfinite_limit_finite
thf(fact_8011_card__of__Plus__infinite,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B2 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Plus_infinite
thf(fact_8012_card__of__Plus__infinite1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% card_of_Plus_infinite1
thf(fact_8013_card__of__Plus__infinite2,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B2 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ).
% card_of_Plus_infinite2
thf(fact_8014_Card__order__Plus__infinite,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),P6: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P6 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ ( field2 @ A @ R2 ) @ ( field2 @ B @ P6 ) ) ) @ R2 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( field2 @ B @ P6 ) @ ( field2 @ A @ R2 ) ) ) @ R2 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ) ).
% Card_order_Plus_infinite
thf(fact_8015_card__of__Times__infinite__simps_I4_J,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu3: B] : A4 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ B @ A ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(4)
thf(fact_8016_card__of__Times__infinite__simps_I2_J,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(2)
thf(fact_8017_Cnotzero__imp__not__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( ( field2 @ A @ R2 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% Cnotzero_imp_not_empty
thf(fact_8018_czeroI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( field2 @ A @ R2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% czeroI
thf(fact_8019_czero__def,axiom,
! [A: $tType] :
( ( bNF_Cardinal_czero @ A )
= ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% czero_def
thf(fact_8020_czeroE,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( field2 @ A @ R2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% czeroE
thf(fact_8021_card__of__ordIso__czero__iff__empty,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_ordIso_czero_iff_empty
thf(fact_8022_cexp__mono2_H,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),Q5: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P22 @ R23 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q5 ) @ Q5 )
=> ( ( ( ( field2 @ A @ P22 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( field2 @ B @ R23 )
= ( bot_bot @ ( set @ B ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q5 @ P22 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q5 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2'
thf(fact_8023_cexp__mono_H,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P12: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R23: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P12 @ R12 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ R23 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( ( field2 @ C @ P22 )
= ( bot_bot @ ( set @ C ) ) )
=> ( ( field2 @ D @ R23 )
= ( bot_bot @ ( set @ D ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P12 @ P22 ) @ ( bNF_Cardinal_cexp @ B @ D @ R12 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ).
% cexp_mono'
thf(fact_8024_Real_Opositive__def,axiom,
( positive2
= ( map_fun @ real @ ( nat > rat ) @ $o @ $o @ rep_real @ ( id @ $o )
@ ^ [X8: nat > rat] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less @ rat @ R5 @ ( X8 @ N2 ) ) ) ) ) ) ).
% Real.positive_def
thf(fact_8025_cmod__plus__Re__le__0__iff,axiom,
! [Z: complex] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( re @ Z ) ) @ ( zero_zero @ real ) )
= ( ( re @ Z )
= ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) ) ) ).
% cmod_plus_Re_le_0_iff
thf(fact_8026_id__funpow,axiom,
! [A: $tType,N: nat] :
( ( compow @ ( A > A ) @ N @ ( id @ A ) )
= ( id @ A ) ) ).
% id_funpow
thf(fact_8027_filtermap__id,axiom,
! [A: $tType] :
( ( filtermap @ A @ A @ ( id @ A ) )
= ( id @ ( filter @ A ) ) ) ).
% filtermap_id
thf(fact_8028_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_8029_drop__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A @ ( zero_zero @ nat ) )
= ( id @ A ) ) ) ).
% drop_bit_0
thf(fact_8030_funpow__simps__right_I1_J,axiom,
! [A: $tType,F3: A > A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F3 )
= ( id @ A ) ) ).
% funpow_simps_right(1)
thf(fact_8031_complex__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq @ real @ ( re @ X ) @ ( real_V7770717601297561774m_norm @ complex @ X ) ) ).
% complex_Re_le_cmod
thf(fact_8032_abs__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( re @ X ) ) @ ( real_V7770717601297561774m_norm @ complex @ X ) ) ).
% abs_Re_le_cmod
thf(fact_8033_Re__csqrt,axiom,
! [Z: complex] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( csqrt @ Z ) ) ) ).
% Re_csqrt
thf(fact_8034_ofilter__subset__embedS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ B2 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
@ ( id @ A ) ) ) ) ) ) ).
% ofilter_subset_embedS
thf(fact_8035_Rat_Opositive__def,axiom,
( positive
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ $o @ $o @ rep_Rat @ ( id @ $o )
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ).
% Rat.positive_def
thf(fact_8036_rotate0,axiom,
! [A: $tType] :
( ( rotate @ A @ ( zero_zero @ nat ) )
= ( id @ ( list @ A ) ) ) ).
% rotate0
thf(fact_8037_embedS__Field,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Wellorder_embedS @ A @ B @ R2 @ R4 @ F3 )
=> ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( field2 @ A @ R2 ) ) @ ( field2 @ B @ R4 ) ) ) ) ).
% embedS_Field
thf(fact_8038_ofilter__subset__embedS__iso,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ( ord_less @ ( set @ A ) @ A4 @ B2 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
@ ( id @ A ) ) )
& ( ( A4 = B2 )
= ( bNF_Wellorder_iso @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
@ ( id @ A ) ) ) ) ) ) ) ).
% ofilter_subset_embedS_iso
thf(fact_8039_complex__abs__le__norm,axiom,
! [Z: complex] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( abs_abs @ real @ ( re @ Z ) ) @ ( abs_abs @ real @ ( im @ Z ) ) ) @ ( times_times @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) ) ).
% complex_abs_le_norm
thf(fact_8040_csqrt__unique,axiom,
! [W2: complex,Z: complex] :
( ( ( power_power @ complex @ W2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ W2 ) )
| ( ( ( re @ W2 )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ W2 ) ) ) )
=> ( ( csqrt @ Z )
= W2 ) ) ) ).
% csqrt_unique
thf(fact_8041_csqrt__of__real__nonneg,axiom,
! [X: complex] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ X ) )
=> ( ( csqrt @ X )
= ( real_Vector_of_real @ complex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).
% csqrt_of_real_nonneg
thf(fact_8042_abs__Im__le__cmod,axiom,
! [X: complex] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( im @ X ) ) @ ( real_V7770717601297561774m_norm @ complex @ X ) ) ).
% abs_Im_le_cmod
thf(fact_8043_cmod__Im__le__iff,axiom,
! [X: complex,Y: complex] :
( ( ( re @ X )
= ( re @ Y ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ complex @ X ) @ ( real_V7770717601297561774m_norm @ complex @ Y ) )
= ( ord_less_eq @ real @ ( abs_abs @ real @ ( im @ X ) ) @ ( abs_abs @ real @ ( im @ Y ) ) ) ) ) ).
% cmod_Im_le_iff
thf(fact_8044_cmod__Re__le__iff,axiom,
! [X: complex,Y: complex] :
( ( ( im @ X )
= ( im @ Y ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ complex @ X ) @ ( real_V7770717601297561774m_norm @ complex @ Y ) )
= ( ord_less_eq @ real @ ( abs_abs @ real @ ( re @ X ) ) @ ( abs_abs @ real @ ( re @ Y ) ) ) ) ) ).
% cmod_Re_le_iff
thf(fact_8045_csqrt__principal,axiom,
! [Z: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( csqrt @ Z ) ) )
| ( ( ( re @ ( csqrt @ Z ) )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% csqrt_principal
thf(fact_8046_cmod__le,axiom,
! [Z: complex] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( plus_plus @ real @ ( abs_abs @ real @ ( re @ Z ) ) @ ( abs_abs @ real @ ( im @ Z ) ) ) ) ).
% cmod_le
thf(fact_8047_complex__neq__0,axiom,
! [Z: complex] :
( ( Z
!= ( zero_zero @ complex ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_neq_0
thf(fact_8048_csqrt__square,axiom,
! [B3: complex] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ B3 ) )
| ( ( ( re @ B3 )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ B3 ) ) ) )
=> ( ( csqrt @ ( power_power @ complex @ B3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= B3 ) ) ).
% csqrt_square
thf(fact_8049_csqrt__of__real__nonpos,axiom,
! [X: complex] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( re @ X ) @ ( zero_zero @ real ) )
=> ( ( csqrt @ X )
= ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sqrt @ ( abs_abs @ real @ ( re @ X ) ) ) ) ) ) ) ) ).
% csqrt_of_real_nonpos
thf(fact_8050_csqrt__minus,axiom,
! [X: complex] :
( ( ( ord_less @ real @ ( im @ X ) @ ( zero_zero @ real ) )
| ( ( ( im @ X )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ X ) ) ) )
=> ( ( csqrt @ ( uminus_uminus @ complex @ X ) )
= ( times_times @ complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% csqrt_minus
thf(fact_8051_ofilter__subset__embed,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
= ( bNF_Wellorder_embed @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
@ ( id @ A ) ) ) ) ) ) ).
% ofilter_subset_embed
thf(fact_8052_ofilter__embed,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R2 ) )
& ( bNF_Wellorder_embed @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ R2
@ ( id @ A ) ) ) ) ) ).
% ofilter_embed
thf(fact_8053_embed__Field,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( bNF_Wellorder_embed @ A @ B @ R2 @ R4 @ F3 )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( field2 @ A @ R2 ) ) @ ( field2 @ B @ R4 ) ) ) ).
% embed_Field
thf(fact_8054_embedS__iff,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Wellorder_embed @ A @ B @ R2 @ R4 @ F3 )
=> ( ( bNF_Wellorder_embedS @ A @ B @ R2 @ R4 @ F3 )
= ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( field2 @ A @ R2 ) ) @ ( field2 @ B @ R4 ) ) ) ) ) ).
% embedS_iff
thf(fact_8055_series__comparison__complex,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G2: nat > complex,N6: nat,F3: nat > A] :
( ( summable @ complex @ G2 )
=> ( ! [N3: nat] : ( member @ complex @ ( G2 @ N3 ) @ ( real_Vector_Reals @ complex ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( G2 @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( real_V7770717601297561774m_norm @ complex @ ( G2 @ N3 ) ) ) )
=> ( summable @ A @ F3 ) ) ) ) ) ) ).
% series_comparison_complex
thf(fact_8056_Rangep__Range__eq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( rangep @ A @ B
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R2 ) )
= ( ^ [X2: B] : ( member @ B @ X2 @ ( range @ A @ B @ R2 ) ) ) ) ).
% Rangep_Range_eq
thf(fact_8057_nonzero__Reals__inverse,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( real_Vector_Reals @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( member @ A @ ( inverse_inverse @ A @ A3 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% nonzero_Reals_inverse
thf(fact_8058_Reals__0,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_0
thf(fact_8059_nonzero__Reals__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [A3: A,B3: A] :
( ( member @ A @ A3 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B3 @ ( real_Vector_Reals @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( member @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ) ).
% nonzero_Reals_divide
thf(fact_8060_Rangep_Ocases,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A3: B] :
( ( rangep @ A @ B @ R2 @ A3 )
=> ~ ! [A7: A] :
~ ( R2 @ A7 @ A3 ) ) ).
% Rangep.cases
thf(fact_8061_Rangep_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( rangep @ A @ B )
= ( ^ [R5: A > B > $o,A5: B] :
? [B5: A,C6: B] :
( ( A5 = C6 )
& ( R5 @ B5 @ C6 ) ) ) ) ).
% Rangep.simps
thf(fact_8062_RangePI,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A3: A,B3: B] :
( ( R2 @ A3 @ B3 )
=> ( rangep @ A @ B @ R2 @ B3 ) ) ).
% RangePI
thf(fact_8063_RangepE,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,B3: B] :
( ( rangep @ A @ B @ R2 @ B3 )
=> ~ ! [A7: A] :
~ ( R2 @ A7 @ B3 ) ) ).
% RangepE
thf(fact_8064_Range__def,axiom,
! [B: $tType,A: $tType] :
( ( range @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ B
@ ( rangep @ A @ B
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R5 ) ) ) ) ) ).
% Range_def
thf(fact_8065_num_Orec__transfer,axiom,
! [A: $tType,B: $tType,S: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( ( num > A > A ) > ( num > A > A ) > num > A ) @ ( ( num > B > B ) > ( num > B > B ) > num > B ) @ S
@ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( ( num > A > A ) > num > A ) @ ( ( num > B > B ) > num > B )
@ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
@ ^ [Y4: num,Z2: num] : Y4 = Z2
@ ( bNF_rel_fun @ A @ B @ A @ B @ S @ S ) )
@ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( num > A ) @ ( num > B )
@ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
@ ^ [Y4: num,Z2: num] : Y4 = Z2
@ ( bNF_rel_fun @ A @ B @ A @ B @ S @ S ) )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y4: num,Z2: num] : Y4 = Z2
@ S ) ) )
@ ( rec_num @ A )
@ ( rec_num @ B ) ) ).
% num.rec_transfer
thf(fact_8066_complex__div__gt__0,axiom,
! [A3: complex,B3: complex] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A3 @ B3 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) ) )
& ( ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A3 @ B3 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) ) ) ) ).
% complex_div_gt_0
thf(fact_8067_Re__complex__div__gt__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A3 @ B3 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) ) ) ).
% Re_complex_div_gt_0
thf(fact_8068_Re__complex__div__lt__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less @ real @ ( re @ ( divide_divide @ complex @ A3 @ B3 ) ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Re_complex_div_lt_0
thf(fact_8069_Re__complex__div__le__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq @ real @ ( re @ ( divide_divide @ complex @ A3 @ B3 ) ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Re_complex_div_le_0
thf(fact_8070_Re__complex__div__ge__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A3 @ B3 ) ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) ) ) ).
% Re_complex_div_ge_0
thf(fact_8071_Im__complex__div__gt__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A3 @ B3 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) ) ) ).
% Im_complex_div_gt_0
thf(fact_8072_Im__complex__div__lt__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less @ real @ ( im @ ( divide_divide @ complex @ A3 @ B3 ) ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Im_complex_div_lt_0
thf(fact_8073_Im__complex__div__le__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq @ real @ ( im @ ( divide_divide @ complex @ A3 @ B3 ) ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Im_complex_div_le_0
thf(fact_8074_Im__complex__div__ge__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A3 @ B3 ) ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) ) ) ).
% Im_complex_div_ge_0
thf(fact_8075_num_Ocase__transfer,axiom,
! [A: $tType,B: $tType,S: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( ( num > A ) > ( num > A ) > num > A ) @ ( ( num > B ) > ( num > B ) > num > B ) @ S
@ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( ( num > A ) > num > A ) @ ( ( num > B ) > num > B )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y4: num,Z2: num] : Y4 = Z2
@ S )
@ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( num > A ) @ ( num > B )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y4: num,Z2: num] : Y4 = Z2
@ S )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y4: num,Z2: num] : Y4 = Z2
@ S ) ) )
@ ( case_num @ A )
@ ( case_num @ B ) ) ).
% num.case_transfer
thf(fact_8076_Zfun__imp__Zfun,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F4: filter @ A,G2: A > C,K4: real] :
( ( zfun @ A @ B @ F3 @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G2 @ X2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) @ K4 ) )
@ F4 )
=> ( zfun @ A @ C @ G2 @ F4 ) ) ) ) ).
% Zfun_imp_Zfun
thf(fact_8077_Zfun__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F4: filter @ A] :
( zfun @ A @ B
@ ^ [X2: A] : ( zero_zero @ B )
@ F4 ) ) ).
% Zfun_zero
thf(fact_8078_num_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F16: A,F25: num > A,F33: num > A,Num: num] :
( ( H @ ( case_num @ A @ F16 @ F25 @ F33 @ Num ) )
= ( case_num @ B @ ( H @ F16 )
@ ^ [X2: num] : ( H @ ( F25 @ X2 ) )
@ ^ [X2: num] : ( H @ ( F33 @ X2 ) )
@ Num ) ) ).
% num.case_distrib
thf(fact_8079_Zfun__le,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G2: A > B,F4: filter @ A,F3: A > C] :
( ( zfun @ A @ B @ G2 @ F4 )
=> ( ! [X3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( F3 @ X3 ) ) @ ( real_V7770717601297561774m_norm @ B @ ( G2 @ X3 ) ) )
=> ( zfun @ A @ C @ F3 @ F4 ) ) ) ) ).
% Zfun_le
thf(fact_8080_Zfun__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ( ( zfun @ A @ B )
= ( ^ [F2: A > B,F8: filter @ A] :
! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) @ R5 )
@ F8 ) ) ) ) ) ).
% Zfun_def
thf(fact_8081_ZfunI,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) @ R3 )
@ F4 ) )
=> ( zfun @ A @ B @ F3 @ F4 ) ) ) ).
% ZfunI
thf(fact_8082_ZfunD,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A,R2: real] :
( ( zfun @ A @ B @ F3 @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) @ R2 )
@ F4 ) ) ) ) ).
% ZfunD
thf(fact_8083_semilattice__order__set_Osubset__imp,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: set @ A,B2: set @ A] :
( ( lattic4895041142388067077er_set @ A @ F3 @ Less_eq @ Less )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( Less_eq @ ( lattic1715443433743089157tice_F @ A @ F3 @ B2 ) @ ( lattic1715443433743089157tice_F @ A @ F3 @ A4 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_8084_MOST__INFM,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( eventually @ A @ P @ ( cofinite @ A ) )
=> ( frequently @ A @ P @ ( cofinite @ A ) ) ) ) ).
% MOST_INFM
thf(fact_8085_not__MOST,axiom,
! [A: $tType,P: A > $o] :
( ( ~ ( eventually @ A @ P @ ( cofinite @ A ) ) )
= ( frequently @ A
@ ^ [X2: A] :
~ ( P @ X2 )
@ ( cofinite @ A ) ) ) ).
% not_MOST
thf(fact_8086_not__INFM,axiom,
! [A: $tType,P: A > $o] :
( ( ~ ( frequently @ A @ P @ ( cofinite @ A ) ) )
= ( eventually @ A
@ ^ [X2: A] :
~ ( P @ X2 )
@ ( cofinite @ A ) ) ) ).
% not_INFM
thf(fact_8087_cofinite__bot,axiom,
! [A: $tType] :
( ( ( cofinite @ A )
= ( bot_bot @ ( filter @ A ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% cofinite_bot
thf(fact_8088_MOST__const,axiom,
! [A: $tType,P: $o] :
( ( eventually @ A
@ ^ [X2: A] : P
@ ( cofinite @ A ) )
= ( P
| ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% MOST_const
thf(fact_8089_MOST__eq_I1_J,axiom,
! [A: $tType,A3: A] :
( ( eventually @ A
@ ^ [X2: A] : X2 = A3
@ ( cofinite @ A ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% MOST_eq(1)
thf(fact_8090_MOST__eq_I2_J,axiom,
! [A: $tType,A3: A] :
( ( eventually @ A
@ ( ^ [Y4: A,Z2: A] : Y4 = Z2
@ A3 )
@ ( cofinite @ A ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% MOST_eq(2)
thf(fact_8091_INFM__neq_I2_J,axiom,
! [A: $tType,A3: A] :
( ( frequently @ A
@ ^ [X2: A] : A3 != X2
@ ( cofinite @ A ) )
= ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% INFM_neq(2)
thf(fact_8092_INFM__neq_I1_J,axiom,
! [A: $tType,A3: A] :
( ( frequently @ A
@ ^ [X2: A] : X2 != A3
@ ( cofinite @ A ) )
= ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% INFM_neq(1)
thf(fact_8093_INFM__const,axiom,
! [A: $tType,P: $o] :
( ( frequently @ A
@ ^ [X2: A] : P
@ ( cofinite @ A ) )
= ( P
& ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% INFM_const
thf(fact_8094_MOST__ge__nat,axiom,
! [M: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ M ) @ ( cofinite @ nat ) ) ).
% MOST_ge_nat
thf(fact_8095_MOST__nat__le,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( cofinite @ nat ) )
= ( ? [M2: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( P @ N2 ) ) ) ) ).
% MOST_nat_le
thf(fact_8096_cofinite__def,axiom,
! [A: $tType] :
( ( cofinite @ A )
= ( abs_filter @ A
@ ^ [P3: A > $o] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
~ ( P3 @ X2 ) ) ) ) ) ).
% cofinite_def
thf(fact_8097_semilattice__order__set_OcoboundedI,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: set @ A,A3: A] :
( ( lattic4895041142388067077er_set @ A @ F3 @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( Less_eq @ ( lattic1715443433743089157tice_F @ A @ F3 @ A4 ) @ A3 ) ) ) ) ).
% semilattice_order_set.coboundedI
thf(fact_8098_Sup__fin__def,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( lattic1715443433743089157tice_F @ A @ ( sup_sup @ A ) ) ) ) ).
% Sup_fin_def
thf(fact_8099_eventually__cofinite,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
= ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) ) ) ).
% eventually_cofinite
thf(fact_8100_MOST__iff__finiteNeg,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
= ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) ) ) ).
% MOST_iff_finiteNeg
thf(fact_8101_MOST__Suc__iff,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
@ ( cofinite @ nat ) )
= ( eventually @ nat @ P @ ( cofinite @ nat ) ) ) ).
% MOST_Suc_iff
thf(fact_8102_MOST__SucI,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( cofinite @ nat ) )
=> ( eventually @ nat
@ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
@ ( cofinite @ nat ) ) ) ).
% MOST_SucI
thf(fact_8103_MOST__SucD,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
@ ( cofinite @ nat ) )
=> ( eventually @ nat @ P @ ( cofinite @ nat ) ) ) ).
% MOST_SucD
thf(fact_8104_Max__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( lattic1715443433743089157tice_F @ A @ ( ord_max @ A ) ) ) ) ).
% Max_def
thf(fact_8105_Min__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( lattic1715443433743089157tice_F @ A @ ( ord_min @ A ) ) ) ) ).
% Min_def
thf(fact_8106_cofinite__eq__sequentially,axiom,
( ( cofinite @ nat )
= ( at_top @ nat ) ) ).
% cofinite_eq_sequentially
thf(fact_8107_semilattice__set_OF_Ocong,axiom,
! [A: $tType] :
( ( lattic1715443433743089157tice_F @ A )
= ( lattic1715443433743089157tice_F @ A ) ) ).
% semilattice_set.F.cong
thf(fact_8108_MOST__neq_I2_J,axiom,
! [A: $tType,A3: A] :
( eventually @ A
@ ^ [X2: A] : A3 != X2
@ ( cofinite @ A ) ) ).
% MOST_neq(2)
thf(fact_8109_MOST__neq_I1_J,axiom,
! [A: $tType,A3: A] :
( eventually @ A
@ ^ [X2: A] : X2 != A3
@ ( cofinite @ A ) ) ).
% MOST_neq(1)
thf(fact_8110_MOST__eq__imp_I2_J,axiom,
! [A: $tType,A3: A,P: A > $o] :
( eventually @ A
@ ^ [X2: A] :
( ( A3 = X2 )
=> ( P @ X2 ) )
@ ( cofinite @ A ) ) ).
% MOST_eq_imp(2)
thf(fact_8111_MOST__eq__imp_I1_J,axiom,
! [A: $tType,A3: A,P: A > $o] :
( eventually @ A
@ ^ [X2: A] :
( ( X2 = A3 )
=> ( P @ X2 ) )
@ ( cofinite @ A ) ) ).
% MOST_eq_imp(1)
thf(fact_8112_MOST__I,axiom,
! [A: $tType,P: A > $o] :
( ! [X_1: A] : ( P @ X_1 )
=> ( eventually @ A @ P @ ( cofinite @ A ) ) ) ).
% MOST_I
thf(fact_8113_ALL__MOST,axiom,
! [A: $tType,P: A > $o] :
( ! [X_1: A] : ( P @ X_1 )
=> ( eventually @ A @ P @ ( cofinite @ A ) ) ) ).
% ALL_MOST
thf(fact_8114_MOST__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( eventually @ A @ Q @ ( cofinite @ A ) ) ) ) ).
% MOST_mono
thf(fact_8115_MOST__conjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
=> ( ( eventually @ A @ Q @ ( cofinite @ A ) )
=> ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ ( cofinite @ A ) ) ) ) ).
% MOST_conjI
thf(fact_8116_MOST__rev__mp,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ ( cofinite @ A ) )
=> ( eventually @ A @ Q @ ( cofinite @ A ) ) ) ) ).
% MOST_rev_mp
thf(fact_8117_MOST__imp__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ ( cofinite @ A ) )
= ( eventually @ A @ Q @ ( cofinite @ A ) ) ) ) ).
% MOST_imp_iff
thf(fact_8118_MOST__conj__distrib,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ ( cofinite @ A ) )
= ( ( eventually @ A @ P @ ( cofinite @ A ) )
& ( eventually @ A @ Q @ ( cofinite @ A ) ) ) ) ).
% MOST_conj_distrib
thf(fact_8119_Inf__fin__def,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( lattic1715443433743089157tice_F @ A @ ( inf_inf @ A ) ) ) ) ).
% Inf_fin_def
thf(fact_8120_MOST__nat,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( cofinite @ nat ) )
= ( ? [M2: nat] :
! [N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( P @ N2 ) ) ) ) ).
% MOST_nat
thf(fact_8121_INFM__nat,axiom,
! [P: nat > $o] :
( ( frequently @ nat @ P @ ( cofinite @ nat ) )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ( P @ N2 ) ) ) ) ).
% INFM_nat
thf(fact_8122_INFM__imp__distrib,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( frequently @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ ( cofinite @ A ) )
= ( ( eventually @ A @ P @ ( cofinite @ A ) )
=> ( frequently @ A @ Q @ ( cofinite @ A ) ) ) ) ).
% INFM_imp_distrib
thf(fact_8123_Alm__all__def,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
= ( ~ ( frequently @ A
@ ^ [X2: A] :
~ ( P @ X2 )
@ ( cofinite @ A ) ) ) ) ).
% Alm_all_def
thf(fact_8124_INFM__conjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( frequently @ A @ P @ ( cofinite @ A ) )
=> ( ( eventually @ A @ Q @ ( cofinite @ A ) )
=> ( frequently @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ ( cofinite @ A ) ) ) ) ).
% INFM_conjI
thf(fact_8125_not__INFM__eq_I2_J,axiom,
! [A: $tType,A3: A] :
~ ( frequently @ A
@ ( ^ [Y4: A,Z2: A] : Y4 = Z2
@ A3 )
@ ( cofinite @ A ) ) ).
% not_INFM_eq(2)
thf(fact_8126_not__INFM__eq_I1_J,axiom,
! [A: $tType,A3: A] :
~ ( frequently @ A
@ ^ [X2: A] : X2 = A3
@ ( cofinite @ A ) ) ).
% not_INFM_eq(1)
thf(fact_8127_INFM__E,axiom,
! [A: $tType,P: A > $o] :
( ( frequently @ A @ P @ ( cofinite @ A ) )
=> ~ ! [X3: A] :
~ ( P @ X3 ) ) ).
% INFM_E
thf(fact_8128_INFM__EX,axiom,
! [A: $tType,P: A > $o] :
( ( frequently @ A @ P @ ( cofinite @ A ) )
=> ? [X_1: A] : ( P @ X_1 ) ) ).
% INFM_EX
thf(fact_8129_INFM__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( frequently @ A @ P @ ( cofinite @ A ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( frequently @ A @ Q @ ( cofinite @ A ) ) ) ) ).
% INFM_mono
thf(fact_8130_INFM__disj__distrib,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( frequently @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) )
@ ( cofinite @ A ) )
= ( ( frequently @ A @ P @ ( cofinite @ A ) )
| ( frequently @ A @ Q @ ( cofinite @ A ) ) ) ) ).
% INFM_disj_distrib
thf(fact_8131_frequently__cofinite,axiom,
! [A: $tType,P: A > $o] :
( ( frequently @ A @ P @ ( cofinite @ A ) )
= ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) ) ) ) ).
% frequently_cofinite
thf(fact_8132_INFM__iff__infinite,axiom,
! [A: $tType,P: A > $o] :
( ( frequently @ A @ P @ ( cofinite @ A ) )
= ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) ) ) ) ).
% INFM_iff_infinite
thf(fact_8133_INFM__nat__le,axiom,
! [P: nat > $o] :
( ( frequently @ nat @ P @ ( cofinite @ nat ) )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
& ( P @ N2 ) ) ) ) ).
% INFM_nat_le
thf(fact_8134_MOST__finite__Ball__distrib,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( eventually @ B
@ ^ [Y3: B] :
! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( P @ X2 @ Y3 ) )
@ ( cofinite @ B ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( eventually @ B @ ( P @ X2 ) @ ( cofinite @ B ) ) ) ) ) ) ).
% MOST_finite_Ball_distrib
thf(fact_8135_MOST__inj,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: B > A] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
=> ( ( inj_on @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
=> ( eventually @ B
@ ^ [X2: B] : ( P @ ( F3 @ X2 ) )
@ ( cofinite @ B ) ) ) ) ).
% MOST_inj
thf(fact_8136_semilattice__order__set_OboundedE,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: set @ A,X: A] :
( ( lattic4895041142388067077er_set @ A @ F3 @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F3 @ A4 ) )
=> ! [A10: A] :
( ( member @ A @ A10 @ A4 )
=> ( Less_eq @ X @ A10 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_8137_semilattice__order__set_OboundedI,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: set @ A,X: A] :
( ( lattic4895041142388067077er_set @ A @ F3 @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A7: A] :
( ( member @ A @ A7 @ A4 )
=> ( Less_eq @ X @ A7 ) )
=> ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F3 @ A4 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_8138_semilattice__order__set_Obounded__iff,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: set @ A,X: A] :
( ( lattic4895041142388067077er_set @ A @ F3 @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F3 @ A4 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( Less_eq @ X @ X2 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_8139_INFM__inj,axiom,
! [A: $tType,B: $tType,P: B > $o,F3: A > B] :
( ( frequently @ A
@ ^ [X2: A] : ( P @ ( F3 @ X2 ) )
@ ( cofinite @ A ) )
=> ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( frequently @ B @ P @ ( cofinite @ B ) ) ) ) ).
% INFM_inj
thf(fact_8140_INFM__finite__Bex__distrib,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( frequently @ B
@ ^ [Y3: B] :
? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( P @ X2 @ Y3 ) )
@ ( cofinite @ B ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( frequently @ B @ ( P @ X2 ) @ ( cofinite @ B ) ) ) ) ) ) ).
% INFM_finite_Bex_distrib
thf(fact_8141_semilattice__set_Oeq__fold_H,axiom,
! [A: $tType,F3: A > A > A,A4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ A4 )
= ( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( F3 @ X2 ) @ Y3 ) )
@ ( none @ A )
@ A4 ) ) ) ) ).
% semilattice_set.eq_fold'
thf(fact_8142_semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F3: A > A > A,A4: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert2 @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert2 @ A @ X @ A4 ) )
= ( F3 @ X @ ( lattic1715443433743089157tice_F @ A @ F3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_8143_semilattice__set_Oin__idem,axiom,
! [A: $tType,F3: A > A > A,A4: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( F3 @ X @ ( lattic1715443433743089157tice_F @ A @ F3 @ A4 ) )
= ( lattic1715443433743089157tice_F @ A @ F3 @ A4 ) ) ) ) ) ).
% semilattice_set.in_idem
thf(fact_8144_semilattice__order__set_Oaxioms_I2_J,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( lattic4895041142388067077er_set @ A @ F3 @ Less_eq @ Less )
=> ( lattic149705377957585745ce_set @ A @ F3 ) ) ).
% semilattice_order_set.axioms(2)
thf(fact_8145_Inf__fin_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( lattic149705377957585745ce_set @ A @ ( inf_inf @ A ) ) ) ).
% Inf_fin.semilattice_set_axioms
thf(fact_8146_Max_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic149705377957585745ce_set @ A @ ( ord_max @ A ) ) ) ).
% Max.semilattice_set_axioms
thf(fact_8147_Min_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic149705377957585745ce_set @ A @ ( ord_min @ A ) ) ) ).
% Min.semilattice_set_axioms
thf(fact_8148_Sup__fin_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic149705377957585745ce_set @ A @ ( sup_sup @ A ) ) ) ).
% Sup_fin.semilattice_set_axioms
thf(fact_8149_semilattice__set_Osingleton,axiom,
! [A: $tType,F3: A > A > A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% semilattice_set.singleton
thf(fact_8150_semilattice__set_Ohom__commute,axiom,
! [A: $tType,F3: A > A > A,H: A > A,N6: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ! [X3: A,Y2: A] :
( ( H @ ( F3 @ X3 @ Y2 ) )
= ( F3 @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic1715443433743089157tice_F @ A @ F3 @ N6 ) )
= ( lattic1715443433743089157tice_F @ A @ F3 @ ( image2 @ A @ A @ H @ N6 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_8151_semilattice__set_Osubset,axiom,
! [A: $tType,F3: A > A > A,A4: set @ A,B2: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( F3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ B2 ) @ ( lattic1715443433743089157tice_F @ A @ F3 @ A4 ) )
= ( lattic1715443433743089157tice_F @ A @ F3 @ A4 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_8152_semilattice__set_Oinsert__not__elem,axiom,
! [A: $tType,F3: A > A > A,A4: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert2 @ A @ X @ A4 ) )
= ( F3 @ X @ ( lattic1715443433743089157tice_F @ A @ F3 @ A4 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_8153_semilattice__set_Oinsert,axiom,
! [A: $tType,F3: A > A > A,A4: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert2 @ A @ X @ A4 ) )
= ( F3 @ X @ ( lattic1715443433743089157tice_F @ A @ F3 @ A4 ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_8154_semilattice__set_Oclosed,axiom,
! [A: $tType,F3: A > A > A,A4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] : ( member @ A @ ( F3 @ X3 @ Y2 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic1715443433743089157tice_F @ A @ F3 @ A4 ) @ A4 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_8155_semilattice__set_Ounion,axiom,
! [A: $tType,F3: A > A > A,A4: set @ A,B2: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( F3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ A4 ) @ ( lattic1715443433743089157tice_F @ A @ F3 @ B2 ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_8156_semilattice__set_Oeq__fold,axiom,
! [A: $tType,F3: A > A > A,A4: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert2 @ A @ X @ A4 ) )
= ( finite_fold @ A @ A @ F3 @ X @ A4 ) ) ) ) ).
% semilattice_set.eq_fold
thf(fact_8157_semilattice__set_Oinfinite,axiom,
! [A: $tType,F3: A > A > A,A4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% semilattice_set.infinite
thf(fact_8158_semilattice__set_Oremove,axiom,
! [A: $tType,F3: A > A > A,A4: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ A4 )
= ( F3 @ X @ ( lattic1715443433743089157tice_F @ A @ F3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_8159_folding__def_H,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding @ A @ B )
= ( finite_folding_on @ A @ B @ ( top_top @ ( set @ A ) ) ) ) ).
% folding_def'
thf(fact_8160_antisymp__antisym__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( antisymp @ A
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) )
= ( antisym @ A @ R2 ) ) ).
% antisymp_antisym_eq
thf(fact_8161_antisym__bot,axiom,
! [A: $tType] : ( antisymp @ A @ ( bot_bot @ ( A > A > $o ) ) ) ).
% antisym_bot
thf(fact_8162_antisymp__less__eq,axiom,
! [A: $tType,R2: A > A > $o,S3: A > A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ R2 @ S3 )
=> ( ( antisymp @ A @ S3 )
=> ( antisymp @ A @ R2 ) ) ) ).
% antisymp_less_eq
thf(fact_8163_folding__def,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding @ A @ B )
= ( ^ [F2: A > B > B] :
! [Y3: A,X2: A] :
( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X2 ) )
= ( comp @ B @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% folding_def
thf(fact_8164_folding_Ocomp__fun__commute,axiom,
! [B: $tType,A: $tType,F3: A > B > B,Y: A,X: A] :
( ( finite_folding @ A @ B @ F3 )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y ) @ ( F3 @ X ) )
= ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ).
% folding.comp_fun_commute
thf(fact_8165_folding_Ointro,axiom,
! [B: $tType,A: $tType,F3: A > B > B] :
( ! [Y2: A,X3: A] :
( ( comp @ B @ B @ B @ ( F3 @ Y2 ) @ ( F3 @ X3 ) )
= ( comp @ B @ B @ B @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( finite_folding @ A @ B @ F3 ) ) ).
% folding.intro
thf(fact_8166_card_Ofolding__axioms,axiom,
! [A: $tType] :
( finite_folding @ A @ nat
@ ^ [Uu3: A] : suc ) ).
% card.folding_axioms
thf(fact_8167_antisymp__equality,axiom,
! [A: $tType] :
( antisymp @ A
@ ^ [Y4: A,Z2: A] : Y4 = Z2 ) ).
% antisymp_equality
thf(fact_8168_antisymp__def,axiom,
! [A: $tType] :
( ( antisymp @ A )
= ( ^ [R5: A > A > $o] :
! [X2: A,Y3: A] :
( ( R5 @ X2 @ Y3 )
=> ( ( R5 @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ) ) ).
% antisymp_def
thf(fact_8169_antisympI,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [X3: A,Y2: A] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ X3 )
=> ( X3 = Y2 ) ) )
=> ( antisymp @ A @ R2 ) ) ).
% antisympI
thf(fact_8170_antisympD,axiom,
! [A: $tType,R2: A > A > $o,A3: A,B3: A] :
( ( antisymp @ A @ R2 )
=> ( ( R2 @ A3 @ B3 )
=> ( ( R2 @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% antisympD
thf(fact_8171_folding__idem_Oaxioms_I1_J,axiom,
! [B: $tType,A: $tType,F3: A > B > B] :
( ( finite_folding_idem @ A @ B @ F3 )
=> ( finite_folding @ A @ B @ F3 ) ) ).
% folding_idem.axioms(1)
thf(fact_8172_folding__idem_Ointro,axiom,
! [B: $tType,A: $tType,F3: A > B > B] :
( ( finite_folding @ A @ B @ F3 )
=> ( ( finite7837460588564673216axioms @ A @ B @ F3 )
=> ( finite_folding_idem @ A @ B @ F3 ) ) ) ).
% folding_idem.intro
thf(fact_8173_folding__idem__def,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding_idem @ A @ B )
= ( ^ [F2: A > B > B] :
( ( finite_folding @ A @ B @ F2 )
& ( finite7837460588564673216axioms @ A @ B @ F2 ) ) ) ) ).
% folding_idem_def
thf(fact_8174_folding__idem__axioms_Ointro,axiom,
! [B: $tType,A: $tType,F3: A > B > B] :
( ! [X3: A] :
( ( comp @ B @ B @ B @ ( F3 @ X3 ) @ ( F3 @ X3 ) )
= ( F3 @ X3 ) )
=> ( finite7837460588564673216axioms @ A @ B @ F3 ) ) ).
% folding_idem_axioms.intro
thf(fact_8175_folding__idem__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite7837460588564673216axioms @ A @ B )
= ( ^ [F2: A > B > B] :
! [X2: A] :
( ( comp @ B @ B @ B @ ( F2 @ X2 ) @ ( F2 @ X2 ) )
= ( F2 @ X2 ) ) ) ) ).
% folding_idem_axioms_def
thf(fact_8176_folding__idem_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType,F3: A > B > B] :
( ( finite_folding_idem @ A @ B @ F3 )
=> ( finite7837460588564673216axioms @ A @ B @ F3 ) ) ).
% folding_idem.axioms(2)
thf(fact_8177_inv__into__Field__embed,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Wellorder_embed @ A @ B @ R2 @ R4 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( field2 @ B @ R4 ) @ ( image2 @ A @ B @ F3 @ ( field2 @ A @ R2 ) ) )
=> ( bNF_Wellorder_embed @ B @ A @ R4 @ R2 @ ( hilbert_inv_into @ A @ B @ ( field2 @ A @ R2 ) @ F3 ) ) ) ) ) ).
% inv_into_Field_embed
thf(fact_8178_Un__csum,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) @ ( bNF_Cardinal_csum @ A @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% Un_csum
thf(fact_8179_inv__into__image__cancel,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A,S: set @ A] :
( ( inj_on @ A @ B @ F3 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ A4 )
=> ( ( image2 @ B @ A @ ( hilbert_inv_into @ A @ B @ A4 @ F3 ) @ ( image2 @ A @ B @ F3 @ S ) )
= S ) ) ) ).
% inv_into_image_cancel
thf(fact_8180_image__inv__into__cancel,axiom,
! [B: $tType,A: $tType,F3: B > A,A4: set @ B,A17: set @ A,B13: set @ A] :
( ( ( image2 @ B @ A @ F3 @ A4 )
= A17 )
=> ( ( ord_less_eq @ ( set @ A ) @ B13 @ A17 )
=> ( ( image2 @ B @ A @ F3 @ ( image2 @ A @ B @ ( hilbert_inv_into @ B @ A @ A4 @ F3 ) @ B13 ) )
= B13 ) ) ) ).
% image_inv_into_cancel
thf(fact_8181_bij__betw__inv__into__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: set @ A,A17: set @ B,B2: set @ A,B13: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A4 @ A17 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( ( image2 @ A @ B @ F3 @ B2 )
= B13 )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A4 @ F3 ) @ B13 @ B2 ) ) ) ) ).
% bij_betw_inv_into_subset
thf(fact_8182_inj__on__inv__into,axiom,
! [B: $tType,A: $tType,B2: set @ A,F3: B > A,A4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F3 @ A4 ) )
=> ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ A4 @ F3 ) @ B2 ) ) ).
% inj_on_inv_into
thf(fact_8183_strict__sorted__equal__Uniq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( uniq @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs3 )
& ( ( set2 @ A @ Xs3 )
= A4 ) ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_8184_subset__singleton__iff__Uniq,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [A5: A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( uniq @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_8185_single__valuedp__iff__Uniq,axiom,
! [B: $tType,A: $tType] :
( ( single_valuedp @ A @ B )
= ( ^ [R5: A > B > $o] :
! [X2: A] : ( uniq @ B @ ( R5 @ X2 ) ) ) ) ).
% single_valuedp_iff_Uniq
% Type constructors (830)
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,axiom,
! [A15: $tType,A29: $tType] :
( ( comple592849572758109894attice @ A29 )
=> ( counta4013691401010221786attice @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A15: $tType,A29: $tType] :
( ( comple6319245703460814977attice @ A29 )
=> ( condit1219197933456340205attice @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
! [A15: $tType,A29: $tType] :
( ( counta3822494911875563373attice @ A29 )
=> ( counta3822494911875563373attice @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A15: $tType,A29: $tType] :
( ( comple592849572758109894attice @ A29 )
=> ( comple592849572758109894attice @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A15: $tType,A29: $tType] :
( ( bounded_lattice @ A29 )
=> ( bounde4967611905675639751up_bot @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A15: $tType,A29: $tType] :
( ( bounded_lattice @ A29 )
=> ( bounde4346867609351753570nf_top @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A15: $tType,A29: $tType] :
( ( comple6319245703460814977attice @ A29 )
=> ( comple6319245703460814977attice @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A15: $tType,A29: $tType] :
( ( boolea8198339166811842893lgebra @ A29 )
=> ( boolea8198339166811842893lgebra @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A15: $tType,A29: $tType] :
( ( bounded_lattice @ A29 )
=> ( bounded_lattice_top @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A15: $tType,A29: $tType] :
( ( bounded_lattice @ A29 )
=> ( bounded_lattice_bot @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A15: $tType,A29: $tType] :
( ( comple6319245703460814977attice @ A29 )
=> ( comple9053668089753744459l_ccpo @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A15: $tType,A29: $tType] :
( ( semilattice_sup @ A29 )
=> ( semilattice_sup @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A15: $tType,A29: $tType] :
( ( semilattice_inf @ A29 )
=> ( semilattice_inf @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A15: $tType,A29: $tType] :
( ( distrib_lattice @ A29 )
=> ( distrib_lattice @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A15: $tType,A29: $tType] :
( ( bounded_lattice @ A29 )
=> ( bounded_lattice @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A15: $tType,A29: $tType] :
( ( order_top @ A29 )
=> ( order_top @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A15: $tType,A29: $tType] :
( ( order_bot @ A29 )
=> ( order_bot @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Countable_Ocountable,axiom,
! [A15: $tType,A29: $tType] :
( ( ( finite_finite @ A15 )
& ( countable @ A29 ) )
=> ( countable @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A15: $tType,A29: $tType] :
( ( preorder @ A29 )
=> ( preorder @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A15: $tType,A29: $tType] :
( ( ( finite_finite @ A15 )
& ( finite_finite @ A29 ) )
=> ( finite_finite @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A15: $tType,A29: $tType] :
( ( lattice @ A29 )
=> ( lattice @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A15: $tType,A29: $tType] :
( ( order @ A29 )
=> ( order @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A15: $tType,A29: $tType] :
( ( top @ A29 )
=> ( top @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A15: $tType,A29: $tType] :
( ( ord @ A29 )
=> ( ord @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A15: $tType,A29: $tType] :
( ( bot @ A29 )
=> ( bot @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A15: $tType,A29: $tType] :
( ( uminus @ A29 )
=> ( uminus @ ( A15 > A29 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A15: $tType,A29: $tType] :
( ( minus @ A29 )
=> ( minus @ ( A15 > A29 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,axiom,
condit1219197933456340205attice @ int ).
thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations @ int ).
thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
topolo4958980785337419405_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
topolo1944317154257567458pology @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Odiscrete__topology,axiom,
topolo8865339358273720382pology @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict @ int ).
thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
topolo5987344860129210374id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
topolo2564578578187576103pology @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri2026040879449505780visors @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring @ int ).
thf(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
topolo4211221413907600880p_mult @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs @ int ).
thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add @ int ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors @ int ).
thf(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
topolo6943815403480290642id_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1802427076303600483id_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord4710134922213307826strict @ int ).
thf(tcon_Int_Oint___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___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Countable_Ocountable_5,axiom,
countable @ int ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_6,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0 @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Lattices_Olattice_7,axiom,
lattice @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
semiring_gcd @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
semiring_Gcd @ int ).
thf(tcon_Int_Oint___Rings_Omult__zero,axiom,
mult_zero @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_8,axiom,
order @ int ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring @ int ).
thf(tcon_Int_Oint___Rings_Osemidom,axiom,
semidom @ int ).
thf(tcon_Int_Oint___Orderings_Oord_9,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_10,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
abs_if @ int ).
thf(tcon_Int_Oint___Groups_Ominus_11,axiom,
minus @ int ).
thf(tcon_Int_Oint___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Groups_Oplus,axiom,
plus @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_12,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_13,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_14,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_15,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_16,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_17,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_18,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_19,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_20,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_21,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_22,axiom,
strict9044650504122735259up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
ordere1170586879665033532d_diff @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_23,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_24,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_25,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_26,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_27,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_28,axiom,
ordere8940638589300402666id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni5634975068530333245id_add @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_29,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_30,axiom,
topolo4958980785337419405_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_31,axiom,
topolo1944317154257567458pology @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_32,axiom,
topolo8865339358273720382pology @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_33,axiom,
topolo5987344860129210374id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_34,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_35,axiom,
topolo2564578578187576103pology @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_36,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_37,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_38,axiom,
topolo4211221413907600880p_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_39,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_40,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_41,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_42,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_43,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_44,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_45,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___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___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_74,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_75,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Countable_Ocountable_76,axiom,
countable @ 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___Groups_Ominus_94,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Power_Opower_95,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_96,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_97,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_98,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_99,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_100,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_101,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_102,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_103,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_104,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Oplus_105,axiom,
plus @ num ).
thf(tcon_Num_Onum___Nat_Osize_106,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_107,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_108,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_109,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_110,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_111,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_112,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_113,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_114,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_115,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_116,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_117,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_118,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_119,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_120,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_121,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_122,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_123,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_124,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_125,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_126,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_127,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_128,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_129,axiom,
linord4710134922213307826strict @ 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_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_155,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_156,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_157,axiom,
ab_group_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0 @ rat ).
thf(tcon_Rat_Orat___Countable_Ocountable_158,axiom,
countable @ 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_Ocomm__ring__1_165,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_166,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_167,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_168,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_169,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_170,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_171,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_172,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_173,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_174,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_175,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_176,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_177,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_178,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom_179,axiom,
semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_180,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_181,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_182,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_183,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Groups_Ominus_184,axiom,
minus @ 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___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_193,axiom,
! [A15: $tType] : ( counta4013691401010221786attice @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_194,axiom,
! [A15: $tType] : ( condit1219197933456340205attice @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_195,axiom,
! [A15: $tType] : ( counta3822494911875563373attice @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_196,axiom,
! [A15: $tType] : ( comple592849572758109894attice @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_197,axiom,
! [A15: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_198,axiom,
! [A15: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_199,axiom,
! [A15: $tType] : ( comple6319245703460814977attice @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_200,axiom,
! [A15: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_201,axiom,
! [A15: $tType] : ( bounded_lattice_top @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_202,axiom,
! [A15: $tType] : ( bounded_lattice_bot @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_203,axiom,
! [A15: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_204,axiom,
! [A15: $tType] : ( semilattice_sup @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_205,axiom,
! [A15: $tType] : ( semilattice_inf @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_206,axiom,
! [A15: $tType] : ( distrib_lattice @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_207,axiom,
! [A15: $tType] : ( bounded_lattice @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_208,axiom,
! [A15: $tType] : ( order_top @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_209,axiom,
! [A15: $tType] : ( order_bot @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Countable_Ocountable_210,axiom,
! [A15: $tType] :
( ( finite_finite @ A15 )
=> ( countable @ ( set @ A15 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_211,axiom,
! [A15: $tType] : ( preorder @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_212,axiom,
! [A15: $tType] :
( ( finite_finite @ A15 )
=> ( finite_finite @ ( set @ A15 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_213,axiom,
! [A15: $tType] : ( lattice @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_214,axiom,
! [A15: $tType] : ( order @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_215,axiom,
! [A15: $tType] : ( top @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_216,axiom,
! [A15: $tType] : ( ord @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_217,axiom,
! [A15: $tType] : ( bot @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_218,axiom,
! [A15: $tType] : ( uminus @ ( set @ A15 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_219,axiom,
! [A15: $tType] : ( minus @ ( set @ A15 ) ) ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_220,axiom,
counta4013691401010221786attice @ $o ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_221,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_222,axiom,
counta3822494911875563373attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_223,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_224,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_225,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_226,axiom,
topolo8865339358273720382pology @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_227,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_228,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_229,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_230,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_231,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_232,axiom,
bounded_lattice_top @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_233,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_234,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_235,axiom,
topological_t1_space @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_236,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_237,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_238,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_239,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_240,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_241,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_242,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Countable_Ocountable_243,axiom,
countable @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_244,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_245,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_246,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_247,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_248,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_249,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_250,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_251,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_252,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_253,axiom,
minus @ $o ).
thf(tcon_List_Olist___Countable_Ocountable_254,axiom,
! [A15: $tType] :
( ( countable @ A15 )
=> ( countable @ ( list @ A15 ) ) ) ).
thf(tcon_List_Olist___Nat_Osize_255,axiom,
! [A15: $tType] : ( size @ ( list @ A15 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_256,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_257,axiom,
condit1219197933456340205attice @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_258,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_259,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_260,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_261,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_262,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_263,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_264,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_265,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_266,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_267,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_268,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_269,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_270,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_271,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_272,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_273,axiom,
linord4140545234300271783up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_274,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_275,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_276,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_277,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_278,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_279,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_280,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_281,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_282,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_283,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_284,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_285,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_286,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_287,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_288,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_289,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_290,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot1__space_291,axiom,
topological_t1_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_292,axiom,
ordere2520102378445227354miring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_293,axiom,
linord6961819062388156250ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_294,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_295,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring_296,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_297,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_298,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder_299,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_300,axiom,
semilattice_sup @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__inf_301,axiom,
semilattice_inf @ real ).
thf(tcon_Real_Oreal___Lattices_Odistrib__lattice_302,axiom,
distrib_lattice @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_303,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_304,axiom,
semiring_1_cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_305,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_306,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field_307,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_308,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_309,axiom,
ordered_ring_abs @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_310,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_311,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_312,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_313,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_314,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order_315,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_316,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_317,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_318,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_319,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring_320,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_321,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_322,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_323,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0_324,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_325,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_326,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_327,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_328,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_329,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_330,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_331,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_332,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_333,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_334,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_335,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_336,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_337,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_338,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_339,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_340,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_341,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_342,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_343,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_344,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse_345,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom_346,axiom,
semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_347,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_348,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_349,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if_350,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Groups_Ominus_351,axiom,
minus @ real ).
thf(tcon_Real_Oreal___Fields_Ofield_352,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_353,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_354,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_355,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_356,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oring_357,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_358,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_359,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_360,axiom,
dvd @ real ).
thf(tcon_String_Ochar___Countable_Ocountable_361,axiom,
countable @ char ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_362,axiom,
finite_finite @ char ).
thf(tcon_String_Ochar___Nat_Osize_363,axiom,
size @ char ).
thf(tcon_Sum__Type_Osum___Countable_Ocountable_364,axiom,
! [A15: $tType,A29: $tType] :
( ( ( countable @ A15 )
& ( countable @ A29 ) )
=> ( countable @ ( sum_sum @ A15 @ A29 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_365,axiom,
! [A15: $tType,A29: $tType] :
( ( ( finite_finite @ A15 )
& ( finite_finite @ A29 ) )
=> ( finite_finite @ ( sum_sum @ A15 @ A29 ) ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_366,axiom,
! [A15: $tType,A29: $tType] : ( size @ ( sum_sum @ A15 @ A29 ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_367,axiom,
! [A15: $tType] : ( condit1219197933456340205attice @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_368,axiom,
! [A15: $tType] : ( counta3822494911875563373attice @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_369,axiom,
! [A15: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_370,axiom,
! [A15: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_371,axiom,
! [A15: $tType] : ( comple6319245703460814977attice @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_372,axiom,
! [A15: $tType] : ( bounded_lattice_top @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_373,axiom,
! [A15: $tType] : ( bounded_lattice_bot @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_374,axiom,
! [A15: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_375,axiom,
! [A15: $tType] : ( semilattice_sup @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_376,axiom,
! [A15: $tType] : ( semilattice_inf @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_377,axiom,
! [A15: $tType] : ( distrib_lattice @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_378,axiom,
! [A15: $tType] : ( bounded_lattice @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_379,axiom,
! [A15: $tType] : ( order_top @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_380,axiom,
! [A15: $tType] : ( order_bot @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_381,axiom,
! [A15: $tType] : ( preorder @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_382,axiom,
! [A15: $tType] : ( lattice @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_383,axiom,
! [A15: $tType] : ( order @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_384,axiom,
! [A15: $tType] : ( top @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_385,axiom,
! [A15: $tType] : ( ord @ ( filter @ A15 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_386,axiom,
! [A15: $tType] : ( bot @ ( filter @ A15 ) ) ).
thf(tcon_Option_Ooption___Countable_Ocountable_387,axiom,
! [A15: $tType] :
( ( countable @ A15 )
=> ( countable @ ( option @ A15 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_388,axiom,
! [A15: $tType] :
( ( finite_finite @ A15 )
=> ( finite_finite @ ( option @ A15 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_389,axiom,
! [A15: $tType] : ( size @ ( option @ A15 ) ) ).
thf(tcon_String_Oliteral___Groups_Osemigroup__add_390,axiom,
semigroup_add @ literal ).
thf(tcon_String_Oliteral___Countable_Ocountable_391,axiom,
countable @ literal ).
thf(tcon_String_Oliteral___Orderings_Opreorder_392,axiom,
preorder @ literal ).
thf(tcon_String_Oliteral___Orderings_Olinorder_393,axiom,
linorder @ literal ).
thf(tcon_String_Oliteral___Groups_Omonoid__add_394,axiom,
monoid_add @ literal ).
thf(tcon_String_Oliteral___Orderings_Oorder_395,axiom,
order @ literal ).
thf(tcon_String_Oliteral___Orderings_Oord_396,axiom,
ord @ literal ).
thf(tcon_String_Oliteral___Groups_Ozero_397,axiom,
zero @ literal ).
thf(tcon_String_Oliteral___Groups_Oplus_398,axiom,
plus @ literal ).
thf(tcon_String_Oliteral___Nat_Osize_399,axiom,
size @ literal ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_400,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_401,axiom,
topolo3112930676232923870pology @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_402,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_403,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_404,axiom,
semiri6575147826004484403cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_405,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_406,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_407,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_408,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_409,axiom,
real_V5047593784448816457lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_410,axiom,
real_V768167426530841204y_dist @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_411,axiom,
topolo5987344860129210374id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_412,axiom,
semiri2026040879449505780visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_413,axiom,
real_V2191834092415804123ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_414,axiom,
real_V8037385150606011577_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_415,axiom,
topolo4211221413907600880p_mult @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_416,axiom,
topolo7287701948861334536_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_417,axiom,
topolo8386298272705272623_space @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_418,axiom,
semiri3467727345109120633visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_419,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_420,axiom,
topolo1287966508704411220up_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_421,axiom,
real_V4867850818363320053vector @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_422,axiom,
cancel2418104881723323429up_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_423,axiom,
ring_15535105094025558882visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_424,axiom,
real_V7773925162809079976_field @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_425,axiom,
topolo6943815403480290642id_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_426,axiom,
cancel1802427076303600483id_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_427,axiom,
topological_t2_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_428,axiom,
topological_t1_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_429,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_430,axiom,
real_Vector_banach @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_431,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_432,axiom,
semiring_1_cancel @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_433,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_434,axiom,
ab_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_435,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_436,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_437,axiom,
comm_semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_438,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_439,axiom,
semidom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_440,axiom,
semiring_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_441,axiom,
semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_442,axiom,
division_ring @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_443,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_444,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_445,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_446,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_447,axiom,
idom_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_448,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_449,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_450,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_451,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_452,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_453,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_454,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_455,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_456,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_457,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring_458,axiom,
semiring @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_459,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom_460,axiom,
semidom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_461,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_462,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ominus_463,axiom,
minus @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_464,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_465,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_466,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_467,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oplus_468,axiom,
plus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring_469,axiom,
ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_470,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_471,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_472,axiom,
dvd @ complex ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_473,axiom,
condit6923001295902523014norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_474,axiom,
counta4013691401010221786attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_475,axiom,
condit1219197933456340205attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_476,axiom,
counta3822494911875563373attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_477,axiom,
comple592849572758109894attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_478,axiom,
strict9044650504122735259up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_479,axiom,
strict7427464778891057005id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_480,axiom,
canoni5634975068530333245id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_481,axiom,
bounde4967611905675639751up_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_482,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_483,axiom,
linord4140545234300271783up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_484,axiom,
comple6319245703460814977attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_485,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_486,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_487,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_488,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__top_489,axiom,
bounded_lattice_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_490,axiom,
bounded_lattice_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_491,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_492,axiom,
comple9053668089753744459l_ccpo @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_493,axiom,
semilattice_sup @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_494,axiom,
semilattice_inf @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_495,axiom,
distrib_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_496,axiom,
bounded_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_497,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_498,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_499,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_500,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_501,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_502,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_503,axiom,
comm_semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_504,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_505,axiom,
semiring_numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_506,axiom,
semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_507,axiom,
zero_less_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_508,axiom,
wellorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_509,axiom,
order_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_510,axiom,
order_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_511,axiom,
semiring_char_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable_Ocountable_512,axiom,
countable @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_513,axiom,
zero_neq_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_514,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_515,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_516,axiom,
monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_517,axiom,
monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_518,axiom,
semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_519,axiom,
semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_520,axiom,
lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_521,axiom,
mult_zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_522,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_523,axiom,
semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Otop_524,axiom,
top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_525,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Obot_526,axiom,
bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ominus_527,axiom,
minus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Power_Opower_528,axiom,
power @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_529,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_530,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oplus_531,axiom,
plus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_532,axiom,
one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Odvd_533,axiom,
dvd @ extended_enat ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_534,axiom,
! [A15: $tType,A29: $tType] :
( ( ( topolo4958980785337419405_space @ A15 )
& ( topolo4958980785337419405_space @ A29 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A15 @ A29 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_535,axiom,
! [A15: $tType,A29: $tType] :
( ( ( topological_t2_space @ A15 )
& ( topological_t2_space @ A29 ) )
=> ( topological_t2_space @ ( product_prod @ A15 @ A29 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_536,axiom,
! [A15: $tType,A29: $tType] :
( ( ( topological_t1_space @ A15 )
& ( topological_t1_space @ A29 ) )
=> ( topological_t1_space @ ( product_prod @ A15 @ A29 ) ) ) ).
thf(tcon_Product__Type_Oprod___Countable_Ocountable_537,axiom,
! [A15: $tType,A29: $tType] :
( ( ( countable @ A15 )
& ( countable @ A29 ) )
=> ( countable @ ( product_prod @ A15 @ A29 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_538,axiom,
! [A15: $tType,A29: $tType] :
( ( ( finite_finite @ A15 )
& ( finite_finite @ A29 ) )
=> ( finite_finite @ ( product_prod @ A15 @ A29 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_539,axiom,
! [A15: $tType,A29: $tType] : ( size @ ( product_prod @ A15 @ A29 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_540,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_541,axiom,
counta4013691401010221786attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_542,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_543,axiom,
counta3822494911875563373attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_544,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_545,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_546,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_547,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_548,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_549,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_550,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_551,axiom,
bounded_lattice_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_552,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_553,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_554,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_555,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_556,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_557,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_558,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_559,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable_Ocountable_560,axiom,
countable @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_561,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_562,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_563,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_564,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_565,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_566,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_567,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_568,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_569,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_570,axiom,
minus @ product_unit ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_571,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_572,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_573,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_574,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_575,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_576,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_577,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_578,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_579,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_580,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_581,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_582,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_583,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_584,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_585,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_586,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_587,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_588,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_589,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_590,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_591,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_592,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_593,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_594,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_595,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_596,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_597,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_598,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_599,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_600,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_601,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_602,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_603,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_604,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_605,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_606,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_607,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_608,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_609,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_610,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_611,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_612,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_613,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_614,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_615,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_616,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_617,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_618,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_619,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_620,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_621,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_622,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_623,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_624,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_625,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_626,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_627,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_628,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_629,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_630,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_631,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_632,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_633,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_634,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_635,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_636,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_637,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_638,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_639,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_640,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_641,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_642,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_643,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_644,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_645,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_646,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_647,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_648,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_649,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_650,axiom,
semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_651,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_652,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_653,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_654,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_655,axiom,
minus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_656,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_657,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_658,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_659,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_660,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_661,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_662,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_663,axiom,
dvd @ code_integer ).
thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_664,axiom,
size @ vEBT_VEBT ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X8: A] : ( P @ X8 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member @ nat @ y @ ( sup_sup @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ t ) @ ( insert2 @ nat @ x @ ( bot_bot @ ( set @ nat ) ) ) ) ).
%------------------------------------------------------------------------------